Research Article Analytic Solutions for Heat Conduction in ...
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Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 816385, 8 pages http://dx.doi.org/10.1155/2013/816385 Research Article Analytic Solutions for Heat Conduction in Functionally Graded Circular Hollow Cylinders with Time-Dependent Boundary Conditions Sen-Yung Lee and Chih-Cheng Huang Department of Mechanical Engineering, National Cheng Kung University, Tainan 701, Taiwan Correspondence should be addressed to Sen-Yung Lee; [email protected] Received 16 May 2013; Revised 27 June 2013; Accepted 1 July 2013 Academic Editor: Abdelouahed Tounsi Copyright © 2013 S.-Y. Lee and C.-C. Huang. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. An analytic solution method, without integral transformation, is developed to find the exact solutions for transient heat conduction in functionally graded (FG) circular hollow cylinders with time-dependent boundary conditions. By introducing suitable shiſting functions, the governing second-order regular singular differential equation with variable coefficients and time-dependent boundary conditions is transformed into a differential equation with homogenous boundary conditions. e exact solution of the system with thermal conductivity and specific heat in power functions with different orders is developed. Finally, limiting studies and numerical analyses are given to illustrate the efficiency and the accuracy of the analysis. 1. Introduction e applications of heat conduction in functionally graded (FG) circular hollow cylinders with time-dependent bound- ary conditions can be widely found in many engineering fields, such as cannon barrels, heat exchanger tubes, time var- ied heating on walls of circular structure, and heat treatment on hollow cylinders. As such, an accurate solution method is very helpful for relevant developments. e problem of heat conduction with time-dependent boundary conditions cannot be solved directly by the method of separation of variables. In most of the analyses, an integral transform has been used to remove the time-dependent term. For the problem of heat conduction in circular hollow uni- form cylinders with time-dependent boundary conditions, the associated governing differential equation is a second- order Bessel differential equation with constant coefficients. Aſter conducting a Laplace transformation, the analytical solution can be obtained and found in the book by ˝ Ozisik [1]. When the structure is an FG circular hollow cylinder, the associated governing differential equation is a second- order regular singular differential equation with variable coefficients. For problems with time-independent boundary conditions, various numerical methods have been developed, such as the perturbation method [2], the finite difference method [3], and the finite element method [4]. Jabbari et al. [5, 6] derived analytical solutions for thermal stresses of FG hollow cylinders whose material properties vary with power law distribution through the thickness due to radially symmetric loads and nonaxisymmetric loads. By using the Laplace transformation and a series expansion of Bessel func- tions, Ootao and Tanigawa [7] analyzed one-dimensional transient thermoelastic problem with the material properties varying with the powerlaw form of the radial coordinate variable. Zhao et al. [8] analyzed the temperature change when the thermal and thermoelastic properties are assumed to vary exponentially in the radial direction. Hosseini et al. [9] considered the material properties to be nonlinear with a power law distribution through the thickness, while the temperature distribution was derived analytically using the Bessel functions. In the study of heat conduction in FG circular hollow cylinders with time-dependent boundary conditions, only limited studies can be found. Ootao et al. [10] studied the transient temperature and thermal stress distribution in an infinitely long nonhomogeneous hollow cylinder due to
Transcript of Research Article Analytic Solutions for Heat Conduction in ...
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 816385 8 pageshttpdxdoiorg1011552013816385
Research ArticleAnalytic Solutions for Heat Conduction in FunctionallyGraded Circular Hollow Cylinders with Time-DependentBoundary Conditions
Sen-Yung Lee and Chih-Cheng Huang
Department of Mechanical Engineering National Cheng Kung University Tainan 701 Taiwan
Correspondence should be addressed to Sen-Yung Lee syleemailnckuedutw
Received 16 May 2013 Revised 27 June 2013 Accepted 1 July 2013
Academic Editor Abdelouahed Tounsi
Copyright copy 2013 S-Y Lee and C-C Huang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
An analytic solutionmethod without integral transformation is developed to find the exact solutions for transient heat conductionin functionally graded (FG) circular hollow cylinders with time-dependent boundary conditions By introducing suitable shiftingfunctions the governing second-order regular singular differential equation with variable coefficients and time-dependentboundary conditions is transformed into a differential equation with homogenous boundary conditions The exact solution of thesystem with thermal conductivity and specific heat in power functions with different orders is developed Finally limiting studiesand numerical analyses are given to illustrate the efficiency and the accuracy of the analysis
1 Introduction
The applications of heat conduction in functionally graded(FG) circular hollow cylinders with time-dependent bound-ary conditions can be widely found in many engineeringfields such as cannon barrels heat exchanger tubes time var-ied heating on walls of circular structure and heat treatmenton hollow cylinders As such an accurate solution method isvery helpful for relevant developments
The problem of heat conduction with time-dependentboundary conditions cannot be solved directly by themethodof separation of variables In most of the analyses an integraltransformhas been used to remove the time-dependent termFor the problem of heat conduction in circular hollow uni-form cylinders with time-dependent boundary conditionsthe associated governing differential equation is a second-order Bessel differential equation with constant coefficientsAfter conducting a Laplace transformation the analyticalsolution can be obtained and found in the book by Ozisik [1]
When the structure is an FG circular hollow cylinderthe associated governing differential equation is a second-order regular singular differential equation with variablecoefficients For problems with time-independent boundary
conditions various numerical methods have been developedsuch as the perturbation method [2] the finite differencemethod [3] and the finite element method [4] Jabbari et al[5 6] derived analytical solutions for thermal stresses ofFG hollow cylinders whose material properties vary withpower law distribution through the thickness due to radiallysymmetric loads and nonaxisymmetric loads By using theLaplace transformation and a series expansion of Bessel func-tions Ootao and Tanigawa [7] analyzed one-dimensionaltransient thermoelastic problem with the material propertiesvarying with the powerlaw form of the radial coordinatevariable Zhao et al [8] analyzed the temperature changewhen the thermal and thermoelastic properties are assumedto vary exponentially in the radial direction Hosseini et al[9] considered the material properties to be nonlinear witha power law distribution through the thickness while thetemperature distribution was derived analytically using theBessel functions
In the study of heat conduction in FG circular hollowcylinders with time-dependent boundary conditions onlylimited studies can be found Ootao et al [10] studiedthe transient temperature and thermal stress distribution inan infinitely long nonhomogeneous hollow cylinder due to
2 Mathematical Problems in Engineering
a moving heat source in the axial direction from the innerandor outer surfaces using the layerwise theory in con-junction with the method of Fourier cosine and the Laplacetransformations Shao and Ma [11] employed the Laplacetransform techniques and the series solving method to studythermomechanical stresses in FG circular hollow cylinderswith linearly increasing boundary temperature Jabbari et al[12] developed an analytical solution for the one-dimensionaltemperature distribution mechanical and thermal stresses inan infinitely long FG hollow cylinder under a moving heatsource which moves across the thickness of the cylinderAsgari andAkhlaghi [13] employed the finite elementmethodto study the transient thermal stresses in a thick hollowcylinder with finite length mad of 2D-FGM that its materialproperties are varied in the radial and axial directions witha power law function The thermal boundary conditions atthe inner and outer radiuses are time dependent Singh et al[14] applied the finite integral transform method and theseparation of variables method to solve time-dependent heatconduction problem in a multilayer annulus MalekzadehandHeydarpour [15] used the differential quadraturemethod(DQM) in conjunctionwith the finite elementmethod (FEM)to study the response of FG cylindrical shells under movingthermomechanical loads Recently Wang and Liu [16] haveemployed the method of separation of variables to developthe analytical solution of transient temperature fields for two-dimensional transient heat conduction in a fiber-reinforcedmultilayer cylindrical composite
In the study of thermal elastic response of FG cylinderswithout mechanical loading the heat conduction problemis not incorporated with the elastic field and can be studiedindependently However the thermal field will be coupledwith the temperature field In this paper one considersthe heat conduction problem of FG cylinders only A newsolution method which is a modification on the methoddeveloped by Lee and Lin [17] and Chen et al [18] is used todevelop the analytical solution for transient heat conductionin FG circular hollow cylinders with time-dependent bound-ary conditions By introducing suitable shifting functionsthe governing second-order differential equation with vari-able coefficients and time-dependent boundary conditions istransformed into a differential equation with homogenousboundary conditionsThe analytic solution of the systemwiththermal conductivity and specific heat in power functionswith different orders is developed Finally limiting studiesand numerical analysis are given to illustrate the efficiencyand accuracy of the solution method
2 Mathematical Modeling
Consider the transient heat conduction in an FG circularhollow cylinder with time-dependent boundary conditionat the inner and outer surfaces as shown in Figure 1 Thegoverning differential equation of the system is
1
119903
1
120597119903[119903119896 (119903)
120597119879 (119903 119905)
120597119903] + 119892 (119903 119905) = 120588119888 (119903)
120597119879 (119903 119905)
120597119905
119886 lt 119903 lt 119887 119905 gt 0
(1)
a
b
InitiallyT0(r)
T(b t) = T2(t)
T(a t) = T1(t)
Figure 1 FG circular hollow cylinder with time-dependent bound-ary conditions
The boundary conditions are
119879 (119886 119905) = 1198791(119905)
119879 (119887 119905) = 1198792(119905)
(2)
and the initial condition is
119879 (119903 0) = 1198790(119903) (3)
Here 119903 is the space variable and 119905 is the time variable 119888(119903)is the specific heat 119896(119903) is the thermal conductivity 119879(119903 119905) isthe temperature 120588 is the mass density and 119892(119903 119905) is the heatsource inside the circular hollow cylinder 119879
1(119905) and 119879
2(119905)
are the time-dependent temperatures at the inner and outersurfaces respectively
In terms of the following dimensionless parameters
119870 (120585) =119896 (119903)
119896 (119886) 119862 (120585) =
119888 (119903)
119888 (119886) 119866 (120585 120591) =
119892 (119903 119905) 1198872
119896 (119886) 119879119903
120585 =119903
119887 119903 =
119886
119887 120579 =
119879
119879119903
1205790=1198790
119879119903
120579119894=119879119894
119879119903
120591 =119896 (119886) 119905
119888 (119886) 1205881198872
(4)
where 119879119903is the reference temperature the boundary value
problem of heat conduction becomes
1
120585
1
120597120585[120585119870 (120585)
120597120579 (120585 120591)
120597120585] + 119866 (120585 120591) = 119862 (120585)
120597120579 (120585 120591)
120597120591
119903 lt 120585 lt 1 120591 gt 0
120579 (119903 120591) minus 1205791= 0
120579 (1 120591) minus 1205792= 0
120579 (120585 0) = 1205790(120585)
(5)
Mathematical Problems in Engineering 3
3 Solution Method
31 Change of Variables To find the solution for the second-order differential equation with nonhomogeneous boundaryconditions the shifting variable method developed by Leeand Lin [17] and Chen et al [18] was extended by taking
120579 (120585 120591) = ] (120585 120591) + 1198911(120591) 1198921(120585) + 119891
2(120591) 1198922(120585) (6)
where 119892119894(120585) 119894 = 1 2 are shifting functions to be specified and
](120585 120591) is the transformed function Substituting (6) into (5)yields the following partial differential equation
1
120585
120597
120597120585[120585119870 (120585)
120597] (120585 120591)
120597120585] minus 119862 (120585)
120597] (120585 120591)
120597120591
= minus119866 (120585 120591) +
2
sum
119894=1
119862 (120585) 119892119894(120585)119889119891119894(120591)
119889120591
minus119891119894(120591)1
120585
119889
119889120585[120585119870 (120585)
119889119892119894(120585)
119889120585]
(7)
the associated boundary conditions
] (119903 120591) + 1198911(120591) 1198921(119903) + 119891
2(120591) 1198922(119903) = 119891
1(120591)
] (1 120591) + 1198911(120591) 1198921(1) + 119891
2(120591) 1198922(1) = 119891
2(120591)
(8)
and the associated initial condition
] (120585 0) = 120579 (120585 0) minus 1198911(0) 1198921(120585) minus 119891
2(0) 1198922(120585) (9)
32 Shifting Functions To simplify the analysis the shiftingfunctions are specifically chosen such that they satisfy thefollowing differential equations and the boundary conditions
1198892119892119894(120585)
1198891205852= 0 119894 = 1 2
1198921(119903) = 1
1198921(1) = 0
1198922(119903) = 0
1198922(1) = 1
(10)
These two shifting functions can be easily determined as
1198921(120585) =
1
1 minus 119903(1 minus 120585)
1198922(120585) =
1
1 minus 119903(minus119903 + 120585)
(11)
33 Reduced Homogenous Problem With these two shift-ing functions (11) the governing differential equation (7)becomes
1
120585
120597
120597120585[120585119870 (120585)
120597] (120585 120591)
120597120585] minus 119862 (120585)
120597] (120585 120591)
120597120591= 119865 (120585 120591) (12)
where
119865 (120585 120591)
= minus119866 (120585 120591)
+
2
sum
119894=1
119862 (120585) 119892119894(120585)119889119891119894(120591)
119889120591minus119891119894(120591)
120585
119889
119889120585[120585119870 (120585)]
119889119892119894(120585)
119889120585
(13)
The two nonhomogenous boundary conditions (8) for thetransformed function ](120585 120591) are reduced to homogenousones
] (119903 120591) = 0
] (1 120591) = 0(14)
The transformed initial condition is
] (120585 0) = 1205790(120585) minus 119891
1(0) 1198921(120585) minus 119891
2(0) 1198922(120585) = ]
0(120585) (15)
34 Solution of Transformed Variable
341 Characteristic Solution To find the solution ](120585 120591) weuse the eigenfunction expansion method and assume thesolution to be in the form
] (120585 120591) = 120601 (120585) 119861 (120591) (16)
The separation equation for the dimensionless time variable119861(120591) is
119889119861 (120591)
119889120591= minus1205822119861 (120591) (17)
and the dimensionless space variable 120601(120585) satisfies the follow-ing self-adjoin operator
1
120585
d119889120585[120585119870 (120585)
119889120601 (120585)
119889120585] + 1205822119862 (120585) 120601 (120585) = 0 120585 isin (119903 1)
(18)
120601 (119903) = 0 (19)
120601 (1) = 0 (20)
Let 119883119895(120585) 119895 = 1 2 be the two linearly independent fun-
damental solutions of the system then the solution of thedifferential equation (18) can be expressed as
120601 (120585) = 11986211198831(120585) minus 119862
21198832(120585) (21)
where 1198621and 119862
2are constants to be determined from the
boundary conditions (19)-(20)After substituting solutions (21) into the boundary con-
ditions (19)-(20) we obtain the following characteristicequation
1198831(1)1198832(119903) minus 119883
1(119903)1198832(1) = 0 (22)
4 Mathematical Problems in Engineering
Consequently the eigenvalues 120582119899 119899 = 1 2 3 can be deter-
minedThe associated 119899th eigenfunction 120601119899(120585) is determined
as
120601119899(120585) = 119883
2(1)1198831198991(120585) minus 119883
1(1)1198831198992(120585) (23)
where 1198831198991(120585) and 119883
1198992(120585) are respectively the fundamental
solutions1198831(120582119899 120585) and119883
2(120582119899 120585) associated with eigenvalues
120582119899 119899 = 1 2 3 They are defined as 119883
1198991(120585) = 119883
1(120582119899 120585)
and 1198831198992(120585) = 119883
2(120582119899 120585) The eigenfunctions 120601
119899(120585) constitute
an orthogonal set in the interval 119903 le 120585 le 1 with respect to aweighting function 120585119862(120585)
int
1
119903
120585119862 (120585) 120601119898(120585) 120601119899(120585) 119889120585 =
0 for 119898 = 119899
120575119899 for 119898 = 119899
(24)
where
120575119899= int
1
119903
120585119862 (120585) 1206012
119899(120585) 119889120585 (25)
In terms of eigenfunctions the transformed variable](120585 120591) can be expressed as
] (120585 120591) =infin
sum
119899=1
120601119899(120585) 119861119899(120591) (26)
Substituting (26) into (12) multiplying it by 120585120601119898 and inte-
grating from 119903 to 1 we obtain
119889119861119899(120591)
119889120591+ 1205822
119899119861119899(120591) = minus120574
119899(120591) (27)
where
120574119899(120591) =
1
120575119899
int
1
119903
120585120601119899(120585) 119865 (120585 120591) 119889120585 (28)
The general solution of (27) is
119861119899(120591) = 119890
minus1205822
119899120591[120572119899minus int
120591
0
1198901205822
119899120594120574119899(120594) 119889120594] (29)
120572119899is determined from the initial condition (15) and
120572119899= 119861119899(0) =
1
120575119899
int
1
119903
120585119862 (120585) 120601119899(120585) ]0(120585) 119889120585 (30)
After substituting the solution of the transformed variable(26) and the shifting functions (11) back to (6) the exactsolution for the general system is obtained
4 Verification and Examples
To illustrate the previous analysis the following examples andlimiting cases are given
Example 1 Consider the heat conduction in an uniformcircular hollow cylinder with time-dependent boundary con-ditions The boundary value problem of the heat conductionin dimension-less form is
1
120585
1
120597120585[120585120597120579 (120585 120591)
120597120585] =
120597120579 (120585 120591)
120597120591 119903 lt 120585 lt 1 120591 gt 0
120579 (119903 120591) =1205951(120591)
119879119903
120579 (1 120591) =1205952(120591)
119879119903
120579 (120585 0) =1205790(120585)
119879119903
(31)
In this case 119870(120585) = 1 119862(120585) = 1 The two shifting func-tions are
1198921(120585) =
1
1 minus 119903(1 minus 120585)
1198922(120585) =
1
1 minus 119903(120585 minus 119903)
(32)
The two linearly independent fundamental solutions are
1198831(120582119899 120585) = 119869
0(120582119899120585)
1198832(120582119899 120585) = 119884
0(120582119899120585)
(33)
This leads to
119865 (120585 120591) =1
(1 minus 119903) 119879119903
[ (1 minus 120585) 1205951015840
1(120591) + (minus119903 + 120585) 120595
1015840
2(120591)
+1205951(120591)
120585minus1205952(120591)
120585]
(34)
where the characteristic equation is
1198831(1)1198832(119903) minus 119883
1(119903)1198832(1) = 0 (35)
The associated 119899th eigenfunction 120601119899(120585) is determined as
120601119899(120585) =
1198690(120582119899120585)
1198690(120582119899)minus1198840(120582119899120585)
1198840(120582119899) (36)
The eigenvalues 120582119899and the associated eigenfunctions 120601
119899(120585)
are obtained from (35) and (36) The two coefficients in (28)-(29) are derived as
120574119899(120591) =
1
120575119899
1
(1 minus 119903) 119879119903
times int
1
119903
120601119899(120585) [ (120585 minus 120585
2) 1205951015840
1(120591) + (minus119903120585 + 120585
2) 1205951015840
2(120591)
+ 1205951(120591) minus 120595
2(120591) ] 119889120585
119861119899(120591) = 119890
minus1205822
119899120591[120572119899minus int
120591
0
1198901205822
119899120594120574119899(120594) 119889120594]
(37)
Mathematical Problems in Engineering 5
where
120575119899= [int
1
119903
120585 1206012
119899(120585) 119889120585]
120572119899=
1
120575119899119879119903
int
1
119903
120585120601119899(120585) 1205790(120585) 119889120585
(38)
As a result the analytic closed solution of the system indimensionless form is
120579 (120585 120591) =
infin
sum
119899=1
[120601119899(120585) 119861119899(120591)]
+ (1 minus 120585
1 minus 119903)1205951(120591)
119879119903
+ (minus119903 + 120585
1 minus 119903)1205952(120591)
119879119903
(39)
when
1205951(120591) = 120595
2(120591) = 0 (40)
The analytic closed solution in dimensionless form is re-duced to
120579 (120585 120591) =
infin
sum
119899=1
[119890minus1205822
119899120591120601119899(120585) 120572119899] (41)
The solution is exactly the same as the one given by Ozisik [1]
Example 2 Consider the heat conduction in an FG circularhollow cylinder with time-dependent boundary conditionsThe coefficients of thermal conductivity and the specificheat are 119870(120585) = 119896
1198981205851205731 and 119862(120585) = 119888
1198981205851205732 respectively
The boundary value problem of the heat conduction indimensionless form is
1
120585
120597
120597120585[120585119870 (120585)
120597120579 (120585 120591)
120597120585] minus 119862 (120585)
120597120579 (120585 120591)
120597120591= 0
119903 lt 120585 lt 1 120591 gt 0
(42)
120579 (119903 120591) = 0 (43)
120579 (1 120591) = (1 minus 119890minus1198620120591) 1205792 (44)
120579 (120585 0) = 0 (45)
In this case two shifting functions are
1198921(120585) =
1
1 minus 119903(1 minus 120585)
1198922(120585) =
1
1 minus 119903(120585 minus 119903)
(46)
The route to two independent fundamental solutions of of(42) lies in the use of the Frobenius method which can berepresented in terms of the Bessel functions
1198831(120582119899 120585) = 120585
minus12057312119869] (120578119899120585
(2+1205732minus1205731)2)
1198832(120582119899 120585) = 120585
minus12057312119884] (120578119899120585
(2+1205732minus1205731)2)
(47)
where
120578119899= radic
119888119898
119896119898
(2120582119899
1205731minus 1205732minus 2)
] =1205731
1205731minus 1205732minus 2
(48)
Now
119865 (120585 120591)
= 119890minus11986201205911205792[11988811989812058512057321198620(119886lowast+ 119887lowast120585) + 119887
lowast(1205731+ 1) 119896
1198981205851205731minus1]
minus 119887lowast[1205792(1205731+ 1) 119896
1198981205851205731minus1]
(49)
where
119886lowast=minus119903
1 minus 119903
119887lowast=
1
1 minus 119903
(50)
The characteristic equation is
1198831(1)1198832(119903) minus 119883
1(119903)1198832(1) = 0 (51)
The associated 119899th eigenfunction 120601119899(120585) is determined as
120601119899(120585) = 119883
2(1)1198831198991(120585) minus 119883
1(1)1198831198992(120585) (52)
The eigenvalues 120582119899and the associated eigenfunctions 120601
119899(120585)
are obtained from (51) and (52) by numerical analysis Twocoefficients in (28)ndash(30) are derived as
120574119899(120591) = 120574
1198991(120585) 119890minus1198620120591+ 1205741198992(120585)
119861119899(120591) = [120572
119899+1205741198991(120585)
1205822
119899minus 1198620
+1205741198992(120585)
1205822
119899
] 119890minus1205822
119899120591
minus1205741198991(120585)
1205822
119899minus 1198620
119890minus1198620120591minus1205741198992(120585)
1205822
119899
(53)
where
120575119899= [int
1
119903
1198881198981205851205732+11206012
119899(120585) 119889120585]
1205741198991(120585) =
1205792
120575119899
int
1
119903
120601119899(120585) [11988811989812058512057321198620(119886lowast+ 119887lowast120585)
+119887lowast(1205731+ 1) 119896
1198981205851205731minus1] 119889120585
1205741198992(120585) =
minus119887lowast1205792
120575119899
[int
1
119903
120601119899(120585) (120573
1+ 1) 119896
1198981205851205731minus1119889120585]
120572119899= 0
(54)
6 Mathematical Problems in Engineering
Table 1 Temperature distribution of FG circular hollow cylinders with constant value of 1205732and various parameters of 120573
1and 119862
0at 120591 = 05
[119896119898= 1 119888119898= 1 120579(1 120591) = (119862
0sin120596120591)120579
2 1205792= 4 and 120596 = 25]
1198620
1205731
1205732
120585
05 06 07 08 09 10
01075
10 4595 2259 1845 1632 0380
1 0 4592 2285 1990 1820 0380
125 0 4486 2279 2133 1975 0380
1075
10 45952 22589 18453 16320 3796
1 0 45920 22848 19895 18202 3796
125 0 44865 22788 21329 19752 3796
10075
10 459520 225888 184526 163201 37959
1 0 459196 228480 198950 182017 37959
125 0 448647 227881 213290 197520 37959
30
25
20
15
10
05
00
120579
120591C0 = 10
1205731 = 1 1205732 = 51205731 = 1 1205732 = 05
1205731 = 1 1205732 = 51205731 = 1 1205732 = 05
C0 = 1
001 01 1 10
Figure 2 Temperature variation of FG circular hollow cylinderswith constant value of 120573
1and various values of 120573
2and119862
0at 120585 = 075
[119896119898= 1 119888119898= 1 120579(1 120591) = (1 minus 119890minus1198620120591)120579
2 1205792= 4]
Consequently the analytic closed solution for the systemcan be derived as
120579 (120585 120591)
=
(1 minus 119890minus1198620120591) 1205792
1 minus 119903(120585 minus 119903)
+
infin
sum
119899=1
1198832(1)1198831198991(120585) minus 119883
1(1)1198831198992(120585)
times
[1205741198991(120585)
1205822
119899minus 1198620
+1205741198992(120585)
1205822
119899
] 119890minus1205822
119899120591
minus1205741198991(120585)
1205822
119899minus 1198620
119890minus1198880120591minus1205741198992(120585)
1205822
119899
(55)
30
35
40
25
20
15
10
05
00
120579
05 06 07 08 09 10120585
C0 = infin
C0 = 10
1205731 = 0 1205732 = 0
1205731 = 1 1205732 = 5
1205731 = 1 1205732 = 1
1205731 = 0 1205732 = 0
C0 = 11205731 = 1 1205732 = 5
1205731 = 1 1205732 = 1
Figure 3 Temperature distribution of FG circular hollow cylinderswith constant value of 120573
1and various values of 120573
2and 119862
0at 120591 = 02
[119896119898= 1 119888119898= 1 120579(1 120591) = (1 minus 119890minus1198620120591)120579
2 1205792= 4]
In Figure 2 the temperature variation of FG circularhollow cylinders with various parameters of 120573
1 1205732 and 119862
0
at 120585 = 075 is shown It can be found that when 1198620is a
positive constant the temperature parameter of the mediumsat 120585 = 075 increases then reaches the associated constanttemperatures over time The temperature increase rate forthe system with a higher value of 119862
0is greater than that
of one with a lower value of 1198620 Figures 3 and 4 show the
temperature distribution of FG circular hollow cylinders withvarious parameters of 120573
1 1205732 and 119862
0at 120591 = 02 From these
figures it can be observed that with constant parameter 1205731
the temperature of the mediums increases as parameter 1205732
Mathematical Problems in Engineering 7
30
35
25
20
15
10
05
00
120579
05 06 07 08 09 10120585
C0 = 10 C0 = 11205731 = 075 1205732 = 11205731 = 1 1205732 = 1
1205731 = 125 1205732 = 1
1205731 = 1 1205732 = 1
1205731 = 125 1205732 = 1
1205731 = 075 1205732 = 1
Figure 4 Temperature distribution of FG circular hollow cylinderswith constant value of 120573
2and various values of 120573
1and 119862
0at 120591 = 02
[119896119898= 1 119888119898= 1 120579(1 120591) = (1 minus 119890minus1198620120591)120579
2 1205792= 4]
400350300250200150100500
minus50minus100minus150minus200minus250minus300minus350minus400
05 10 15 20 25 30 35 40 45
120579
120591
C0 = 10
1205731 = 1 1205732 = 075
1205731 = 1 1205732 = 1
1205731 = 1 1205732 = 125
Figure 5 Temperature variation of FG circular hollow cylinderswith constant value of 120573
1and various values of 120573
2at 120585 = 075
[119896119898= 1 119888
119898= 1 120579(1 120591) = (119862
0sin120596120591)120579
2 1205792= 4 119862
0= 10 and
120596 = 25]
is increased With constant parameter 1205732 the temperature of
the mediums increases as parameter 1205731is increased
Example 3 Consider the same physical system as discussedin Example 2 In this case the time dependent boundarycondition at 120585 = 1 (44) is changed to the form
120579 (1 120591) = (1198620sin120596120591) 120579
2 (56)
In this case the eigenvalues and eigenfunctions are the sameas those given in Example 2
Now
119865 (120585 120591) =minus12057921198620
1 minus 119903[(1205731+ 1) 119896
1198981205851205731 sin120596120591
minus (120585 minus 119903) 1205961198881198981205851205732 cos120596120591]
(57)
Following the same solution procedures as shown theexact solution for the general system can be derived as
120579 (120585 120591) =1198620sin1205961205911 minus 119903
1205792(120585 minus 119903)
+
infin
sum
119899=1
[1198832(1)1198831198991(120585) minus 119883
1(1)1198831198992(120585)] 119861
119899(120591)
(58)
where
119861119899(120591) =
1
120596 [(1205822
119899120596)2
+ 1]
times
(1205822
119899
120596cos120596120591 + sin120596120591 minus
1205822
119899
120596119890minus1205822
119899120591)1205741198991(120585)
minus(1205822
119899
120596sin120596120591 minus cos120596120591 minus 119890minus120582
2
119899120591)1205741198992(120585)
120575119899= [int
1
119903
1198881198981205851205732+11206012
119899(120585) 119889120585]
1205741198991(120585) =
minus12057921198620
(1 minus 119903) 120575119899
[int
1
0
120601119899(120585) (120585 minus 119903) 120596119888
1198981205851205732+1119889120585]
1205741198992(120585) =
minus12057921198620
(1 minus 119903) 120575119899
[int
1
0
120601119899(120585) (120573
1+ 1) 119896
1198981205851205731+1119889120585]
120572119899= 0
(59)
Figure 5 shows the harmonic temperature variation of FGcircular hollow cylinders with constant value of 120573
1 various
parameters of 1205732at 120585 = 075 119862
0= 10 and 120596 = 25 It can
be observed that with constant value of 1205731 the amplitude of
temperature oscillation for the system with a higher value of1205732will be more than that of one with a lower value of 120573
2
In Table 1 the temperature variations of FG circularhollow cylinders with constant value of 120573
2and various values
of 1205731and 119862
0at 120591 = 05 are given It can be observed that with
constant parameter 1205732 the temperature of the mediums will
decrease as parameter 1205731is decreased
5 Conclusions
Theproblemof heat conductionwith general time-dependentboundary conditions cannot be solved directly by themethodof separation of variables In most of the analyses an integraltransform was used to remove the time-dependent term
8 Mathematical Problems in Engineering
In this paper a new analytic solution method is developedto find the analytic closed solutions for the transient heatconduction in FG circular hollow cylinders with generaltime-dependent boundary conditions The developed solu-tion method is free of any kind of integral transformation
By introducing suitable shifting functions the governingsecond-order regular singular differential equation with vari-able coefficients and time-dependent boundary conditions istransformed into a differential equation with homogenousboundary conditionsThe analytic solution of the systemwiththermal conductivity and specific heat in power functionswith different orders is developed Finally limiting studiesand numerical analyses are given to illustrate the efficiencyand the accuracy of the analysis The proposed solutionmethod can also be extended to the problems with variouskinds of FG materials and time-dependent boundary condi-tions
Acknowledgment
It is gratefully acknowledged that this research was supportedby the National Science Council of Taiwan Taiwan underGrant NSC 99-2221-E-006-021
References
[1] M N Ozisik Boundary Value Problems of Heat ConductionInternational Textbook Company Scranton Pa USA 1st edi-tion 1968
[2] Y Obata and N Noda ldquoSteady thermal stresses in a hollowcircular cylinder and a hollow sphere of a functionally gradientmaterialrdquo Journal of Thermal Stresses vol 17 no 3 pp 471ndash4871994
[3] H Awaji and R Sivakumar ldquoTemperature and stress distribu-tions in a hollow cylinder of functionally graded material thecase of temperature-independent material propertiesrdquo Journalof the American Ceramic Society vol 84 no 5 pp 1059ndash10652001
[4] G N Praveen and J N Reddy ldquoNonlinear transient ther-moelastic analysis of functionally graded ceramic-metal platesrdquoInternational Journal of Solids and Structures vol 35 no 33 pp4457ndash4476 1998
[5] M Jabbari S Sohrabpour and M R Eslami ldquoMechanical andthermal stresses in a functionally graded hollow cylinder dueto radially symmetric loadsrdquo International Journal of PressureVessels and Piping vol 79 no 7 pp 493ndash497 2002
[6] M Jabbari S Sohrabpour and M R Eslami ldquoGeneral solutionfor mechanical and thermal stresses in a functionally gradedhollow cylinder due to nonaxisymmetric steady-state loadsrdquoASME Transactions Journal of Applied Mechanics vol 70 no1 pp 111ndash118 2003
[7] Y Ootao and Y Tanigawa ldquoTransient thermoelastic analysisfor a functionally graded hollow cylinderrdquo Journal of ThermalStresses vol 29 no 11 pp 1031ndash1046 2006
[8] J Zhao X Ai Y Li and Y Zhou ldquoThermal shock resistanceof functionally gradient solid cylindersrdquo Materials Science andEngineering A vol 418 no 1-2 pp 99ndash110 2006
[9] S M Hosseini M Akhlaghi and M Shakeri ldquoTransient heatconduction in functionally graded thick hollow cylinders by
analytical methodrdquo Heat and Mass Transfer vol 43 no 7 pp669ndash675 2007
[10] Y Ootao T Akai and Y Tanigawa ldquoThree-dimensional tran-sient thermal stress analysis of a nonhomogeneous hollowcircular cylinder due to a moving heat source in the axialdirectionrdquo Journal ofThermal Stresses vol 18 no 5 pp 497ndash5121995
[11] Z S Shao and GWMa ldquoThermo-mechanical stresses in func-tionally graded circular hollow cylinder with linearly increasingboundary temperaturerdquo Composite Structures vol 83 no 3 pp259ndash265 2008
[12] M Jabbari A H Mohazzab and A Bahtui ldquoOne-dimensionalmoving heat source in a hollow FGM cylinderrdquo ASME Transac-tions Journal of Pressure Vessel Technology vol 131 no 2 article021202 7 pages 2009
[13] M Asgari andM Akhlaghi ldquoTransient thermal stresses in two-dimensional functionally graded thick hollow cylinder withfinite lengthrdquo Archive of Applied Mechanics vol 80 no 4 pp353ndash376 2010
[14] S Singh P K Jain and R-U Rizwan-Uddin ldquoFinite integraltransform method to solve asymmetric heat conduction in amultilayer annulus with time-dependent boundary conditionsrdquoNuclear Engineering andDesign vol 241 no 1 pp 144ndash154 2011
[15] P Malekzadeh and Y Heydarpour ldquoResponse of functionallygraded cylindrical shells under moving thermo-mechanicalloadsrdquoThin-Walled Structures vol 58 pp 51ndash66 2012
[16] H M Wang and C B Liu ldquoAnalytical solution of two-dimensional transient heat conduction in fiber-reinforcedcylindrical compositesrdquo International Journal of Thermal Sci-ences vol 69 pp 43ndash52 2013
[17] S Y Lee and S M Lin ldquoDynamic analysis of nonuniformbeams with time-dependent elastic boundary conditionsrdquoASME Transactions Journal of Applied Mechanics vol 63 no2 pp 474ndash478 1996
[18] H T Chen S L Sun H C Huang and S Y Lee ldquoAnalyticclosed solution for the heat conduction with time dependentheat convection coefficient at one boundaryrdquo Computer Model-ing in Engineering and Sciences vol 59 no 2 pp 107ndash126 2010
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Mathematical Problems in Engineering
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Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
a moving heat source in the axial direction from the innerandor outer surfaces using the layerwise theory in con-junction with the method of Fourier cosine and the Laplacetransformations Shao and Ma [11] employed the Laplacetransform techniques and the series solving method to studythermomechanical stresses in FG circular hollow cylinderswith linearly increasing boundary temperature Jabbari et al[12] developed an analytical solution for the one-dimensionaltemperature distribution mechanical and thermal stresses inan infinitely long FG hollow cylinder under a moving heatsource which moves across the thickness of the cylinderAsgari andAkhlaghi [13] employed the finite elementmethodto study the transient thermal stresses in a thick hollowcylinder with finite length mad of 2D-FGM that its materialproperties are varied in the radial and axial directions witha power law function The thermal boundary conditions atthe inner and outer radiuses are time dependent Singh et al[14] applied the finite integral transform method and theseparation of variables method to solve time-dependent heatconduction problem in a multilayer annulus MalekzadehandHeydarpour [15] used the differential quadraturemethod(DQM) in conjunctionwith the finite elementmethod (FEM)to study the response of FG cylindrical shells under movingthermomechanical loads Recently Wang and Liu [16] haveemployed the method of separation of variables to developthe analytical solution of transient temperature fields for two-dimensional transient heat conduction in a fiber-reinforcedmultilayer cylindrical composite
In the study of thermal elastic response of FG cylinderswithout mechanical loading the heat conduction problemis not incorporated with the elastic field and can be studiedindependently However the thermal field will be coupledwith the temperature field In this paper one considersthe heat conduction problem of FG cylinders only A newsolution method which is a modification on the methoddeveloped by Lee and Lin [17] and Chen et al [18] is used todevelop the analytical solution for transient heat conductionin FG circular hollow cylinders with time-dependent bound-ary conditions By introducing suitable shifting functionsthe governing second-order differential equation with vari-able coefficients and time-dependent boundary conditions istransformed into a differential equation with homogenousboundary conditionsThe analytic solution of the systemwiththermal conductivity and specific heat in power functionswith different orders is developed Finally limiting studiesand numerical analysis are given to illustrate the efficiencyand accuracy of the solution method
2 Mathematical Modeling
Consider the transient heat conduction in an FG circularhollow cylinder with time-dependent boundary conditionat the inner and outer surfaces as shown in Figure 1 Thegoverning differential equation of the system is
1
119903
1
120597119903[119903119896 (119903)
120597119879 (119903 119905)
120597119903] + 119892 (119903 119905) = 120588119888 (119903)
120597119879 (119903 119905)
120597119905
119886 lt 119903 lt 119887 119905 gt 0
(1)
a
b
InitiallyT0(r)
T(b t) = T2(t)
T(a t) = T1(t)
Figure 1 FG circular hollow cylinder with time-dependent bound-ary conditions
The boundary conditions are
119879 (119886 119905) = 1198791(119905)
119879 (119887 119905) = 1198792(119905)
(2)
and the initial condition is
119879 (119903 0) = 1198790(119903) (3)
Here 119903 is the space variable and 119905 is the time variable 119888(119903)is the specific heat 119896(119903) is the thermal conductivity 119879(119903 119905) isthe temperature 120588 is the mass density and 119892(119903 119905) is the heatsource inside the circular hollow cylinder 119879
1(119905) and 119879
2(119905)
are the time-dependent temperatures at the inner and outersurfaces respectively
In terms of the following dimensionless parameters
119870 (120585) =119896 (119903)
119896 (119886) 119862 (120585) =
119888 (119903)
119888 (119886) 119866 (120585 120591) =
119892 (119903 119905) 1198872
119896 (119886) 119879119903
120585 =119903
119887 119903 =
119886
119887 120579 =
119879
119879119903
1205790=1198790
119879119903
120579119894=119879119894
119879119903
120591 =119896 (119886) 119905
119888 (119886) 1205881198872
(4)
where 119879119903is the reference temperature the boundary value
problem of heat conduction becomes
1
120585
1
120597120585[120585119870 (120585)
120597120579 (120585 120591)
120597120585] + 119866 (120585 120591) = 119862 (120585)
120597120579 (120585 120591)
120597120591
119903 lt 120585 lt 1 120591 gt 0
120579 (119903 120591) minus 1205791= 0
120579 (1 120591) minus 1205792= 0
120579 (120585 0) = 1205790(120585)
(5)
Mathematical Problems in Engineering 3
3 Solution Method
31 Change of Variables To find the solution for the second-order differential equation with nonhomogeneous boundaryconditions the shifting variable method developed by Leeand Lin [17] and Chen et al [18] was extended by taking
120579 (120585 120591) = ] (120585 120591) + 1198911(120591) 1198921(120585) + 119891
2(120591) 1198922(120585) (6)
where 119892119894(120585) 119894 = 1 2 are shifting functions to be specified and
](120585 120591) is the transformed function Substituting (6) into (5)yields the following partial differential equation
1
120585
120597
120597120585[120585119870 (120585)
120597] (120585 120591)
120597120585] minus 119862 (120585)
120597] (120585 120591)
120597120591
= minus119866 (120585 120591) +
2
sum
119894=1
119862 (120585) 119892119894(120585)119889119891119894(120591)
119889120591
minus119891119894(120591)1
120585
119889
119889120585[120585119870 (120585)
119889119892119894(120585)
119889120585]
(7)
the associated boundary conditions
] (119903 120591) + 1198911(120591) 1198921(119903) + 119891
2(120591) 1198922(119903) = 119891
1(120591)
] (1 120591) + 1198911(120591) 1198921(1) + 119891
2(120591) 1198922(1) = 119891
2(120591)
(8)
and the associated initial condition
] (120585 0) = 120579 (120585 0) minus 1198911(0) 1198921(120585) minus 119891
2(0) 1198922(120585) (9)
32 Shifting Functions To simplify the analysis the shiftingfunctions are specifically chosen such that they satisfy thefollowing differential equations and the boundary conditions
1198892119892119894(120585)
1198891205852= 0 119894 = 1 2
1198921(119903) = 1
1198921(1) = 0
1198922(119903) = 0
1198922(1) = 1
(10)
These two shifting functions can be easily determined as
1198921(120585) =
1
1 minus 119903(1 minus 120585)
1198922(120585) =
1
1 minus 119903(minus119903 + 120585)
(11)
33 Reduced Homogenous Problem With these two shift-ing functions (11) the governing differential equation (7)becomes
1
120585
120597
120597120585[120585119870 (120585)
120597] (120585 120591)
120597120585] minus 119862 (120585)
120597] (120585 120591)
120597120591= 119865 (120585 120591) (12)
where
119865 (120585 120591)
= minus119866 (120585 120591)
+
2
sum
119894=1
119862 (120585) 119892119894(120585)119889119891119894(120591)
119889120591minus119891119894(120591)
120585
119889
119889120585[120585119870 (120585)]
119889119892119894(120585)
119889120585
(13)
The two nonhomogenous boundary conditions (8) for thetransformed function ](120585 120591) are reduced to homogenousones
] (119903 120591) = 0
] (1 120591) = 0(14)
The transformed initial condition is
] (120585 0) = 1205790(120585) minus 119891
1(0) 1198921(120585) minus 119891
2(0) 1198922(120585) = ]
0(120585) (15)
34 Solution of Transformed Variable
341 Characteristic Solution To find the solution ](120585 120591) weuse the eigenfunction expansion method and assume thesolution to be in the form
] (120585 120591) = 120601 (120585) 119861 (120591) (16)
The separation equation for the dimensionless time variable119861(120591) is
119889119861 (120591)
119889120591= minus1205822119861 (120591) (17)
and the dimensionless space variable 120601(120585) satisfies the follow-ing self-adjoin operator
1
120585
d119889120585[120585119870 (120585)
119889120601 (120585)
119889120585] + 1205822119862 (120585) 120601 (120585) = 0 120585 isin (119903 1)
(18)
120601 (119903) = 0 (19)
120601 (1) = 0 (20)
Let 119883119895(120585) 119895 = 1 2 be the two linearly independent fun-
damental solutions of the system then the solution of thedifferential equation (18) can be expressed as
120601 (120585) = 11986211198831(120585) minus 119862
21198832(120585) (21)
where 1198621and 119862
2are constants to be determined from the
boundary conditions (19)-(20)After substituting solutions (21) into the boundary con-
ditions (19)-(20) we obtain the following characteristicequation
1198831(1)1198832(119903) minus 119883
1(119903)1198832(1) = 0 (22)
4 Mathematical Problems in Engineering
Consequently the eigenvalues 120582119899 119899 = 1 2 3 can be deter-
minedThe associated 119899th eigenfunction 120601119899(120585) is determined
as
120601119899(120585) = 119883
2(1)1198831198991(120585) minus 119883
1(1)1198831198992(120585) (23)
where 1198831198991(120585) and 119883
1198992(120585) are respectively the fundamental
solutions1198831(120582119899 120585) and119883
2(120582119899 120585) associated with eigenvalues
120582119899 119899 = 1 2 3 They are defined as 119883
1198991(120585) = 119883
1(120582119899 120585)
and 1198831198992(120585) = 119883
2(120582119899 120585) The eigenfunctions 120601
119899(120585) constitute
an orthogonal set in the interval 119903 le 120585 le 1 with respect to aweighting function 120585119862(120585)
int
1
119903
120585119862 (120585) 120601119898(120585) 120601119899(120585) 119889120585 =
0 for 119898 = 119899
120575119899 for 119898 = 119899
(24)
where
120575119899= int
1
119903
120585119862 (120585) 1206012
119899(120585) 119889120585 (25)
In terms of eigenfunctions the transformed variable](120585 120591) can be expressed as
] (120585 120591) =infin
sum
119899=1
120601119899(120585) 119861119899(120591) (26)
Substituting (26) into (12) multiplying it by 120585120601119898 and inte-
grating from 119903 to 1 we obtain
119889119861119899(120591)
119889120591+ 1205822
119899119861119899(120591) = minus120574
119899(120591) (27)
where
120574119899(120591) =
1
120575119899
int
1
119903
120585120601119899(120585) 119865 (120585 120591) 119889120585 (28)
The general solution of (27) is
119861119899(120591) = 119890
minus1205822
119899120591[120572119899minus int
120591
0
1198901205822
119899120594120574119899(120594) 119889120594] (29)
120572119899is determined from the initial condition (15) and
120572119899= 119861119899(0) =
1
120575119899
int
1
119903
120585119862 (120585) 120601119899(120585) ]0(120585) 119889120585 (30)
After substituting the solution of the transformed variable(26) and the shifting functions (11) back to (6) the exactsolution for the general system is obtained
4 Verification and Examples
To illustrate the previous analysis the following examples andlimiting cases are given
Example 1 Consider the heat conduction in an uniformcircular hollow cylinder with time-dependent boundary con-ditions The boundary value problem of the heat conductionin dimension-less form is
1
120585
1
120597120585[120585120597120579 (120585 120591)
120597120585] =
120597120579 (120585 120591)
120597120591 119903 lt 120585 lt 1 120591 gt 0
120579 (119903 120591) =1205951(120591)
119879119903
120579 (1 120591) =1205952(120591)
119879119903
120579 (120585 0) =1205790(120585)
119879119903
(31)
In this case 119870(120585) = 1 119862(120585) = 1 The two shifting func-tions are
1198921(120585) =
1
1 minus 119903(1 minus 120585)
1198922(120585) =
1
1 minus 119903(120585 minus 119903)
(32)
The two linearly independent fundamental solutions are
1198831(120582119899 120585) = 119869
0(120582119899120585)
1198832(120582119899 120585) = 119884
0(120582119899120585)
(33)
This leads to
119865 (120585 120591) =1
(1 minus 119903) 119879119903
[ (1 minus 120585) 1205951015840
1(120591) + (minus119903 + 120585) 120595
1015840
2(120591)
+1205951(120591)
120585minus1205952(120591)
120585]
(34)
where the characteristic equation is
1198831(1)1198832(119903) minus 119883
1(119903)1198832(1) = 0 (35)
The associated 119899th eigenfunction 120601119899(120585) is determined as
120601119899(120585) =
1198690(120582119899120585)
1198690(120582119899)minus1198840(120582119899120585)
1198840(120582119899) (36)
The eigenvalues 120582119899and the associated eigenfunctions 120601
119899(120585)
are obtained from (35) and (36) The two coefficients in (28)-(29) are derived as
120574119899(120591) =
1
120575119899
1
(1 minus 119903) 119879119903
times int
1
119903
120601119899(120585) [ (120585 minus 120585
2) 1205951015840
1(120591) + (minus119903120585 + 120585
2) 1205951015840
2(120591)
+ 1205951(120591) minus 120595
2(120591) ] 119889120585
119861119899(120591) = 119890
minus1205822
119899120591[120572119899minus int
120591
0
1198901205822
119899120594120574119899(120594) 119889120594]
(37)
Mathematical Problems in Engineering 5
where
120575119899= [int
1
119903
120585 1206012
119899(120585) 119889120585]
120572119899=
1
120575119899119879119903
int
1
119903
120585120601119899(120585) 1205790(120585) 119889120585
(38)
As a result the analytic closed solution of the system indimensionless form is
120579 (120585 120591) =
infin
sum
119899=1
[120601119899(120585) 119861119899(120591)]
+ (1 minus 120585
1 minus 119903)1205951(120591)
119879119903
+ (minus119903 + 120585
1 minus 119903)1205952(120591)
119879119903
(39)
when
1205951(120591) = 120595
2(120591) = 0 (40)
The analytic closed solution in dimensionless form is re-duced to
120579 (120585 120591) =
infin
sum
119899=1
[119890minus1205822
119899120591120601119899(120585) 120572119899] (41)
The solution is exactly the same as the one given by Ozisik [1]
Example 2 Consider the heat conduction in an FG circularhollow cylinder with time-dependent boundary conditionsThe coefficients of thermal conductivity and the specificheat are 119870(120585) = 119896
1198981205851205731 and 119862(120585) = 119888
1198981205851205732 respectively
The boundary value problem of the heat conduction indimensionless form is
1
120585
120597
120597120585[120585119870 (120585)
120597120579 (120585 120591)
120597120585] minus 119862 (120585)
120597120579 (120585 120591)
120597120591= 0
119903 lt 120585 lt 1 120591 gt 0
(42)
120579 (119903 120591) = 0 (43)
120579 (1 120591) = (1 minus 119890minus1198620120591) 1205792 (44)
120579 (120585 0) = 0 (45)
In this case two shifting functions are
1198921(120585) =
1
1 minus 119903(1 minus 120585)
1198922(120585) =
1
1 minus 119903(120585 minus 119903)
(46)
The route to two independent fundamental solutions of of(42) lies in the use of the Frobenius method which can berepresented in terms of the Bessel functions
1198831(120582119899 120585) = 120585
minus12057312119869] (120578119899120585
(2+1205732minus1205731)2)
1198832(120582119899 120585) = 120585
minus12057312119884] (120578119899120585
(2+1205732minus1205731)2)
(47)
where
120578119899= radic
119888119898
119896119898
(2120582119899
1205731minus 1205732minus 2)
] =1205731
1205731minus 1205732minus 2
(48)
Now
119865 (120585 120591)
= 119890minus11986201205911205792[11988811989812058512057321198620(119886lowast+ 119887lowast120585) + 119887
lowast(1205731+ 1) 119896
1198981205851205731minus1]
minus 119887lowast[1205792(1205731+ 1) 119896
1198981205851205731minus1]
(49)
where
119886lowast=minus119903
1 minus 119903
119887lowast=
1
1 minus 119903
(50)
The characteristic equation is
1198831(1)1198832(119903) minus 119883
1(119903)1198832(1) = 0 (51)
The associated 119899th eigenfunction 120601119899(120585) is determined as
120601119899(120585) = 119883
2(1)1198831198991(120585) minus 119883
1(1)1198831198992(120585) (52)
The eigenvalues 120582119899and the associated eigenfunctions 120601
119899(120585)
are obtained from (51) and (52) by numerical analysis Twocoefficients in (28)ndash(30) are derived as
120574119899(120591) = 120574
1198991(120585) 119890minus1198620120591+ 1205741198992(120585)
119861119899(120591) = [120572
119899+1205741198991(120585)
1205822
119899minus 1198620
+1205741198992(120585)
1205822
119899
] 119890minus1205822
119899120591
minus1205741198991(120585)
1205822
119899minus 1198620
119890minus1198620120591minus1205741198992(120585)
1205822
119899
(53)
where
120575119899= [int
1
119903
1198881198981205851205732+11206012
119899(120585) 119889120585]
1205741198991(120585) =
1205792
120575119899
int
1
119903
120601119899(120585) [11988811989812058512057321198620(119886lowast+ 119887lowast120585)
+119887lowast(1205731+ 1) 119896
1198981205851205731minus1] 119889120585
1205741198992(120585) =
minus119887lowast1205792
120575119899
[int
1
119903
120601119899(120585) (120573
1+ 1) 119896
1198981205851205731minus1119889120585]
120572119899= 0
(54)
6 Mathematical Problems in Engineering
Table 1 Temperature distribution of FG circular hollow cylinders with constant value of 1205732and various parameters of 120573
1and 119862
0at 120591 = 05
[119896119898= 1 119888119898= 1 120579(1 120591) = (119862
0sin120596120591)120579
2 1205792= 4 and 120596 = 25]
1198620
1205731
1205732
120585
05 06 07 08 09 10
01075
10 4595 2259 1845 1632 0380
1 0 4592 2285 1990 1820 0380
125 0 4486 2279 2133 1975 0380
1075
10 45952 22589 18453 16320 3796
1 0 45920 22848 19895 18202 3796
125 0 44865 22788 21329 19752 3796
10075
10 459520 225888 184526 163201 37959
1 0 459196 228480 198950 182017 37959
125 0 448647 227881 213290 197520 37959
30
25
20
15
10
05
00
120579
120591C0 = 10
1205731 = 1 1205732 = 51205731 = 1 1205732 = 05
1205731 = 1 1205732 = 51205731 = 1 1205732 = 05
C0 = 1
001 01 1 10
Figure 2 Temperature variation of FG circular hollow cylinderswith constant value of 120573
1and various values of 120573
2and119862
0at 120585 = 075
[119896119898= 1 119888119898= 1 120579(1 120591) = (1 minus 119890minus1198620120591)120579
2 1205792= 4]
Consequently the analytic closed solution for the systemcan be derived as
120579 (120585 120591)
=
(1 minus 119890minus1198620120591) 1205792
1 minus 119903(120585 minus 119903)
+
infin
sum
119899=1
1198832(1)1198831198991(120585) minus 119883
1(1)1198831198992(120585)
times
[1205741198991(120585)
1205822
119899minus 1198620
+1205741198992(120585)
1205822
119899
] 119890minus1205822
119899120591
minus1205741198991(120585)
1205822
119899minus 1198620
119890minus1198880120591minus1205741198992(120585)
1205822
119899
(55)
30
35
40
25
20
15
10
05
00
120579
05 06 07 08 09 10120585
C0 = infin
C0 = 10
1205731 = 0 1205732 = 0
1205731 = 1 1205732 = 5
1205731 = 1 1205732 = 1
1205731 = 0 1205732 = 0
C0 = 11205731 = 1 1205732 = 5
1205731 = 1 1205732 = 1
Figure 3 Temperature distribution of FG circular hollow cylinderswith constant value of 120573
1and various values of 120573
2and 119862
0at 120591 = 02
[119896119898= 1 119888119898= 1 120579(1 120591) = (1 minus 119890minus1198620120591)120579
2 1205792= 4]
In Figure 2 the temperature variation of FG circularhollow cylinders with various parameters of 120573
1 1205732 and 119862
0
at 120585 = 075 is shown It can be found that when 1198620is a
positive constant the temperature parameter of the mediumsat 120585 = 075 increases then reaches the associated constanttemperatures over time The temperature increase rate forthe system with a higher value of 119862
0is greater than that
of one with a lower value of 1198620 Figures 3 and 4 show the
temperature distribution of FG circular hollow cylinders withvarious parameters of 120573
1 1205732 and 119862
0at 120591 = 02 From these
figures it can be observed that with constant parameter 1205731
the temperature of the mediums increases as parameter 1205732
Mathematical Problems in Engineering 7
30
35
25
20
15
10
05
00
120579
05 06 07 08 09 10120585
C0 = 10 C0 = 11205731 = 075 1205732 = 11205731 = 1 1205732 = 1
1205731 = 125 1205732 = 1
1205731 = 1 1205732 = 1
1205731 = 125 1205732 = 1
1205731 = 075 1205732 = 1
Figure 4 Temperature distribution of FG circular hollow cylinderswith constant value of 120573
2and various values of 120573
1and 119862
0at 120591 = 02
[119896119898= 1 119888119898= 1 120579(1 120591) = (1 minus 119890minus1198620120591)120579
2 1205792= 4]
400350300250200150100500
minus50minus100minus150minus200minus250minus300minus350minus400
05 10 15 20 25 30 35 40 45
120579
120591
C0 = 10
1205731 = 1 1205732 = 075
1205731 = 1 1205732 = 1
1205731 = 1 1205732 = 125
Figure 5 Temperature variation of FG circular hollow cylinderswith constant value of 120573
1and various values of 120573
2at 120585 = 075
[119896119898= 1 119888
119898= 1 120579(1 120591) = (119862
0sin120596120591)120579
2 1205792= 4 119862
0= 10 and
120596 = 25]
is increased With constant parameter 1205732 the temperature of
the mediums increases as parameter 1205731is increased
Example 3 Consider the same physical system as discussedin Example 2 In this case the time dependent boundarycondition at 120585 = 1 (44) is changed to the form
120579 (1 120591) = (1198620sin120596120591) 120579
2 (56)
In this case the eigenvalues and eigenfunctions are the sameas those given in Example 2
Now
119865 (120585 120591) =minus12057921198620
1 minus 119903[(1205731+ 1) 119896
1198981205851205731 sin120596120591
minus (120585 minus 119903) 1205961198881198981205851205732 cos120596120591]
(57)
Following the same solution procedures as shown theexact solution for the general system can be derived as
120579 (120585 120591) =1198620sin1205961205911 minus 119903
1205792(120585 minus 119903)
+
infin
sum
119899=1
[1198832(1)1198831198991(120585) minus 119883
1(1)1198831198992(120585)] 119861
119899(120591)
(58)
where
119861119899(120591) =
1
120596 [(1205822
119899120596)2
+ 1]
times
(1205822
119899
120596cos120596120591 + sin120596120591 minus
1205822
119899
120596119890minus1205822
119899120591)1205741198991(120585)
minus(1205822
119899
120596sin120596120591 minus cos120596120591 minus 119890minus120582
2
119899120591)1205741198992(120585)
120575119899= [int
1
119903
1198881198981205851205732+11206012
119899(120585) 119889120585]
1205741198991(120585) =
minus12057921198620
(1 minus 119903) 120575119899
[int
1
0
120601119899(120585) (120585 minus 119903) 120596119888
1198981205851205732+1119889120585]
1205741198992(120585) =
minus12057921198620
(1 minus 119903) 120575119899
[int
1
0
120601119899(120585) (120573
1+ 1) 119896
1198981205851205731+1119889120585]
120572119899= 0
(59)
Figure 5 shows the harmonic temperature variation of FGcircular hollow cylinders with constant value of 120573
1 various
parameters of 1205732at 120585 = 075 119862
0= 10 and 120596 = 25 It can
be observed that with constant value of 1205731 the amplitude of
temperature oscillation for the system with a higher value of1205732will be more than that of one with a lower value of 120573
2
In Table 1 the temperature variations of FG circularhollow cylinders with constant value of 120573
2and various values
of 1205731and 119862
0at 120591 = 05 are given It can be observed that with
constant parameter 1205732 the temperature of the mediums will
decrease as parameter 1205731is decreased
5 Conclusions
Theproblemof heat conductionwith general time-dependentboundary conditions cannot be solved directly by themethodof separation of variables In most of the analyses an integraltransform was used to remove the time-dependent term
8 Mathematical Problems in Engineering
In this paper a new analytic solution method is developedto find the analytic closed solutions for the transient heatconduction in FG circular hollow cylinders with generaltime-dependent boundary conditions The developed solu-tion method is free of any kind of integral transformation
By introducing suitable shifting functions the governingsecond-order regular singular differential equation with vari-able coefficients and time-dependent boundary conditions istransformed into a differential equation with homogenousboundary conditionsThe analytic solution of the systemwiththermal conductivity and specific heat in power functionswith different orders is developed Finally limiting studiesand numerical analyses are given to illustrate the efficiencyand the accuracy of the analysis The proposed solutionmethod can also be extended to the problems with variouskinds of FG materials and time-dependent boundary condi-tions
Acknowledgment
It is gratefully acknowledged that this research was supportedby the National Science Council of Taiwan Taiwan underGrant NSC 99-2221-E-006-021
References
[1] M N Ozisik Boundary Value Problems of Heat ConductionInternational Textbook Company Scranton Pa USA 1st edi-tion 1968
[2] Y Obata and N Noda ldquoSteady thermal stresses in a hollowcircular cylinder and a hollow sphere of a functionally gradientmaterialrdquo Journal of Thermal Stresses vol 17 no 3 pp 471ndash4871994
[3] H Awaji and R Sivakumar ldquoTemperature and stress distribu-tions in a hollow cylinder of functionally graded material thecase of temperature-independent material propertiesrdquo Journalof the American Ceramic Society vol 84 no 5 pp 1059ndash10652001
[4] G N Praveen and J N Reddy ldquoNonlinear transient ther-moelastic analysis of functionally graded ceramic-metal platesrdquoInternational Journal of Solids and Structures vol 35 no 33 pp4457ndash4476 1998
[5] M Jabbari S Sohrabpour and M R Eslami ldquoMechanical andthermal stresses in a functionally graded hollow cylinder dueto radially symmetric loadsrdquo International Journal of PressureVessels and Piping vol 79 no 7 pp 493ndash497 2002
[6] M Jabbari S Sohrabpour and M R Eslami ldquoGeneral solutionfor mechanical and thermal stresses in a functionally gradedhollow cylinder due to nonaxisymmetric steady-state loadsrdquoASME Transactions Journal of Applied Mechanics vol 70 no1 pp 111ndash118 2003
[7] Y Ootao and Y Tanigawa ldquoTransient thermoelastic analysisfor a functionally graded hollow cylinderrdquo Journal of ThermalStresses vol 29 no 11 pp 1031ndash1046 2006
[8] J Zhao X Ai Y Li and Y Zhou ldquoThermal shock resistanceof functionally gradient solid cylindersrdquo Materials Science andEngineering A vol 418 no 1-2 pp 99ndash110 2006
[9] S M Hosseini M Akhlaghi and M Shakeri ldquoTransient heatconduction in functionally graded thick hollow cylinders by
analytical methodrdquo Heat and Mass Transfer vol 43 no 7 pp669ndash675 2007
[10] Y Ootao T Akai and Y Tanigawa ldquoThree-dimensional tran-sient thermal stress analysis of a nonhomogeneous hollowcircular cylinder due to a moving heat source in the axialdirectionrdquo Journal ofThermal Stresses vol 18 no 5 pp 497ndash5121995
[11] Z S Shao and GWMa ldquoThermo-mechanical stresses in func-tionally graded circular hollow cylinder with linearly increasingboundary temperaturerdquo Composite Structures vol 83 no 3 pp259ndash265 2008
[12] M Jabbari A H Mohazzab and A Bahtui ldquoOne-dimensionalmoving heat source in a hollow FGM cylinderrdquo ASME Transac-tions Journal of Pressure Vessel Technology vol 131 no 2 article021202 7 pages 2009
[13] M Asgari andM Akhlaghi ldquoTransient thermal stresses in two-dimensional functionally graded thick hollow cylinder withfinite lengthrdquo Archive of Applied Mechanics vol 80 no 4 pp353ndash376 2010
[14] S Singh P K Jain and R-U Rizwan-Uddin ldquoFinite integraltransform method to solve asymmetric heat conduction in amultilayer annulus with time-dependent boundary conditionsrdquoNuclear Engineering andDesign vol 241 no 1 pp 144ndash154 2011
[15] P Malekzadeh and Y Heydarpour ldquoResponse of functionallygraded cylindrical shells under moving thermo-mechanicalloadsrdquoThin-Walled Structures vol 58 pp 51ndash66 2012
[16] H M Wang and C B Liu ldquoAnalytical solution of two-dimensional transient heat conduction in fiber-reinforcedcylindrical compositesrdquo International Journal of Thermal Sci-ences vol 69 pp 43ndash52 2013
[17] S Y Lee and S M Lin ldquoDynamic analysis of nonuniformbeams with time-dependent elastic boundary conditionsrdquoASME Transactions Journal of Applied Mechanics vol 63 no2 pp 474ndash478 1996
[18] H T Chen S L Sun H C Huang and S Y Lee ldquoAnalyticclosed solution for the heat conduction with time dependentheat convection coefficient at one boundaryrdquo Computer Model-ing in Engineering and Sciences vol 59 no 2 pp 107ndash126 2010
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Mathematical Problems in Engineering
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
3 Solution Method
31 Change of Variables To find the solution for the second-order differential equation with nonhomogeneous boundaryconditions the shifting variable method developed by Leeand Lin [17] and Chen et al [18] was extended by taking
120579 (120585 120591) = ] (120585 120591) + 1198911(120591) 1198921(120585) + 119891
2(120591) 1198922(120585) (6)
where 119892119894(120585) 119894 = 1 2 are shifting functions to be specified and
](120585 120591) is the transformed function Substituting (6) into (5)yields the following partial differential equation
1
120585
120597
120597120585[120585119870 (120585)
120597] (120585 120591)
120597120585] minus 119862 (120585)
120597] (120585 120591)
120597120591
= minus119866 (120585 120591) +
2
sum
119894=1
119862 (120585) 119892119894(120585)119889119891119894(120591)
119889120591
minus119891119894(120591)1
120585
119889
119889120585[120585119870 (120585)
119889119892119894(120585)
119889120585]
(7)
the associated boundary conditions
] (119903 120591) + 1198911(120591) 1198921(119903) + 119891
2(120591) 1198922(119903) = 119891
1(120591)
] (1 120591) + 1198911(120591) 1198921(1) + 119891
2(120591) 1198922(1) = 119891
2(120591)
(8)
and the associated initial condition
] (120585 0) = 120579 (120585 0) minus 1198911(0) 1198921(120585) minus 119891
2(0) 1198922(120585) (9)
32 Shifting Functions To simplify the analysis the shiftingfunctions are specifically chosen such that they satisfy thefollowing differential equations and the boundary conditions
1198892119892119894(120585)
1198891205852= 0 119894 = 1 2
1198921(119903) = 1
1198921(1) = 0
1198922(119903) = 0
1198922(1) = 1
(10)
These two shifting functions can be easily determined as
1198921(120585) =
1
1 minus 119903(1 minus 120585)
1198922(120585) =
1
1 minus 119903(minus119903 + 120585)
(11)
33 Reduced Homogenous Problem With these two shift-ing functions (11) the governing differential equation (7)becomes
1
120585
120597
120597120585[120585119870 (120585)
120597] (120585 120591)
120597120585] minus 119862 (120585)
120597] (120585 120591)
120597120591= 119865 (120585 120591) (12)
where
119865 (120585 120591)
= minus119866 (120585 120591)
+
2
sum
119894=1
119862 (120585) 119892119894(120585)119889119891119894(120591)
119889120591minus119891119894(120591)
120585
119889
119889120585[120585119870 (120585)]
119889119892119894(120585)
119889120585
(13)
The two nonhomogenous boundary conditions (8) for thetransformed function ](120585 120591) are reduced to homogenousones
] (119903 120591) = 0
] (1 120591) = 0(14)
The transformed initial condition is
] (120585 0) = 1205790(120585) minus 119891
1(0) 1198921(120585) minus 119891
2(0) 1198922(120585) = ]
0(120585) (15)
34 Solution of Transformed Variable
341 Characteristic Solution To find the solution ](120585 120591) weuse the eigenfunction expansion method and assume thesolution to be in the form
] (120585 120591) = 120601 (120585) 119861 (120591) (16)
The separation equation for the dimensionless time variable119861(120591) is
119889119861 (120591)
119889120591= minus1205822119861 (120591) (17)
and the dimensionless space variable 120601(120585) satisfies the follow-ing self-adjoin operator
1
120585
d119889120585[120585119870 (120585)
119889120601 (120585)
119889120585] + 1205822119862 (120585) 120601 (120585) = 0 120585 isin (119903 1)
(18)
120601 (119903) = 0 (19)
120601 (1) = 0 (20)
Let 119883119895(120585) 119895 = 1 2 be the two linearly independent fun-
damental solutions of the system then the solution of thedifferential equation (18) can be expressed as
120601 (120585) = 11986211198831(120585) minus 119862
21198832(120585) (21)
where 1198621and 119862
2are constants to be determined from the
boundary conditions (19)-(20)After substituting solutions (21) into the boundary con-
ditions (19)-(20) we obtain the following characteristicequation
1198831(1)1198832(119903) minus 119883
1(119903)1198832(1) = 0 (22)
4 Mathematical Problems in Engineering
Consequently the eigenvalues 120582119899 119899 = 1 2 3 can be deter-
minedThe associated 119899th eigenfunction 120601119899(120585) is determined
as
120601119899(120585) = 119883
2(1)1198831198991(120585) minus 119883
1(1)1198831198992(120585) (23)
where 1198831198991(120585) and 119883
1198992(120585) are respectively the fundamental
solutions1198831(120582119899 120585) and119883
2(120582119899 120585) associated with eigenvalues
120582119899 119899 = 1 2 3 They are defined as 119883
1198991(120585) = 119883
1(120582119899 120585)
and 1198831198992(120585) = 119883
2(120582119899 120585) The eigenfunctions 120601
119899(120585) constitute
an orthogonal set in the interval 119903 le 120585 le 1 with respect to aweighting function 120585119862(120585)
int
1
119903
120585119862 (120585) 120601119898(120585) 120601119899(120585) 119889120585 =
0 for 119898 = 119899
120575119899 for 119898 = 119899
(24)
where
120575119899= int
1
119903
120585119862 (120585) 1206012
119899(120585) 119889120585 (25)
In terms of eigenfunctions the transformed variable](120585 120591) can be expressed as
] (120585 120591) =infin
sum
119899=1
120601119899(120585) 119861119899(120591) (26)
Substituting (26) into (12) multiplying it by 120585120601119898 and inte-
grating from 119903 to 1 we obtain
119889119861119899(120591)
119889120591+ 1205822
119899119861119899(120591) = minus120574
119899(120591) (27)
where
120574119899(120591) =
1
120575119899
int
1
119903
120585120601119899(120585) 119865 (120585 120591) 119889120585 (28)
The general solution of (27) is
119861119899(120591) = 119890
minus1205822
119899120591[120572119899minus int
120591
0
1198901205822
119899120594120574119899(120594) 119889120594] (29)
120572119899is determined from the initial condition (15) and
120572119899= 119861119899(0) =
1
120575119899
int
1
119903
120585119862 (120585) 120601119899(120585) ]0(120585) 119889120585 (30)
After substituting the solution of the transformed variable(26) and the shifting functions (11) back to (6) the exactsolution for the general system is obtained
4 Verification and Examples
To illustrate the previous analysis the following examples andlimiting cases are given
Example 1 Consider the heat conduction in an uniformcircular hollow cylinder with time-dependent boundary con-ditions The boundary value problem of the heat conductionin dimension-less form is
1
120585
1
120597120585[120585120597120579 (120585 120591)
120597120585] =
120597120579 (120585 120591)
120597120591 119903 lt 120585 lt 1 120591 gt 0
120579 (119903 120591) =1205951(120591)
119879119903
120579 (1 120591) =1205952(120591)
119879119903
120579 (120585 0) =1205790(120585)
119879119903
(31)
In this case 119870(120585) = 1 119862(120585) = 1 The two shifting func-tions are
1198921(120585) =
1
1 minus 119903(1 minus 120585)
1198922(120585) =
1
1 minus 119903(120585 minus 119903)
(32)
The two linearly independent fundamental solutions are
1198831(120582119899 120585) = 119869
0(120582119899120585)
1198832(120582119899 120585) = 119884
0(120582119899120585)
(33)
This leads to
119865 (120585 120591) =1
(1 minus 119903) 119879119903
[ (1 minus 120585) 1205951015840
1(120591) + (minus119903 + 120585) 120595
1015840
2(120591)
+1205951(120591)
120585minus1205952(120591)
120585]
(34)
where the characteristic equation is
1198831(1)1198832(119903) minus 119883
1(119903)1198832(1) = 0 (35)
The associated 119899th eigenfunction 120601119899(120585) is determined as
120601119899(120585) =
1198690(120582119899120585)
1198690(120582119899)minus1198840(120582119899120585)
1198840(120582119899) (36)
The eigenvalues 120582119899and the associated eigenfunctions 120601
119899(120585)
are obtained from (35) and (36) The two coefficients in (28)-(29) are derived as
120574119899(120591) =
1
120575119899
1
(1 minus 119903) 119879119903
times int
1
119903
120601119899(120585) [ (120585 minus 120585
2) 1205951015840
1(120591) + (minus119903120585 + 120585
2) 1205951015840
2(120591)
+ 1205951(120591) minus 120595
2(120591) ] 119889120585
119861119899(120591) = 119890
minus1205822
119899120591[120572119899minus int
120591
0
1198901205822
119899120594120574119899(120594) 119889120594]
(37)
Mathematical Problems in Engineering 5
where
120575119899= [int
1
119903
120585 1206012
119899(120585) 119889120585]
120572119899=
1
120575119899119879119903
int
1
119903
120585120601119899(120585) 1205790(120585) 119889120585
(38)
As a result the analytic closed solution of the system indimensionless form is
120579 (120585 120591) =
infin
sum
119899=1
[120601119899(120585) 119861119899(120591)]
+ (1 minus 120585
1 minus 119903)1205951(120591)
119879119903
+ (minus119903 + 120585
1 minus 119903)1205952(120591)
119879119903
(39)
when
1205951(120591) = 120595
2(120591) = 0 (40)
The analytic closed solution in dimensionless form is re-duced to
120579 (120585 120591) =
infin
sum
119899=1
[119890minus1205822
119899120591120601119899(120585) 120572119899] (41)
The solution is exactly the same as the one given by Ozisik [1]
Example 2 Consider the heat conduction in an FG circularhollow cylinder with time-dependent boundary conditionsThe coefficients of thermal conductivity and the specificheat are 119870(120585) = 119896
1198981205851205731 and 119862(120585) = 119888
1198981205851205732 respectively
The boundary value problem of the heat conduction indimensionless form is
1
120585
120597
120597120585[120585119870 (120585)
120597120579 (120585 120591)
120597120585] minus 119862 (120585)
120597120579 (120585 120591)
120597120591= 0
119903 lt 120585 lt 1 120591 gt 0
(42)
120579 (119903 120591) = 0 (43)
120579 (1 120591) = (1 minus 119890minus1198620120591) 1205792 (44)
120579 (120585 0) = 0 (45)
In this case two shifting functions are
1198921(120585) =
1
1 minus 119903(1 minus 120585)
1198922(120585) =
1
1 minus 119903(120585 minus 119903)
(46)
The route to two independent fundamental solutions of of(42) lies in the use of the Frobenius method which can berepresented in terms of the Bessel functions
1198831(120582119899 120585) = 120585
minus12057312119869] (120578119899120585
(2+1205732minus1205731)2)
1198832(120582119899 120585) = 120585
minus12057312119884] (120578119899120585
(2+1205732minus1205731)2)
(47)
where
120578119899= radic
119888119898
119896119898
(2120582119899
1205731minus 1205732minus 2)
] =1205731
1205731minus 1205732minus 2
(48)
Now
119865 (120585 120591)
= 119890minus11986201205911205792[11988811989812058512057321198620(119886lowast+ 119887lowast120585) + 119887
lowast(1205731+ 1) 119896
1198981205851205731minus1]
minus 119887lowast[1205792(1205731+ 1) 119896
1198981205851205731minus1]
(49)
where
119886lowast=minus119903
1 minus 119903
119887lowast=
1
1 minus 119903
(50)
The characteristic equation is
1198831(1)1198832(119903) minus 119883
1(119903)1198832(1) = 0 (51)
The associated 119899th eigenfunction 120601119899(120585) is determined as
120601119899(120585) = 119883
2(1)1198831198991(120585) minus 119883
1(1)1198831198992(120585) (52)
The eigenvalues 120582119899and the associated eigenfunctions 120601
119899(120585)
are obtained from (51) and (52) by numerical analysis Twocoefficients in (28)ndash(30) are derived as
120574119899(120591) = 120574
1198991(120585) 119890minus1198620120591+ 1205741198992(120585)
119861119899(120591) = [120572
119899+1205741198991(120585)
1205822
119899minus 1198620
+1205741198992(120585)
1205822
119899
] 119890minus1205822
119899120591
minus1205741198991(120585)
1205822
119899minus 1198620
119890minus1198620120591minus1205741198992(120585)
1205822
119899
(53)
where
120575119899= [int
1
119903
1198881198981205851205732+11206012
119899(120585) 119889120585]
1205741198991(120585) =
1205792
120575119899
int
1
119903
120601119899(120585) [11988811989812058512057321198620(119886lowast+ 119887lowast120585)
+119887lowast(1205731+ 1) 119896
1198981205851205731minus1] 119889120585
1205741198992(120585) =
minus119887lowast1205792
120575119899
[int
1
119903
120601119899(120585) (120573
1+ 1) 119896
1198981205851205731minus1119889120585]
120572119899= 0
(54)
6 Mathematical Problems in Engineering
Table 1 Temperature distribution of FG circular hollow cylinders with constant value of 1205732and various parameters of 120573
1and 119862
0at 120591 = 05
[119896119898= 1 119888119898= 1 120579(1 120591) = (119862
0sin120596120591)120579
2 1205792= 4 and 120596 = 25]
1198620
1205731
1205732
120585
05 06 07 08 09 10
01075
10 4595 2259 1845 1632 0380
1 0 4592 2285 1990 1820 0380
125 0 4486 2279 2133 1975 0380
1075
10 45952 22589 18453 16320 3796
1 0 45920 22848 19895 18202 3796
125 0 44865 22788 21329 19752 3796
10075
10 459520 225888 184526 163201 37959
1 0 459196 228480 198950 182017 37959
125 0 448647 227881 213290 197520 37959
30
25
20
15
10
05
00
120579
120591C0 = 10
1205731 = 1 1205732 = 51205731 = 1 1205732 = 05
1205731 = 1 1205732 = 51205731 = 1 1205732 = 05
C0 = 1
001 01 1 10
Figure 2 Temperature variation of FG circular hollow cylinderswith constant value of 120573
1and various values of 120573
2and119862
0at 120585 = 075
[119896119898= 1 119888119898= 1 120579(1 120591) = (1 minus 119890minus1198620120591)120579
2 1205792= 4]
Consequently the analytic closed solution for the systemcan be derived as
120579 (120585 120591)
=
(1 minus 119890minus1198620120591) 1205792
1 minus 119903(120585 minus 119903)
+
infin
sum
119899=1
1198832(1)1198831198991(120585) minus 119883
1(1)1198831198992(120585)
times
[1205741198991(120585)
1205822
119899minus 1198620
+1205741198992(120585)
1205822
119899
] 119890minus1205822
119899120591
minus1205741198991(120585)
1205822
119899minus 1198620
119890minus1198880120591minus1205741198992(120585)
1205822
119899
(55)
30
35
40
25
20
15
10
05
00
120579
05 06 07 08 09 10120585
C0 = infin
C0 = 10
1205731 = 0 1205732 = 0
1205731 = 1 1205732 = 5
1205731 = 1 1205732 = 1
1205731 = 0 1205732 = 0
C0 = 11205731 = 1 1205732 = 5
1205731 = 1 1205732 = 1
Figure 3 Temperature distribution of FG circular hollow cylinderswith constant value of 120573
1and various values of 120573
2and 119862
0at 120591 = 02
[119896119898= 1 119888119898= 1 120579(1 120591) = (1 minus 119890minus1198620120591)120579
2 1205792= 4]
In Figure 2 the temperature variation of FG circularhollow cylinders with various parameters of 120573
1 1205732 and 119862
0
at 120585 = 075 is shown It can be found that when 1198620is a
positive constant the temperature parameter of the mediumsat 120585 = 075 increases then reaches the associated constanttemperatures over time The temperature increase rate forthe system with a higher value of 119862
0is greater than that
of one with a lower value of 1198620 Figures 3 and 4 show the
temperature distribution of FG circular hollow cylinders withvarious parameters of 120573
1 1205732 and 119862
0at 120591 = 02 From these
figures it can be observed that with constant parameter 1205731
the temperature of the mediums increases as parameter 1205732
Mathematical Problems in Engineering 7
30
35
25
20
15
10
05
00
120579
05 06 07 08 09 10120585
C0 = 10 C0 = 11205731 = 075 1205732 = 11205731 = 1 1205732 = 1
1205731 = 125 1205732 = 1
1205731 = 1 1205732 = 1
1205731 = 125 1205732 = 1
1205731 = 075 1205732 = 1
Figure 4 Temperature distribution of FG circular hollow cylinderswith constant value of 120573
2and various values of 120573
1and 119862
0at 120591 = 02
[119896119898= 1 119888119898= 1 120579(1 120591) = (1 minus 119890minus1198620120591)120579
2 1205792= 4]
400350300250200150100500
minus50minus100minus150minus200minus250minus300minus350minus400
05 10 15 20 25 30 35 40 45
120579
120591
C0 = 10
1205731 = 1 1205732 = 075
1205731 = 1 1205732 = 1
1205731 = 1 1205732 = 125
Figure 5 Temperature variation of FG circular hollow cylinderswith constant value of 120573
1and various values of 120573
2at 120585 = 075
[119896119898= 1 119888
119898= 1 120579(1 120591) = (119862
0sin120596120591)120579
2 1205792= 4 119862
0= 10 and
120596 = 25]
is increased With constant parameter 1205732 the temperature of
the mediums increases as parameter 1205731is increased
Example 3 Consider the same physical system as discussedin Example 2 In this case the time dependent boundarycondition at 120585 = 1 (44) is changed to the form
120579 (1 120591) = (1198620sin120596120591) 120579
2 (56)
In this case the eigenvalues and eigenfunctions are the sameas those given in Example 2
Now
119865 (120585 120591) =minus12057921198620
1 minus 119903[(1205731+ 1) 119896
1198981205851205731 sin120596120591
minus (120585 minus 119903) 1205961198881198981205851205732 cos120596120591]
(57)
Following the same solution procedures as shown theexact solution for the general system can be derived as
120579 (120585 120591) =1198620sin1205961205911 minus 119903
1205792(120585 minus 119903)
+
infin
sum
119899=1
[1198832(1)1198831198991(120585) minus 119883
1(1)1198831198992(120585)] 119861
119899(120591)
(58)
where
119861119899(120591) =
1
120596 [(1205822
119899120596)2
+ 1]
times
(1205822
119899
120596cos120596120591 + sin120596120591 minus
1205822
119899
120596119890minus1205822
119899120591)1205741198991(120585)
minus(1205822
119899
120596sin120596120591 minus cos120596120591 minus 119890minus120582
2
119899120591)1205741198992(120585)
120575119899= [int
1
119903
1198881198981205851205732+11206012
119899(120585) 119889120585]
1205741198991(120585) =
minus12057921198620
(1 minus 119903) 120575119899
[int
1
0
120601119899(120585) (120585 minus 119903) 120596119888
1198981205851205732+1119889120585]
1205741198992(120585) =
minus12057921198620
(1 minus 119903) 120575119899
[int
1
0
120601119899(120585) (120573
1+ 1) 119896
1198981205851205731+1119889120585]
120572119899= 0
(59)
Figure 5 shows the harmonic temperature variation of FGcircular hollow cylinders with constant value of 120573
1 various
parameters of 1205732at 120585 = 075 119862
0= 10 and 120596 = 25 It can
be observed that with constant value of 1205731 the amplitude of
temperature oscillation for the system with a higher value of1205732will be more than that of one with a lower value of 120573
2
In Table 1 the temperature variations of FG circularhollow cylinders with constant value of 120573
2and various values
of 1205731and 119862
0at 120591 = 05 are given It can be observed that with
constant parameter 1205732 the temperature of the mediums will
decrease as parameter 1205731is decreased
5 Conclusions
Theproblemof heat conductionwith general time-dependentboundary conditions cannot be solved directly by themethodof separation of variables In most of the analyses an integraltransform was used to remove the time-dependent term
8 Mathematical Problems in Engineering
In this paper a new analytic solution method is developedto find the analytic closed solutions for the transient heatconduction in FG circular hollow cylinders with generaltime-dependent boundary conditions The developed solu-tion method is free of any kind of integral transformation
By introducing suitable shifting functions the governingsecond-order regular singular differential equation with vari-able coefficients and time-dependent boundary conditions istransformed into a differential equation with homogenousboundary conditionsThe analytic solution of the systemwiththermal conductivity and specific heat in power functionswith different orders is developed Finally limiting studiesand numerical analyses are given to illustrate the efficiencyand the accuracy of the analysis The proposed solutionmethod can also be extended to the problems with variouskinds of FG materials and time-dependent boundary condi-tions
Acknowledgment
It is gratefully acknowledged that this research was supportedby the National Science Council of Taiwan Taiwan underGrant NSC 99-2221-E-006-021
References
[1] M N Ozisik Boundary Value Problems of Heat ConductionInternational Textbook Company Scranton Pa USA 1st edi-tion 1968
[2] Y Obata and N Noda ldquoSteady thermal stresses in a hollowcircular cylinder and a hollow sphere of a functionally gradientmaterialrdquo Journal of Thermal Stresses vol 17 no 3 pp 471ndash4871994
[3] H Awaji and R Sivakumar ldquoTemperature and stress distribu-tions in a hollow cylinder of functionally graded material thecase of temperature-independent material propertiesrdquo Journalof the American Ceramic Society vol 84 no 5 pp 1059ndash10652001
[4] G N Praveen and J N Reddy ldquoNonlinear transient ther-moelastic analysis of functionally graded ceramic-metal platesrdquoInternational Journal of Solids and Structures vol 35 no 33 pp4457ndash4476 1998
[5] M Jabbari S Sohrabpour and M R Eslami ldquoMechanical andthermal stresses in a functionally graded hollow cylinder dueto radially symmetric loadsrdquo International Journal of PressureVessels and Piping vol 79 no 7 pp 493ndash497 2002
[6] M Jabbari S Sohrabpour and M R Eslami ldquoGeneral solutionfor mechanical and thermal stresses in a functionally gradedhollow cylinder due to nonaxisymmetric steady-state loadsrdquoASME Transactions Journal of Applied Mechanics vol 70 no1 pp 111ndash118 2003
[7] Y Ootao and Y Tanigawa ldquoTransient thermoelastic analysisfor a functionally graded hollow cylinderrdquo Journal of ThermalStresses vol 29 no 11 pp 1031ndash1046 2006
[8] J Zhao X Ai Y Li and Y Zhou ldquoThermal shock resistanceof functionally gradient solid cylindersrdquo Materials Science andEngineering A vol 418 no 1-2 pp 99ndash110 2006
[9] S M Hosseini M Akhlaghi and M Shakeri ldquoTransient heatconduction in functionally graded thick hollow cylinders by
analytical methodrdquo Heat and Mass Transfer vol 43 no 7 pp669ndash675 2007
[10] Y Ootao T Akai and Y Tanigawa ldquoThree-dimensional tran-sient thermal stress analysis of a nonhomogeneous hollowcircular cylinder due to a moving heat source in the axialdirectionrdquo Journal ofThermal Stresses vol 18 no 5 pp 497ndash5121995
[11] Z S Shao and GWMa ldquoThermo-mechanical stresses in func-tionally graded circular hollow cylinder with linearly increasingboundary temperaturerdquo Composite Structures vol 83 no 3 pp259ndash265 2008
[12] M Jabbari A H Mohazzab and A Bahtui ldquoOne-dimensionalmoving heat source in a hollow FGM cylinderrdquo ASME Transac-tions Journal of Pressure Vessel Technology vol 131 no 2 article021202 7 pages 2009
[13] M Asgari andM Akhlaghi ldquoTransient thermal stresses in two-dimensional functionally graded thick hollow cylinder withfinite lengthrdquo Archive of Applied Mechanics vol 80 no 4 pp353ndash376 2010
[14] S Singh P K Jain and R-U Rizwan-Uddin ldquoFinite integraltransform method to solve asymmetric heat conduction in amultilayer annulus with time-dependent boundary conditionsrdquoNuclear Engineering andDesign vol 241 no 1 pp 144ndash154 2011
[15] P Malekzadeh and Y Heydarpour ldquoResponse of functionallygraded cylindrical shells under moving thermo-mechanicalloadsrdquoThin-Walled Structures vol 58 pp 51ndash66 2012
[16] H M Wang and C B Liu ldquoAnalytical solution of two-dimensional transient heat conduction in fiber-reinforcedcylindrical compositesrdquo International Journal of Thermal Sci-ences vol 69 pp 43ndash52 2013
[17] S Y Lee and S M Lin ldquoDynamic analysis of nonuniformbeams with time-dependent elastic boundary conditionsrdquoASME Transactions Journal of Applied Mechanics vol 63 no2 pp 474ndash478 1996
[18] H T Chen S L Sun H C Huang and S Y Lee ldquoAnalyticclosed solution for the heat conduction with time dependentheat convection coefficient at one boundaryrdquo Computer Model-ing in Engineering and Sciences vol 59 no 2 pp 107ndash126 2010
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Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
Consequently the eigenvalues 120582119899 119899 = 1 2 3 can be deter-
minedThe associated 119899th eigenfunction 120601119899(120585) is determined
as
120601119899(120585) = 119883
2(1)1198831198991(120585) minus 119883
1(1)1198831198992(120585) (23)
where 1198831198991(120585) and 119883
1198992(120585) are respectively the fundamental
solutions1198831(120582119899 120585) and119883
2(120582119899 120585) associated with eigenvalues
120582119899 119899 = 1 2 3 They are defined as 119883
1198991(120585) = 119883
1(120582119899 120585)
and 1198831198992(120585) = 119883
2(120582119899 120585) The eigenfunctions 120601
119899(120585) constitute
an orthogonal set in the interval 119903 le 120585 le 1 with respect to aweighting function 120585119862(120585)
int
1
119903
120585119862 (120585) 120601119898(120585) 120601119899(120585) 119889120585 =
0 for 119898 = 119899
120575119899 for 119898 = 119899
(24)
where
120575119899= int
1
119903
120585119862 (120585) 1206012
119899(120585) 119889120585 (25)
In terms of eigenfunctions the transformed variable](120585 120591) can be expressed as
] (120585 120591) =infin
sum
119899=1
120601119899(120585) 119861119899(120591) (26)
Substituting (26) into (12) multiplying it by 120585120601119898 and inte-
grating from 119903 to 1 we obtain
119889119861119899(120591)
119889120591+ 1205822
119899119861119899(120591) = minus120574
119899(120591) (27)
where
120574119899(120591) =
1
120575119899
int
1
119903
120585120601119899(120585) 119865 (120585 120591) 119889120585 (28)
The general solution of (27) is
119861119899(120591) = 119890
minus1205822
119899120591[120572119899minus int
120591
0
1198901205822
119899120594120574119899(120594) 119889120594] (29)
120572119899is determined from the initial condition (15) and
120572119899= 119861119899(0) =
1
120575119899
int
1
119903
120585119862 (120585) 120601119899(120585) ]0(120585) 119889120585 (30)
After substituting the solution of the transformed variable(26) and the shifting functions (11) back to (6) the exactsolution for the general system is obtained
4 Verification and Examples
To illustrate the previous analysis the following examples andlimiting cases are given
Example 1 Consider the heat conduction in an uniformcircular hollow cylinder with time-dependent boundary con-ditions The boundary value problem of the heat conductionin dimension-less form is
1
120585
1
120597120585[120585120597120579 (120585 120591)
120597120585] =
120597120579 (120585 120591)
120597120591 119903 lt 120585 lt 1 120591 gt 0
120579 (119903 120591) =1205951(120591)
119879119903
120579 (1 120591) =1205952(120591)
119879119903
120579 (120585 0) =1205790(120585)
119879119903
(31)
In this case 119870(120585) = 1 119862(120585) = 1 The two shifting func-tions are
1198921(120585) =
1
1 minus 119903(1 minus 120585)
1198922(120585) =
1
1 minus 119903(120585 minus 119903)
(32)
The two linearly independent fundamental solutions are
1198831(120582119899 120585) = 119869
0(120582119899120585)
1198832(120582119899 120585) = 119884
0(120582119899120585)
(33)
This leads to
119865 (120585 120591) =1
(1 minus 119903) 119879119903
[ (1 minus 120585) 1205951015840
1(120591) + (minus119903 + 120585) 120595
1015840
2(120591)
+1205951(120591)
120585minus1205952(120591)
120585]
(34)
where the characteristic equation is
1198831(1)1198832(119903) minus 119883
1(119903)1198832(1) = 0 (35)
The associated 119899th eigenfunction 120601119899(120585) is determined as
120601119899(120585) =
1198690(120582119899120585)
1198690(120582119899)minus1198840(120582119899120585)
1198840(120582119899) (36)
The eigenvalues 120582119899and the associated eigenfunctions 120601
119899(120585)
are obtained from (35) and (36) The two coefficients in (28)-(29) are derived as
120574119899(120591) =
1
120575119899
1
(1 minus 119903) 119879119903
times int
1
119903
120601119899(120585) [ (120585 minus 120585
2) 1205951015840
1(120591) + (minus119903120585 + 120585
2) 1205951015840
2(120591)
+ 1205951(120591) minus 120595
2(120591) ] 119889120585
119861119899(120591) = 119890
minus1205822
119899120591[120572119899minus int
120591
0
1198901205822
119899120594120574119899(120594) 119889120594]
(37)
Mathematical Problems in Engineering 5
where
120575119899= [int
1
119903
120585 1206012
119899(120585) 119889120585]
120572119899=
1
120575119899119879119903
int
1
119903
120585120601119899(120585) 1205790(120585) 119889120585
(38)
As a result the analytic closed solution of the system indimensionless form is
120579 (120585 120591) =
infin
sum
119899=1
[120601119899(120585) 119861119899(120591)]
+ (1 minus 120585
1 minus 119903)1205951(120591)
119879119903
+ (minus119903 + 120585
1 minus 119903)1205952(120591)
119879119903
(39)
when
1205951(120591) = 120595
2(120591) = 0 (40)
The analytic closed solution in dimensionless form is re-duced to
120579 (120585 120591) =
infin
sum
119899=1
[119890minus1205822
119899120591120601119899(120585) 120572119899] (41)
The solution is exactly the same as the one given by Ozisik [1]
Example 2 Consider the heat conduction in an FG circularhollow cylinder with time-dependent boundary conditionsThe coefficients of thermal conductivity and the specificheat are 119870(120585) = 119896
1198981205851205731 and 119862(120585) = 119888
1198981205851205732 respectively
The boundary value problem of the heat conduction indimensionless form is
1
120585
120597
120597120585[120585119870 (120585)
120597120579 (120585 120591)
120597120585] minus 119862 (120585)
120597120579 (120585 120591)
120597120591= 0
119903 lt 120585 lt 1 120591 gt 0
(42)
120579 (119903 120591) = 0 (43)
120579 (1 120591) = (1 minus 119890minus1198620120591) 1205792 (44)
120579 (120585 0) = 0 (45)
In this case two shifting functions are
1198921(120585) =
1
1 minus 119903(1 minus 120585)
1198922(120585) =
1
1 minus 119903(120585 minus 119903)
(46)
The route to two independent fundamental solutions of of(42) lies in the use of the Frobenius method which can berepresented in terms of the Bessel functions
1198831(120582119899 120585) = 120585
minus12057312119869] (120578119899120585
(2+1205732minus1205731)2)
1198832(120582119899 120585) = 120585
minus12057312119884] (120578119899120585
(2+1205732minus1205731)2)
(47)
where
120578119899= radic
119888119898
119896119898
(2120582119899
1205731minus 1205732minus 2)
] =1205731
1205731minus 1205732minus 2
(48)
Now
119865 (120585 120591)
= 119890minus11986201205911205792[11988811989812058512057321198620(119886lowast+ 119887lowast120585) + 119887
lowast(1205731+ 1) 119896
1198981205851205731minus1]
minus 119887lowast[1205792(1205731+ 1) 119896
1198981205851205731minus1]
(49)
where
119886lowast=minus119903
1 minus 119903
119887lowast=
1
1 minus 119903
(50)
The characteristic equation is
1198831(1)1198832(119903) minus 119883
1(119903)1198832(1) = 0 (51)
The associated 119899th eigenfunction 120601119899(120585) is determined as
120601119899(120585) = 119883
2(1)1198831198991(120585) minus 119883
1(1)1198831198992(120585) (52)
The eigenvalues 120582119899and the associated eigenfunctions 120601
119899(120585)
are obtained from (51) and (52) by numerical analysis Twocoefficients in (28)ndash(30) are derived as
120574119899(120591) = 120574
1198991(120585) 119890minus1198620120591+ 1205741198992(120585)
119861119899(120591) = [120572
119899+1205741198991(120585)
1205822
119899minus 1198620
+1205741198992(120585)
1205822
119899
] 119890minus1205822
119899120591
minus1205741198991(120585)
1205822
119899minus 1198620
119890minus1198620120591minus1205741198992(120585)
1205822
119899
(53)
where
120575119899= [int
1
119903
1198881198981205851205732+11206012
119899(120585) 119889120585]
1205741198991(120585) =
1205792
120575119899
int
1
119903
120601119899(120585) [11988811989812058512057321198620(119886lowast+ 119887lowast120585)
+119887lowast(1205731+ 1) 119896
1198981205851205731minus1] 119889120585
1205741198992(120585) =
minus119887lowast1205792
120575119899
[int
1
119903
120601119899(120585) (120573
1+ 1) 119896
1198981205851205731minus1119889120585]
120572119899= 0
(54)
6 Mathematical Problems in Engineering
Table 1 Temperature distribution of FG circular hollow cylinders with constant value of 1205732and various parameters of 120573
1and 119862
0at 120591 = 05
[119896119898= 1 119888119898= 1 120579(1 120591) = (119862
0sin120596120591)120579
2 1205792= 4 and 120596 = 25]
1198620
1205731
1205732
120585
05 06 07 08 09 10
01075
10 4595 2259 1845 1632 0380
1 0 4592 2285 1990 1820 0380
125 0 4486 2279 2133 1975 0380
1075
10 45952 22589 18453 16320 3796
1 0 45920 22848 19895 18202 3796
125 0 44865 22788 21329 19752 3796
10075
10 459520 225888 184526 163201 37959
1 0 459196 228480 198950 182017 37959
125 0 448647 227881 213290 197520 37959
30
25
20
15
10
05
00
120579
120591C0 = 10
1205731 = 1 1205732 = 51205731 = 1 1205732 = 05
1205731 = 1 1205732 = 51205731 = 1 1205732 = 05
C0 = 1
001 01 1 10
Figure 2 Temperature variation of FG circular hollow cylinderswith constant value of 120573
1and various values of 120573
2and119862
0at 120585 = 075
[119896119898= 1 119888119898= 1 120579(1 120591) = (1 minus 119890minus1198620120591)120579
2 1205792= 4]
Consequently the analytic closed solution for the systemcan be derived as
120579 (120585 120591)
=
(1 minus 119890minus1198620120591) 1205792
1 minus 119903(120585 minus 119903)
+
infin
sum
119899=1
1198832(1)1198831198991(120585) minus 119883
1(1)1198831198992(120585)
times
[1205741198991(120585)
1205822
119899minus 1198620
+1205741198992(120585)
1205822
119899
] 119890minus1205822
119899120591
minus1205741198991(120585)
1205822
119899minus 1198620
119890minus1198880120591minus1205741198992(120585)
1205822
119899
(55)
30
35
40
25
20
15
10
05
00
120579
05 06 07 08 09 10120585
C0 = infin
C0 = 10
1205731 = 0 1205732 = 0
1205731 = 1 1205732 = 5
1205731 = 1 1205732 = 1
1205731 = 0 1205732 = 0
C0 = 11205731 = 1 1205732 = 5
1205731 = 1 1205732 = 1
Figure 3 Temperature distribution of FG circular hollow cylinderswith constant value of 120573
1and various values of 120573
2and 119862
0at 120591 = 02
[119896119898= 1 119888119898= 1 120579(1 120591) = (1 minus 119890minus1198620120591)120579
2 1205792= 4]
In Figure 2 the temperature variation of FG circularhollow cylinders with various parameters of 120573
1 1205732 and 119862
0
at 120585 = 075 is shown It can be found that when 1198620is a
positive constant the temperature parameter of the mediumsat 120585 = 075 increases then reaches the associated constanttemperatures over time The temperature increase rate forthe system with a higher value of 119862
0is greater than that
of one with a lower value of 1198620 Figures 3 and 4 show the
temperature distribution of FG circular hollow cylinders withvarious parameters of 120573
1 1205732 and 119862
0at 120591 = 02 From these
figures it can be observed that with constant parameter 1205731
the temperature of the mediums increases as parameter 1205732
Mathematical Problems in Engineering 7
30
35
25
20
15
10
05
00
120579
05 06 07 08 09 10120585
C0 = 10 C0 = 11205731 = 075 1205732 = 11205731 = 1 1205732 = 1
1205731 = 125 1205732 = 1
1205731 = 1 1205732 = 1
1205731 = 125 1205732 = 1
1205731 = 075 1205732 = 1
Figure 4 Temperature distribution of FG circular hollow cylinderswith constant value of 120573
2and various values of 120573
1and 119862
0at 120591 = 02
[119896119898= 1 119888119898= 1 120579(1 120591) = (1 minus 119890minus1198620120591)120579
2 1205792= 4]
400350300250200150100500
minus50minus100minus150minus200minus250minus300minus350minus400
05 10 15 20 25 30 35 40 45
120579
120591
C0 = 10
1205731 = 1 1205732 = 075
1205731 = 1 1205732 = 1
1205731 = 1 1205732 = 125
Figure 5 Temperature variation of FG circular hollow cylinderswith constant value of 120573
1and various values of 120573
2at 120585 = 075
[119896119898= 1 119888
119898= 1 120579(1 120591) = (119862
0sin120596120591)120579
2 1205792= 4 119862
0= 10 and
120596 = 25]
is increased With constant parameter 1205732 the temperature of
the mediums increases as parameter 1205731is increased
Example 3 Consider the same physical system as discussedin Example 2 In this case the time dependent boundarycondition at 120585 = 1 (44) is changed to the form
120579 (1 120591) = (1198620sin120596120591) 120579
2 (56)
In this case the eigenvalues and eigenfunctions are the sameas those given in Example 2
Now
119865 (120585 120591) =minus12057921198620
1 minus 119903[(1205731+ 1) 119896
1198981205851205731 sin120596120591
minus (120585 minus 119903) 1205961198881198981205851205732 cos120596120591]
(57)
Following the same solution procedures as shown theexact solution for the general system can be derived as
120579 (120585 120591) =1198620sin1205961205911 minus 119903
1205792(120585 minus 119903)
+
infin
sum
119899=1
[1198832(1)1198831198991(120585) minus 119883
1(1)1198831198992(120585)] 119861
119899(120591)
(58)
where
119861119899(120591) =
1
120596 [(1205822
119899120596)2
+ 1]
times
(1205822
119899
120596cos120596120591 + sin120596120591 minus
1205822
119899
120596119890minus1205822
119899120591)1205741198991(120585)
minus(1205822
119899
120596sin120596120591 minus cos120596120591 minus 119890minus120582
2
119899120591)1205741198992(120585)
120575119899= [int
1
119903
1198881198981205851205732+11206012
119899(120585) 119889120585]
1205741198991(120585) =
minus12057921198620
(1 minus 119903) 120575119899
[int
1
0
120601119899(120585) (120585 minus 119903) 120596119888
1198981205851205732+1119889120585]
1205741198992(120585) =
minus12057921198620
(1 minus 119903) 120575119899
[int
1
0
120601119899(120585) (120573
1+ 1) 119896
1198981205851205731+1119889120585]
120572119899= 0
(59)
Figure 5 shows the harmonic temperature variation of FGcircular hollow cylinders with constant value of 120573
1 various
parameters of 1205732at 120585 = 075 119862
0= 10 and 120596 = 25 It can
be observed that with constant value of 1205731 the amplitude of
temperature oscillation for the system with a higher value of1205732will be more than that of one with a lower value of 120573
2
In Table 1 the temperature variations of FG circularhollow cylinders with constant value of 120573
2and various values
of 1205731and 119862
0at 120591 = 05 are given It can be observed that with
constant parameter 1205732 the temperature of the mediums will
decrease as parameter 1205731is decreased
5 Conclusions
Theproblemof heat conductionwith general time-dependentboundary conditions cannot be solved directly by themethodof separation of variables In most of the analyses an integraltransform was used to remove the time-dependent term
8 Mathematical Problems in Engineering
In this paper a new analytic solution method is developedto find the analytic closed solutions for the transient heatconduction in FG circular hollow cylinders with generaltime-dependent boundary conditions The developed solu-tion method is free of any kind of integral transformation
By introducing suitable shifting functions the governingsecond-order regular singular differential equation with vari-able coefficients and time-dependent boundary conditions istransformed into a differential equation with homogenousboundary conditionsThe analytic solution of the systemwiththermal conductivity and specific heat in power functionswith different orders is developed Finally limiting studiesand numerical analyses are given to illustrate the efficiencyand the accuracy of the analysis The proposed solutionmethod can also be extended to the problems with variouskinds of FG materials and time-dependent boundary condi-tions
Acknowledgment
It is gratefully acknowledged that this research was supportedby the National Science Council of Taiwan Taiwan underGrant NSC 99-2221-E-006-021
References
[1] M N Ozisik Boundary Value Problems of Heat ConductionInternational Textbook Company Scranton Pa USA 1st edi-tion 1968
[2] Y Obata and N Noda ldquoSteady thermal stresses in a hollowcircular cylinder and a hollow sphere of a functionally gradientmaterialrdquo Journal of Thermal Stresses vol 17 no 3 pp 471ndash4871994
[3] H Awaji and R Sivakumar ldquoTemperature and stress distribu-tions in a hollow cylinder of functionally graded material thecase of temperature-independent material propertiesrdquo Journalof the American Ceramic Society vol 84 no 5 pp 1059ndash10652001
[4] G N Praveen and J N Reddy ldquoNonlinear transient ther-moelastic analysis of functionally graded ceramic-metal platesrdquoInternational Journal of Solids and Structures vol 35 no 33 pp4457ndash4476 1998
[5] M Jabbari S Sohrabpour and M R Eslami ldquoMechanical andthermal stresses in a functionally graded hollow cylinder dueto radially symmetric loadsrdquo International Journal of PressureVessels and Piping vol 79 no 7 pp 493ndash497 2002
[6] M Jabbari S Sohrabpour and M R Eslami ldquoGeneral solutionfor mechanical and thermal stresses in a functionally gradedhollow cylinder due to nonaxisymmetric steady-state loadsrdquoASME Transactions Journal of Applied Mechanics vol 70 no1 pp 111ndash118 2003
[7] Y Ootao and Y Tanigawa ldquoTransient thermoelastic analysisfor a functionally graded hollow cylinderrdquo Journal of ThermalStresses vol 29 no 11 pp 1031ndash1046 2006
[8] J Zhao X Ai Y Li and Y Zhou ldquoThermal shock resistanceof functionally gradient solid cylindersrdquo Materials Science andEngineering A vol 418 no 1-2 pp 99ndash110 2006
[9] S M Hosseini M Akhlaghi and M Shakeri ldquoTransient heatconduction in functionally graded thick hollow cylinders by
analytical methodrdquo Heat and Mass Transfer vol 43 no 7 pp669ndash675 2007
[10] Y Ootao T Akai and Y Tanigawa ldquoThree-dimensional tran-sient thermal stress analysis of a nonhomogeneous hollowcircular cylinder due to a moving heat source in the axialdirectionrdquo Journal ofThermal Stresses vol 18 no 5 pp 497ndash5121995
[11] Z S Shao and GWMa ldquoThermo-mechanical stresses in func-tionally graded circular hollow cylinder with linearly increasingboundary temperaturerdquo Composite Structures vol 83 no 3 pp259ndash265 2008
[12] M Jabbari A H Mohazzab and A Bahtui ldquoOne-dimensionalmoving heat source in a hollow FGM cylinderrdquo ASME Transac-tions Journal of Pressure Vessel Technology vol 131 no 2 article021202 7 pages 2009
[13] M Asgari andM Akhlaghi ldquoTransient thermal stresses in two-dimensional functionally graded thick hollow cylinder withfinite lengthrdquo Archive of Applied Mechanics vol 80 no 4 pp353ndash376 2010
[14] S Singh P K Jain and R-U Rizwan-Uddin ldquoFinite integraltransform method to solve asymmetric heat conduction in amultilayer annulus with time-dependent boundary conditionsrdquoNuclear Engineering andDesign vol 241 no 1 pp 144ndash154 2011
[15] P Malekzadeh and Y Heydarpour ldquoResponse of functionallygraded cylindrical shells under moving thermo-mechanicalloadsrdquoThin-Walled Structures vol 58 pp 51ndash66 2012
[16] H M Wang and C B Liu ldquoAnalytical solution of two-dimensional transient heat conduction in fiber-reinforcedcylindrical compositesrdquo International Journal of Thermal Sci-ences vol 69 pp 43ndash52 2013
[17] S Y Lee and S M Lin ldquoDynamic analysis of nonuniformbeams with time-dependent elastic boundary conditionsrdquoASME Transactions Journal of Applied Mechanics vol 63 no2 pp 474ndash478 1996
[18] H T Chen S L Sun H C Huang and S Y Lee ldquoAnalyticclosed solution for the heat conduction with time dependentheat convection coefficient at one boundaryrdquo Computer Model-ing in Engineering and Sciences vol 59 no 2 pp 107ndash126 2010
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Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
where
120575119899= [int
1
119903
120585 1206012
119899(120585) 119889120585]
120572119899=
1
120575119899119879119903
int
1
119903
120585120601119899(120585) 1205790(120585) 119889120585
(38)
As a result the analytic closed solution of the system indimensionless form is
120579 (120585 120591) =
infin
sum
119899=1
[120601119899(120585) 119861119899(120591)]
+ (1 minus 120585
1 minus 119903)1205951(120591)
119879119903
+ (minus119903 + 120585
1 minus 119903)1205952(120591)
119879119903
(39)
when
1205951(120591) = 120595
2(120591) = 0 (40)
The analytic closed solution in dimensionless form is re-duced to
120579 (120585 120591) =
infin
sum
119899=1
[119890minus1205822
119899120591120601119899(120585) 120572119899] (41)
The solution is exactly the same as the one given by Ozisik [1]
Example 2 Consider the heat conduction in an FG circularhollow cylinder with time-dependent boundary conditionsThe coefficients of thermal conductivity and the specificheat are 119870(120585) = 119896
1198981205851205731 and 119862(120585) = 119888
1198981205851205732 respectively
The boundary value problem of the heat conduction indimensionless form is
1
120585
120597
120597120585[120585119870 (120585)
120597120579 (120585 120591)
120597120585] minus 119862 (120585)
120597120579 (120585 120591)
120597120591= 0
119903 lt 120585 lt 1 120591 gt 0
(42)
120579 (119903 120591) = 0 (43)
120579 (1 120591) = (1 minus 119890minus1198620120591) 1205792 (44)
120579 (120585 0) = 0 (45)
In this case two shifting functions are
1198921(120585) =
1
1 minus 119903(1 minus 120585)
1198922(120585) =
1
1 minus 119903(120585 minus 119903)
(46)
The route to two independent fundamental solutions of of(42) lies in the use of the Frobenius method which can berepresented in terms of the Bessel functions
1198831(120582119899 120585) = 120585
minus12057312119869] (120578119899120585
(2+1205732minus1205731)2)
1198832(120582119899 120585) = 120585
minus12057312119884] (120578119899120585
(2+1205732minus1205731)2)
(47)
where
120578119899= radic
119888119898
119896119898
(2120582119899
1205731minus 1205732minus 2)
] =1205731
1205731minus 1205732minus 2
(48)
Now
119865 (120585 120591)
= 119890minus11986201205911205792[11988811989812058512057321198620(119886lowast+ 119887lowast120585) + 119887
lowast(1205731+ 1) 119896
1198981205851205731minus1]
minus 119887lowast[1205792(1205731+ 1) 119896
1198981205851205731minus1]
(49)
where
119886lowast=minus119903
1 minus 119903
119887lowast=
1
1 minus 119903
(50)
The characteristic equation is
1198831(1)1198832(119903) minus 119883
1(119903)1198832(1) = 0 (51)
The associated 119899th eigenfunction 120601119899(120585) is determined as
120601119899(120585) = 119883
2(1)1198831198991(120585) minus 119883
1(1)1198831198992(120585) (52)
The eigenvalues 120582119899and the associated eigenfunctions 120601
119899(120585)
are obtained from (51) and (52) by numerical analysis Twocoefficients in (28)ndash(30) are derived as
120574119899(120591) = 120574
1198991(120585) 119890minus1198620120591+ 1205741198992(120585)
119861119899(120591) = [120572
119899+1205741198991(120585)
1205822
119899minus 1198620
+1205741198992(120585)
1205822
119899
] 119890minus1205822
119899120591
minus1205741198991(120585)
1205822
119899minus 1198620
119890minus1198620120591minus1205741198992(120585)
1205822
119899
(53)
where
120575119899= [int
1
119903
1198881198981205851205732+11206012
119899(120585) 119889120585]
1205741198991(120585) =
1205792
120575119899
int
1
119903
120601119899(120585) [11988811989812058512057321198620(119886lowast+ 119887lowast120585)
+119887lowast(1205731+ 1) 119896
1198981205851205731minus1] 119889120585
1205741198992(120585) =
minus119887lowast1205792
120575119899
[int
1
119903
120601119899(120585) (120573
1+ 1) 119896
1198981205851205731minus1119889120585]
120572119899= 0
(54)
6 Mathematical Problems in Engineering
Table 1 Temperature distribution of FG circular hollow cylinders with constant value of 1205732and various parameters of 120573
1and 119862
0at 120591 = 05
[119896119898= 1 119888119898= 1 120579(1 120591) = (119862
0sin120596120591)120579
2 1205792= 4 and 120596 = 25]
1198620
1205731
1205732
120585
05 06 07 08 09 10
01075
10 4595 2259 1845 1632 0380
1 0 4592 2285 1990 1820 0380
125 0 4486 2279 2133 1975 0380
1075
10 45952 22589 18453 16320 3796
1 0 45920 22848 19895 18202 3796
125 0 44865 22788 21329 19752 3796
10075
10 459520 225888 184526 163201 37959
1 0 459196 228480 198950 182017 37959
125 0 448647 227881 213290 197520 37959
30
25
20
15
10
05
00
120579
120591C0 = 10
1205731 = 1 1205732 = 51205731 = 1 1205732 = 05
1205731 = 1 1205732 = 51205731 = 1 1205732 = 05
C0 = 1
001 01 1 10
Figure 2 Temperature variation of FG circular hollow cylinderswith constant value of 120573
1and various values of 120573
2and119862
0at 120585 = 075
[119896119898= 1 119888119898= 1 120579(1 120591) = (1 minus 119890minus1198620120591)120579
2 1205792= 4]
Consequently the analytic closed solution for the systemcan be derived as
120579 (120585 120591)
=
(1 minus 119890minus1198620120591) 1205792
1 minus 119903(120585 minus 119903)
+
infin
sum
119899=1
1198832(1)1198831198991(120585) minus 119883
1(1)1198831198992(120585)
times
[1205741198991(120585)
1205822
119899minus 1198620
+1205741198992(120585)
1205822
119899
] 119890minus1205822
119899120591
minus1205741198991(120585)
1205822
119899minus 1198620
119890minus1198880120591minus1205741198992(120585)
1205822
119899
(55)
30
35
40
25
20
15
10
05
00
120579
05 06 07 08 09 10120585
C0 = infin
C0 = 10
1205731 = 0 1205732 = 0
1205731 = 1 1205732 = 5
1205731 = 1 1205732 = 1
1205731 = 0 1205732 = 0
C0 = 11205731 = 1 1205732 = 5
1205731 = 1 1205732 = 1
Figure 3 Temperature distribution of FG circular hollow cylinderswith constant value of 120573
1and various values of 120573
2and 119862
0at 120591 = 02
[119896119898= 1 119888119898= 1 120579(1 120591) = (1 minus 119890minus1198620120591)120579
2 1205792= 4]
In Figure 2 the temperature variation of FG circularhollow cylinders with various parameters of 120573
1 1205732 and 119862
0
at 120585 = 075 is shown It can be found that when 1198620is a
positive constant the temperature parameter of the mediumsat 120585 = 075 increases then reaches the associated constanttemperatures over time The temperature increase rate forthe system with a higher value of 119862
0is greater than that
of one with a lower value of 1198620 Figures 3 and 4 show the
temperature distribution of FG circular hollow cylinders withvarious parameters of 120573
1 1205732 and 119862
0at 120591 = 02 From these
figures it can be observed that with constant parameter 1205731
the temperature of the mediums increases as parameter 1205732
Mathematical Problems in Engineering 7
30
35
25
20
15
10
05
00
120579
05 06 07 08 09 10120585
C0 = 10 C0 = 11205731 = 075 1205732 = 11205731 = 1 1205732 = 1
1205731 = 125 1205732 = 1
1205731 = 1 1205732 = 1
1205731 = 125 1205732 = 1
1205731 = 075 1205732 = 1
Figure 4 Temperature distribution of FG circular hollow cylinderswith constant value of 120573
2and various values of 120573
1and 119862
0at 120591 = 02
[119896119898= 1 119888119898= 1 120579(1 120591) = (1 minus 119890minus1198620120591)120579
2 1205792= 4]
400350300250200150100500
minus50minus100minus150minus200minus250minus300minus350minus400
05 10 15 20 25 30 35 40 45
120579
120591
C0 = 10
1205731 = 1 1205732 = 075
1205731 = 1 1205732 = 1
1205731 = 1 1205732 = 125
Figure 5 Temperature variation of FG circular hollow cylinderswith constant value of 120573
1and various values of 120573
2at 120585 = 075
[119896119898= 1 119888
119898= 1 120579(1 120591) = (119862
0sin120596120591)120579
2 1205792= 4 119862
0= 10 and
120596 = 25]
is increased With constant parameter 1205732 the temperature of
the mediums increases as parameter 1205731is increased
Example 3 Consider the same physical system as discussedin Example 2 In this case the time dependent boundarycondition at 120585 = 1 (44) is changed to the form
120579 (1 120591) = (1198620sin120596120591) 120579
2 (56)
In this case the eigenvalues and eigenfunctions are the sameas those given in Example 2
Now
119865 (120585 120591) =minus12057921198620
1 minus 119903[(1205731+ 1) 119896
1198981205851205731 sin120596120591
minus (120585 minus 119903) 1205961198881198981205851205732 cos120596120591]
(57)
Following the same solution procedures as shown theexact solution for the general system can be derived as
120579 (120585 120591) =1198620sin1205961205911 minus 119903
1205792(120585 minus 119903)
+
infin
sum
119899=1
[1198832(1)1198831198991(120585) minus 119883
1(1)1198831198992(120585)] 119861
119899(120591)
(58)
where
119861119899(120591) =
1
120596 [(1205822
119899120596)2
+ 1]
times
(1205822
119899
120596cos120596120591 + sin120596120591 minus
1205822
119899
120596119890minus1205822
119899120591)1205741198991(120585)
minus(1205822
119899
120596sin120596120591 minus cos120596120591 minus 119890minus120582
2
119899120591)1205741198992(120585)
120575119899= [int
1
119903
1198881198981205851205732+11206012
119899(120585) 119889120585]
1205741198991(120585) =
minus12057921198620
(1 minus 119903) 120575119899
[int
1
0
120601119899(120585) (120585 minus 119903) 120596119888
1198981205851205732+1119889120585]
1205741198992(120585) =
minus12057921198620
(1 minus 119903) 120575119899
[int
1
0
120601119899(120585) (120573
1+ 1) 119896
1198981205851205731+1119889120585]
120572119899= 0
(59)
Figure 5 shows the harmonic temperature variation of FGcircular hollow cylinders with constant value of 120573
1 various
parameters of 1205732at 120585 = 075 119862
0= 10 and 120596 = 25 It can
be observed that with constant value of 1205731 the amplitude of
temperature oscillation for the system with a higher value of1205732will be more than that of one with a lower value of 120573
2
In Table 1 the temperature variations of FG circularhollow cylinders with constant value of 120573
2and various values
of 1205731and 119862
0at 120591 = 05 are given It can be observed that with
constant parameter 1205732 the temperature of the mediums will
decrease as parameter 1205731is decreased
5 Conclusions
Theproblemof heat conductionwith general time-dependentboundary conditions cannot be solved directly by themethodof separation of variables In most of the analyses an integraltransform was used to remove the time-dependent term
8 Mathematical Problems in Engineering
In this paper a new analytic solution method is developedto find the analytic closed solutions for the transient heatconduction in FG circular hollow cylinders with generaltime-dependent boundary conditions The developed solu-tion method is free of any kind of integral transformation
By introducing suitable shifting functions the governingsecond-order regular singular differential equation with vari-able coefficients and time-dependent boundary conditions istransformed into a differential equation with homogenousboundary conditionsThe analytic solution of the systemwiththermal conductivity and specific heat in power functionswith different orders is developed Finally limiting studiesand numerical analyses are given to illustrate the efficiencyand the accuracy of the analysis The proposed solutionmethod can also be extended to the problems with variouskinds of FG materials and time-dependent boundary condi-tions
Acknowledgment
It is gratefully acknowledged that this research was supportedby the National Science Council of Taiwan Taiwan underGrant NSC 99-2221-E-006-021
References
[1] M N Ozisik Boundary Value Problems of Heat ConductionInternational Textbook Company Scranton Pa USA 1st edi-tion 1968
[2] Y Obata and N Noda ldquoSteady thermal stresses in a hollowcircular cylinder and a hollow sphere of a functionally gradientmaterialrdquo Journal of Thermal Stresses vol 17 no 3 pp 471ndash4871994
[3] H Awaji and R Sivakumar ldquoTemperature and stress distribu-tions in a hollow cylinder of functionally graded material thecase of temperature-independent material propertiesrdquo Journalof the American Ceramic Society vol 84 no 5 pp 1059ndash10652001
[4] G N Praveen and J N Reddy ldquoNonlinear transient ther-moelastic analysis of functionally graded ceramic-metal platesrdquoInternational Journal of Solids and Structures vol 35 no 33 pp4457ndash4476 1998
[5] M Jabbari S Sohrabpour and M R Eslami ldquoMechanical andthermal stresses in a functionally graded hollow cylinder dueto radially symmetric loadsrdquo International Journal of PressureVessels and Piping vol 79 no 7 pp 493ndash497 2002
[6] M Jabbari S Sohrabpour and M R Eslami ldquoGeneral solutionfor mechanical and thermal stresses in a functionally gradedhollow cylinder due to nonaxisymmetric steady-state loadsrdquoASME Transactions Journal of Applied Mechanics vol 70 no1 pp 111ndash118 2003
[7] Y Ootao and Y Tanigawa ldquoTransient thermoelastic analysisfor a functionally graded hollow cylinderrdquo Journal of ThermalStresses vol 29 no 11 pp 1031ndash1046 2006
[8] J Zhao X Ai Y Li and Y Zhou ldquoThermal shock resistanceof functionally gradient solid cylindersrdquo Materials Science andEngineering A vol 418 no 1-2 pp 99ndash110 2006
[9] S M Hosseini M Akhlaghi and M Shakeri ldquoTransient heatconduction in functionally graded thick hollow cylinders by
analytical methodrdquo Heat and Mass Transfer vol 43 no 7 pp669ndash675 2007
[10] Y Ootao T Akai and Y Tanigawa ldquoThree-dimensional tran-sient thermal stress analysis of a nonhomogeneous hollowcircular cylinder due to a moving heat source in the axialdirectionrdquo Journal ofThermal Stresses vol 18 no 5 pp 497ndash5121995
[11] Z S Shao and GWMa ldquoThermo-mechanical stresses in func-tionally graded circular hollow cylinder with linearly increasingboundary temperaturerdquo Composite Structures vol 83 no 3 pp259ndash265 2008
[12] M Jabbari A H Mohazzab and A Bahtui ldquoOne-dimensionalmoving heat source in a hollow FGM cylinderrdquo ASME Transac-tions Journal of Pressure Vessel Technology vol 131 no 2 article021202 7 pages 2009
[13] M Asgari andM Akhlaghi ldquoTransient thermal stresses in two-dimensional functionally graded thick hollow cylinder withfinite lengthrdquo Archive of Applied Mechanics vol 80 no 4 pp353ndash376 2010
[14] S Singh P K Jain and R-U Rizwan-Uddin ldquoFinite integraltransform method to solve asymmetric heat conduction in amultilayer annulus with time-dependent boundary conditionsrdquoNuclear Engineering andDesign vol 241 no 1 pp 144ndash154 2011
[15] P Malekzadeh and Y Heydarpour ldquoResponse of functionallygraded cylindrical shells under moving thermo-mechanicalloadsrdquoThin-Walled Structures vol 58 pp 51ndash66 2012
[16] H M Wang and C B Liu ldquoAnalytical solution of two-dimensional transient heat conduction in fiber-reinforcedcylindrical compositesrdquo International Journal of Thermal Sci-ences vol 69 pp 43ndash52 2013
[17] S Y Lee and S M Lin ldquoDynamic analysis of nonuniformbeams with time-dependent elastic boundary conditionsrdquoASME Transactions Journal of Applied Mechanics vol 63 no2 pp 474ndash478 1996
[18] H T Chen S L Sun H C Huang and S Y Lee ldquoAnalyticclosed solution for the heat conduction with time dependentheat convection coefficient at one boundaryrdquo Computer Model-ing in Engineering and Sciences vol 59 no 2 pp 107ndash126 2010
Submit your manuscripts athttpwwwhindawicom
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
Table 1 Temperature distribution of FG circular hollow cylinders with constant value of 1205732and various parameters of 120573
1and 119862
0at 120591 = 05
[119896119898= 1 119888119898= 1 120579(1 120591) = (119862
0sin120596120591)120579
2 1205792= 4 and 120596 = 25]
1198620
1205731
1205732
120585
05 06 07 08 09 10
01075
10 4595 2259 1845 1632 0380
1 0 4592 2285 1990 1820 0380
125 0 4486 2279 2133 1975 0380
1075
10 45952 22589 18453 16320 3796
1 0 45920 22848 19895 18202 3796
125 0 44865 22788 21329 19752 3796
10075
10 459520 225888 184526 163201 37959
1 0 459196 228480 198950 182017 37959
125 0 448647 227881 213290 197520 37959
30
25
20
15
10
05
00
120579
120591C0 = 10
1205731 = 1 1205732 = 51205731 = 1 1205732 = 05
1205731 = 1 1205732 = 51205731 = 1 1205732 = 05
C0 = 1
001 01 1 10
Figure 2 Temperature variation of FG circular hollow cylinderswith constant value of 120573
1and various values of 120573
2and119862
0at 120585 = 075
[119896119898= 1 119888119898= 1 120579(1 120591) = (1 minus 119890minus1198620120591)120579
2 1205792= 4]
Consequently the analytic closed solution for the systemcan be derived as
120579 (120585 120591)
=
(1 minus 119890minus1198620120591) 1205792
1 minus 119903(120585 minus 119903)
+
infin
sum
119899=1
1198832(1)1198831198991(120585) minus 119883
1(1)1198831198992(120585)
times
[1205741198991(120585)
1205822
119899minus 1198620
+1205741198992(120585)
1205822
119899
] 119890minus1205822
119899120591
minus1205741198991(120585)
1205822
119899minus 1198620
119890minus1198880120591minus1205741198992(120585)
1205822
119899
(55)
30
35
40
25
20
15
10
05
00
120579
05 06 07 08 09 10120585
C0 = infin
C0 = 10
1205731 = 0 1205732 = 0
1205731 = 1 1205732 = 5
1205731 = 1 1205732 = 1
1205731 = 0 1205732 = 0
C0 = 11205731 = 1 1205732 = 5
1205731 = 1 1205732 = 1
Figure 3 Temperature distribution of FG circular hollow cylinderswith constant value of 120573
1and various values of 120573
2and 119862
0at 120591 = 02
[119896119898= 1 119888119898= 1 120579(1 120591) = (1 minus 119890minus1198620120591)120579
2 1205792= 4]
In Figure 2 the temperature variation of FG circularhollow cylinders with various parameters of 120573
1 1205732 and 119862
0
at 120585 = 075 is shown It can be found that when 1198620is a
positive constant the temperature parameter of the mediumsat 120585 = 075 increases then reaches the associated constanttemperatures over time The temperature increase rate forthe system with a higher value of 119862
0is greater than that
of one with a lower value of 1198620 Figures 3 and 4 show the
temperature distribution of FG circular hollow cylinders withvarious parameters of 120573
1 1205732 and 119862
0at 120591 = 02 From these
figures it can be observed that with constant parameter 1205731
the temperature of the mediums increases as parameter 1205732
Mathematical Problems in Engineering 7
30
35
25
20
15
10
05
00
120579
05 06 07 08 09 10120585
C0 = 10 C0 = 11205731 = 075 1205732 = 11205731 = 1 1205732 = 1
1205731 = 125 1205732 = 1
1205731 = 1 1205732 = 1
1205731 = 125 1205732 = 1
1205731 = 075 1205732 = 1
Figure 4 Temperature distribution of FG circular hollow cylinderswith constant value of 120573
2and various values of 120573
1and 119862
0at 120591 = 02
[119896119898= 1 119888119898= 1 120579(1 120591) = (1 minus 119890minus1198620120591)120579
2 1205792= 4]
400350300250200150100500
minus50minus100minus150minus200minus250minus300minus350minus400
05 10 15 20 25 30 35 40 45
120579
120591
C0 = 10
1205731 = 1 1205732 = 075
1205731 = 1 1205732 = 1
1205731 = 1 1205732 = 125
Figure 5 Temperature variation of FG circular hollow cylinderswith constant value of 120573
1and various values of 120573
2at 120585 = 075
[119896119898= 1 119888
119898= 1 120579(1 120591) = (119862
0sin120596120591)120579
2 1205792= 4 119862
0= 10 and
120596 = 25]
is increased With constant parameter 1205732 the temperature of
the mediums increases as parameter 1205731is increased
Example 3 Consider the same physical system as discussedin Example 2 In this case the time dependent boundarycondition at 120585 = 1 (44) is changed to the form
120579 (1 120591) = (1198620sin120596120591) 120579
2 (56)
In this case the eigenvalues and eigenfunctions are the sameas those given in Example 2
Now
119865 (120585 120591) =minus12057921198620
1 minus 119903[(1205731+ 1) 119896
1198981205851205731 sin120596120591
minus (120585 minus 119903) 1205961198881198981205851205732 cos120596120591]
(57)
Following the same solution procedures as shown theexact solution for the general system can be derived as
120579 (120585 120591) =1198620sin1205961205911 minus 119903
1205792(120585 minus 119903)
+
infin
sum
119899=1
[1198832(1)1198831198991(120585) minus 119883
1(1)1198831198992(120585)] 119861
119899(120591)
(58)
where
119861119899(120591) =
1
120596 [(1205822
119899120596)2
+ 1]
times
(1205822
119899
120596cos120596120591 + sin120596120591 minus
1205822
119899
120596119890minus1205822
119899120591)1205741198991(120585)
minus(1205822
119899
120596sin120596120591 minus cos120596120591 minus 119890minus120582
2
119899120591)1205741198992(120585)
120575119899= [int
1
119903
1198881198981205851205732+11206012
119899(120585) 119889120585]
1205741198991(120585) =
minus12057921198620
(1 minus 119903) 120575119899
[int
1
0
120601119899(120585) (120585 minus 119903) 120596119888
1198981205851205732+1119889120585]
1205741198992(120585) =
minus12057921198620
(1 minus 119903) 120575119899
[int
1
0
120601119899(120585) (120573
1+ 1) 119896
1198981205851205731+1119889120585]
120572119899= 0
(59)
Figure 5 shows the harmonic temperature variation of FGcircular hollow cylinders with constant value of 120573
1 various
parameters of 1205732at 120585 = 075 119862
0= 10 and 120596 = 25 It can
be observed that with constant value of 1205731 the amplitude of
temperature oscillation for the system with a higher value of1205732will be more than that of one with a lower value of 120573
2
In Table 1 the temperature variations of FG circularhollow cylinders with constant value of 120573
2and various values
of 1205731and 119862
0at 120591 = 05 are given It can be observed that with
constant parameter 1205732 the temperature of the mediums will
decrease as parameter 1205731is decreased
5 Conclusions
Theproblemof heat conductionwith general time-dependentboundary conditions cannot be solved directly by themethodof separation of variables In most of the analyses an integraltransform was used to remove the time-dependent term
8 Mathematical Problems in Engineering
In this paper a new analytic solution method is developedto find the analytic closed solutions for the transient heatconduction in FG circular hollow cylinders with generaltime-dependent boundary conditions The developed solu-tion method is free of any kind of integral transformation
By introducing suitable shifting functions the governingsecond-order regular singular differential equation with vari-able coefficients and time-dependent boundary conditions istransformed into a differential equation with homogenousboundary conditionsThe analytic solution of the systemwiththermal conductivity and specific heat in power functionswith different orders is developed Finally limiting studiesand numerical analyses are given to illustrate the efficiencyand the accuracy of the analysis The proposed solutionmethod can also be extended to the problems with variouskinds of FG materials and time-dependent boundary condi-tions
Acknowledgment
It is gratefully acknowledged that this research was supportedby the National Science Council of Taiwan Taiwan underGrant NSC 99-2221-E-006-021
References
[1] M N Ozisik Boundary Value Problems of Heat ConductionInternational Textbook Company Scranton Pa USA 1st edi-tion 1968
[2] Y Obata and N Noda ldquoSteady thermal stresses in a hollowcircular cylinder and a hollow sphere of a functionally gradientmaterialrdquo Journal of Thermal Stresses vol 17 no 3 pp 471ndash4871994
[3] H Awaji and R Sivakumar ldquoTemperature and stress distribu-tions in a hollow cylinder of functionally graded material thecase of temperature-independent material propertiesrdquo Journalof the American Ceramic Society vol 84 no 5 pp 1059ndash10652001
[4] G N Praveen and J N Reddy ldquoNonlinear transient ther-moelastic analysis of functionally graded ceramic-metal platesrdquoInternational Journal of Solids and Structures vol 35 no 33 pp4457ndash4476 1998
[5] M Jabbari S Sohrabpour and M R Eslami ldquoMechanical andthermal stresses in a functionally graded hollow cylinder dueto radially symmetric loadsrdquo International Journal of PressureVessels and Piping vol 79 no 7 pp 493ndash497 2002
[6] M Jabbari S Sohrabpour and M R Eslami ldquoGeneral solutionfor mechanical and thermal stresses in a functionally gradedhollow cylinder due to nonaxisymmetric steady-state loadsrdquoASME Transactions Journal of Applied Mechanics vol 70 no1 pp 111ndash118 2003
[7] Y Ootao and Y Tanigawa ldquoTransient thermoelastic analysisfor a functionally graded hollow cylinderrdquo Journal of ThermalStresses vol 29 no 11 pp 1031ndash1046 2006
[8] J Zhao X Ai Y Li and Y Zhou ldquoThermal shock resistanceof functionally gradient solid cylindersrdquo Materials Science andEngineering A vol 418 no 1-2 pp 99ndash110 2006
[9] S M Hosseini M Akhlaghi and M Shakeri ldquoTransient heatconduction in functionally graded thick hollow cylinders by
analytical methodrdquo Heat and Mass Transfer vol 43 no 7 pp669ndash675 2007
[10] Y Ootao T Akai and Y Tanigawa ldquoThree-dimensional tran-sient thermal stress analysis of a nonhomogeneous hollowcircular cylinder due to a moving heat source in the axialdirectionrdquo Journal ofThermal Stresses vol 18 no 5 pp 497ndash5121995
[11] Z S Shao and GWMa ldquoThermo-mechanical stresses in func-tionally graded circular hollow cylinder with linearly increasingboundary temperaturerdquo Composite Structures vol 83 no 3 pp259ndash265 2008
[12] M Jabbari A H Mohazzab and A Bahtui ldquoOne-dimensionalmoving heat source in a hollow FGM cylinderrdquo ASME Transac-tions Journal of Pressure Vessel Technology vol 131 no 2 article021202 7 pages 2009
[13] M Asgari andM Akhlaghi ldquoTransient thermal stresses in two-dimensional functionally graded thick hollow cylinder withfinite lengthrdquo Archive of Applied Mechanics vol 80 no 4 pp353ndash376 2010
[14] S Singh P K Jain and R-U Rizwan-Uddin ldquoFinite integraltransform method to solve asymmetric heat conduction in amultilayer annulus with time-dependent boundary conditionsrdquoNuclear Engineering andDesign vol 241 no 1 pp 144ndash154 2011
[15] P Malekzadeh and Y Heydarpour ldquoResponse of functionallygraded cylindrical shells under moving thermo-mechanicalloadsrdquoThin-Walled Structures vol 58 pp 51ndash66 2012
[16] H M Wang and C B Liu ldquoAnalytical solution of two-dimensional transient heat conduction in fiber-reinforcedcylindrical compositesrdquo International Journal of Thermal Sci-ences vol 69 pp 43ndash52 2013
[17] S Y Lee and S M Lin ldquoDynamic analysis of nonuniformbeams with time-dependent elastic boundary conditionsrdquoASME Transactions Journal of Applied Mechanics vol 63 no2 pp 474ndash478 1996
[18] H T Chen S L Sun H C Huang and S Y Lee ldquoAnalyticclosed solution for the heat conduction with time dependentheat convection coefficient at one boundaryrdquo Computer Model-ing in Engineering and Sciences vol 59 no 2 pp 107ndash126 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
30
35
25
20
15
10
05
00
120579
05 06 07 08 09 10120585
C0 = 10 C0 = 11205731 = 075 1205732 = 11205731 = 1 1205732 = 1
1205731 = 125 1205732 = 1
1205731 = 1 1205732 = 1
1205731 = 125 1205732 = 1
1205731 = 075 1205732 = 1
Figure 4 Temperature distribution of FG circular hollow cylinderswith constant value of 120573
2and various values of 120573
1and 119862
0at 120591 = 02
[119896119898= 1 119888119898= 1 120579(1 120591) = (1 minus 119890minus1198620120591)120579
2 1205792= 4]
400350300250200150100500
minus50minus100minus150minus200minus250minus300minus350minus400
05 10 15 20 25 30 35 40 45
120579
120591
C0 = 10
1205731 = 1 1205732 = 075
1205731 = 1 1205732 = 1
1205731 = 1 1205732 = 125
Figure 5 Temperature variation of FG circular hollow cylinderswith constant value of 120573
1and various values of 120573
2at 120585 = 075
[119896119898= 1 119888
119898= 1 120579(1 120591) = (119862
0sin120596120591)120579
2 1205792= 4 119862
0= 10 and
120596 = 25]
is increased With constant parameter 1205732 the temperature of
the mediums increases as parameter 1205731is increased
Example 3 Consider the same physical system as discussedin Example 2 In this case the time dependent boundarycondition at 120585 = 1 (44) is changed to the form
120579 (1 120591) = (1198620sin120596120591) 120579
2 (56)
In this case the eigenvalues and eigenfunctions are the sameas those given in Example 2
Now
119865 (120585 120591) =minus12057921198620
1 minus 119903[(1205731+ 1) 119896
1198981205851205731 sin120596120591
minus (120585 minus 119903) 1205961198881198981205851205732 cos120596120591]
(57)
Following the same solution procedures as shown theexact solution for the general system can be derived as
120579 (120585 120591) =1198620sin1205961205911 minus 119903
1205792(120585 minus 119903)
+
infin
sum
119899=1
[1198832(1)1198831198991(120585) minus 119883
1(1)1198831198992(120585)] 119861
119899(120591)
(58)
where
119861119899(120591) =
1
120596 [(1205822
119899120596)2
+ 1]
times
(1205822
119899
120596cos120596120591 + sin120596120591 minus
1205822
119899
120596119890minus1205822
119899120591)1205741198991(120585)
minus(1205822
119899
120596sin120596120591 minus cos120596120591 minus 119890minus120582
2
119899120591)1205741198992(120585)
120575119899= [int
1
119903
1198881198981205851205732+11206012
119899(120585) 119889120585]
1205741198991(120585) =
minus12057921198620
(1 minus 119903) 120575119899
[int
1
0
120601119899(120585) (120585 minus 119903) 120596119888
1198981205851205732+1119889120585]
1205741198992(120585) =
minus12057921198620
(1 minus 119903) 120575119899
[int
1
0
120601119899(120585) (120573
1+ 1) 119896
1198981205851205731+1119889120585]
120572119899= 0
(59)
Figure 5 shows the harmonic temperature variation of FGcircular hollow cylinders with constant value of 120573
1 various
parameters of 1205732at 120585 = 075 119862
0= 10 and 120596 = 25 It can
be observed that with constant value of 1205731 the amplitude of
temperature oscillation for the system with a higher value of1205732will be more than that of one with a lower value of 120573
2
In Table 1 the temperature variations of FG circularhollow cylinders with constant value of 120573
2and various values
of 1205731and 119862
0at 120591 = 05 are given It can be observed that with
constant parameter 1205732 the temperature of the mediums will
decrease as parameter 1205731is decreased
5 Conclusions
Theproblemof heat conductionwith general time-dependentboundary conditions cannot be solved directly by themethodof separation of variables In most of the analyses an integraltransform was used to remove the time-dependent term
8 Mathematical Problems in Engineering
In this paper a new analytic solution method is developedto find the analytic closed solutions for the transient heatconduction in FG circular hollow cylinders with generaltime-dependent boundary conditions The developed solu-tion method is free of any kind of integral transformation
By introducing suitable shifting functions the governingsecond-order regular singular differential equation with vari-able coefficients and time-dependent boundary conditions istransformed into a differential equation with homogenousboundary conditionsThe analytic solution of the systemwiththermal conductivity and specific heat in power functionswith different orders is developed Finally limiting studiesand numerical analyses are given to illustrate the efficiencyand the accuracy of the analysis The proposed solutionmethod can also be extended to the problems with variouskinds of FG materials and time-dependent boundary condi-tions
Acknowledgment
It is gratefully acknowledged that this research was supportedby the National Science Council of Taiwan Taiwan underGrant NSC 99-2221-E-006-021
References
[1] M N Ozisik Boundary Value Problems of Heat ConductionInternational Textbook Company Scranton Pa USA 1st edi-tion 1968
[2] Y Obata and N Noda ldquoSteady thermal stresses in a hollowcircular cylinder and a hollow sphere of a functionally gradientmaterialrdquo Journal of Thermal Stresses vol 17 no 3 pp 471ndash4871994
[3] H Awaji and R Sivakumar ldquoTemperature and stress distribu-tions in a hollow cylinder of functionally graded material thecase of temperature-independent material propertiesrdquo Journalof the American Ceramic Society vol 84 no 5 pp 1059ndash10652001
[4] G N Praveen and J N Reddy ldquoNonlinear transient ther-moelastic analysis of functionally graded ceramic-metal platesrdquoInternational Journal of Solids and Structures vol 35 no 33 pp4457ndash4476 1998
[5] M Jabbari S Sohrabpour and M R Eslami ldquoMechanical andthermal stresses in a functionally graded hollow cylinder dueto radially symmetric loadsrdquo International Journal of PressureVessels and Piping vol 79 no 7 pp 493ndash497 2002
[6] M Jabbari S Sohrabpour and M R Eslami ldquoGeneral solutionfor mechanical and thermal stresses in a functionally gradedhollow cylinder due to nonaxisymmetric steady-state loadsrdquoASME Transactions Journal of Applied Mechanics vol 70 no1 pp 111ndash118 2003
[7] Y Ootao and Y Tanigawa ldquoTransient thermoelastic analysisfor a functionally graded hollow cylinderrdquo Journal of ThermalStresses vol 29 no 11 pp 1031ndash1046 2006
[8] J Zhao X Ai Y Li and Y Zhou ldquoThermal shock resistanceof functionally gradient solid cylindersrdquo Materials Science andEngineering A vol 418 no 1-2 pp 99ndash110 2006
[9] S M Hosseini M Akhlaghi and M Shakeri ldquoTransient heatconduction in functionally graded thick hollow cylinders by
analytical methodrdquo Heat and Mass Transfer vol 43 no 7 pp669ndash675 2007
[10] Y Ootao T Akai and Y Tanigawa ldquoThree-dimensional tran-sient thermal stress analysis of a nonhomogeneous hollowcircular cylinder due to a moving heat source in the axialdirectionrdquo Journal ofThermal Stresses vol 18 no 5 pp 497ndash5121995
[11] Z S Shao and GWMa ldquoThermo-mechanical stresses in func-tionally graded circular hollow cylinder with linearly increasingboundary temperaturerdquo Composite Structures vol 83 no 3 pp259ndash265 2008
[12] M Jabbari A H Mohazzab and A Bahtui ldquoOne-dimensionalmoving heat source in a hollow FGM cylinderrdquo ASME Transac-tions Journal of Pressure Vessel Technology vol 131 no 2 article021202 7 pages 2009
[13] M Asgari andM Akhlaghi ldquoTransient thermal stresses in two-dimensional functionally graded thick hollow cylinder withfinite lengthrdquo Archive of Applied Mechanics vol 80 no 4 pp353ndash376 2010
[14] S Singh P K Jain and R-U Rizwan-Uddin ldquoFinite integraltransform method to solve asymmetric heat conduction in amultilayer annulus with time-dependent boundary conditionsrdquoNuclear Engineering andDesign vol 241 no 1 pp 144ndash154 2011
[15] P Malekzadeh and Y Heydarpour ldquoResponse of functionallygraded cylindrical shells under moving thermo-mechanicalloadsrdquoThin-Walled Structures vol 58 pp 51ndash66 2012
[16] H M Wang and C B Liu ldquoAnalytical solution of two-dimensional transient heat conduction in fiber-reinforcedcylindrical compositesrdquo International Journal of Thermal Sci-ences vol 69 pp 43ndash52 2013
[17] S Y Lee and S M Lin ldquoDynamic analysis of nonuniformbeams with time-dependent elastic boundary conditionsrdquoASME Transactions Journal of Applied Mechanics vol 63 no2 pp 474ndash478 1996
[18] H T Chen S L Sun H C Huang and S Y Lee ldquoAnalyticclosed solution for the heat conduction with time dependentheat convection coefficient at one boundaryrdquo Computer Model-ing in Engineering and Sciences vol 59 no 2 pp 107ndash126 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
In this paper a new analytic solution method is developedto find the analytic closed solutions for the transient heatconduction in FG circular hollow cylinders with generaltime-dependent boundary conditions The developed solu-tion method is free of any kind of integral transformation
By introducing suitable shifting functions the governingsecond-order regular singular differential equation with vari-able coefficients and time-dependent boundary conditions istransformed into a differential equation with homogenousboundary conditionsThe analytic solution of the systemwiththermal conductivity and specific heat in power functionswith different orders is developed Finally limiting studiesand numerical analyses are given to illustrate the efficiencyand the accuracy of the analysis The proposed solutionmethod can also be extended to the problems with variouskinds of FG materials and time-dependent boundary condi-tions
Acknowledgment
It is gratefully acknowledged that this research was supportedby the National Science Council of Taiwan Taiwan underGrant NSC 99-2221-E-006-021
References
[1] M N Ozisik Boundary Value Problems of Heat ConductionInternational Textbook Company Scranton Pa USA 1st edi-tion 1968
[2] Y Obata and N Noda ldquoSteady thermal stresses in a hollowcircular cylinder and a hollow sphere of a functionally gradientmaterialrdquo Journal of Thermal Stresses vol 17 no 3 pp 471ndash4871994
[3] H Awaji and R Sivakumar ldquoTemperature and stress distribu-tions in a hollow cylinder of functionally graded material thecase of temperature-independent material propertiesrdquo Journalof the American Ceramic Society vol 84 no 5 pp 1059ndash10652001
[4] G N Praveen and J N Reddy ldquoNonlinear transient ther-moelastic analysis of functionally graded ceramic-metal platesrdquoInternational Journal of Solids and Structures vol 35 no 33 pp4457ndash4476 1998
[5] M Jabbari S Sohrabpour and M R Eslami ldquoMechanical andthermal stresses in a functionally graded hollow cylinder dueto radially symmetric loadsrdquo International Journal of PressureVessels and Piping vol 79 no 7 pp 493ndash497 2002
[6] M Jabbari S Sohrabpour and M R Eslami ldquoGeneral solutionfor mechanical and thermal stresses in a functionally gradedhollow cylinder due to nonaxisymmetric steady-state loadsrdquoASME Transactions Journal of Applied Mechanics vol 70 no1 pp 111ndash118 2003
[7] Y Ootao and Y Tanigawa ldquoTransient thermoelastic analysisfor a functionally graded hollow cylinderrdquo Journal of ThermalStresses vol 29 no 11 pp 1031ndash1046 2006
[8] J Zhao X Ai Y Li and Y Zhou ldquoThermal shock resistanceof functionally gradient solid cylindersrdquo Materials Science andEngineering A vol 418 no 1-2 pp 99ndash110 2006
[9] S M Hosseini M Akhlaghi and M Shakeri ldquoTransient heatconduction in functionally graded thick hollow cylinders by
analytical methodrdquo Heat and Mass Transfer vol 43 no 7 pp669ndash675 2007
[10] Y Ootao T Akai and Y Tanigawa ldquoThree-dimensional tran-sient thermal stress analysis of a nonhomogeneous hollowcircular cylinder due to a moving heat source in the axialdirectionrdquo Journal ofThermal Stresses vol 18 no 5 pp 497ndash5121995
[11] Z S Shao and GWMa ldquoThermo-mechanical stresses in func-tionally graded circular hollow cylinder with linearly increasingboundary temperaturerdquo Composite Structures vol 83 no 3 pp259ndash265 2008
[12] M Jabbari A H Mohazzab and A Bahtui ldquoOne-dimensionalmoving heat source in a hollow FGM cylinderrdquo ASME Transac-tions Journal of Pressure Vessel Technology vol 131 no 2 article021202 7 pages 2009
[13] M Asgari andM Akhlaghi ldquoTransient thermal stresses in two-dimensional functionally graded thick hollow cylinder withfinite lengthrdquo Archive of Applied Mechanics vol 80 no 4 pp353ndash376 2010
[14] S Singh P K Jain and R-U Rizwan-Uddin ldquoFinite integraltransform method to solve asymmetric heat conduction in amultilayer annulus with time-dependent boundary conditionsrdquoNuclear Engineering andDesign vol 241 no 1 pp 144ndash154 2011
[15] P Malekzadeh and Y Heydarpour ldquoResponse of functionallygraded cylindrical shells under moving thermo-mechanicalloadsrdquoThin-Walled Structures vol 58 pp 51ndash66 2012
[16] H M Wang and C B Liu ldquoAnalytical solution of two-dimensional transient heat conduction in fiber-reinforcedcylindrical compositesrdquo International Journal of Thermal Sci-ences vol 69 pp 43ndash52 2013
[17] S Y Lee and S M Lin ldquoDynamic analysis of nonuniformbeams with time-dependent elastic boundary conditionsrdquoASME Transactions Journal of Applied Mechanics vol 63 no2 pp 474ndash478 1996
[18] H T Chen S L Sun H C Huang and S Y Lee ldquoAnalyticclosed solution for the heat conduction with time dependentheat convection coefficient at one boundaryrdquo Computer Model-ing in Engineering and Sciences vol 59 no 2 pp 107ndash126 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
