International Journal of Pure and Applied Mathematics

Volume 106 No. 4 2016, 1127-1150ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.eudoi: 10.12732/ijpam.v106i4.14 ijpam.eu

SOLUTION OF HEAT CONDUCTION PROBLEM WITH

DISCONTINUOUS BOUNDARY CONDITIONS IN

NONHOMOGENOUS MOVING CYLINDER USING MAPLE

G.R. Gasimov1 §, Z.A. Abusutash2

1Faculty of Applied MathematicsBaku State University

23, Zahid Khalil, Baku, AZERBAIJAN2Department of Mathematics

Faculty of ScienceDamietta University

New Damietta, 34517, EGYPT

Abstract: This paper presents a heat conduction problem with discontinuous boundaryconditions in nonhomogeneous moving entire cylinder, which moves on the axis oz with anymovement law z = S(t). The temperature field is determined in the cylindrical coordinatessystem linked with motionless cylinder as a system in a single movement. Using the methodinversion of the unit function η(t − S−1(z)), method a sequence of integral transformationslike Fourier transform with respect to z, Hankel transform with respect to r, Bessel functionstheory,general integral transforms theory, a solution in the form of the series is obtained.

And in order to illustrate theoretical results in this paper,we wrote special programming

in Maple program and for a special cases, where numerical solutions were presented with

explained graphics and discussed.

AMS Subject Classification: 35K05, 35A22, 33C10, 44A05, 42A38, 68N15

Key Words: heat conduction problem, Maple program, general integral transforms theory,

sequential integral transformation method,bessel functions theory

Received: December 11, 2015

Published: March 13, 2016

c© 2016 Academic Publications, Ltd.

url: www.acadpubl.eu

§Correspondence author

1128 G.R. Gasimov, Z.A. Abusutash

1. Introduction

The thermal properties of a solid very with position, exact solutions can befound in a limited number of special,in this paper studying the case of oneof these cases. Thermal conductivity problem of the semi-infinite nonhomo-geneous solids,it has been studied by Carslaw and Jaeger [2]. Combining thetraditional presentation of fundamental mathematical concepts with the con-temporary computational benefits of Maple software submitted by Articolo [4].In recent years,a number of the papers investigated analytical heat flow initiatewithin a hollow infinite cylinder using Maple program and the numerical solu-tions obtained and showed graphic animation, has been submitted by Shahoutet al [7]. Solution of the problems thermal conductivity of the homogeneousmoving bodies with finite size is obtained by Lotarev [8].Kholodovskii [9] con-siders boundary value problems for linear differential in piecewise homogenouscylinders into half cylinders by multilayer film. Studied thermal conductivity inthe moving composite cylinder is obtained by [6]. Our motivation in this paperis to investigate the solution of heat conduction problem in physically nonhomo-geneous moving entire cylinder,on the direction of the radius with discontinuousboundary conditions on the surface of the cylinder, which moves along the axisoz. Using the method insertion of the unit function,general integral transformstheory, method sequential of integral transformations like Fourier and Hankeltransformation with corresponding inverse transformations respectively, a solu-tion to the problem of the spread heat in the form of the series is obtained.Asnumerical solutions of the problem under study are achieved by using Maple18package, where results are obtained and clear graphics with discussion. Andthese problems have a great importance in many engineering fields which inter-vention in the design of internal combustion engines, material in aviation, andthe factories of the production of military.

2. Formulation of the Problem

Consider the temperature field of moving cylinder with discontinuous boundaryconditions and initially at temperature ϕ(r, z) for entire cylinder,which movesalong the z-axis with any movement law.The temperature field is determined inthe cylindrical coordinates system linked with motionless cylinder as a systemin a single movement. This coordinate system moves from one environment inother environment with different thermophyscial descriptions.

SOLUTION OF HEAT CONDUCTION PROBLEM WITH... 1129

The formulation of this problem is given as:

∂

∂r(k(r)

∂

∂rT (r, z, t)) +

∂2

∂z2T (r, z, t)− ρc ∂

∂tT (r, z, t) = 0,

0 < r < a , 0 < z < `, z = S(t) , t > 0 (1)

T (r, z, 0) = ϕ(r, z), (2)

[α1∂

∂rT + α2T ]|r=a =

f0(z, t), if z < S(t),

f1(z, t), if z > S(t),(3)

T (r, z, t)|r=0 <∞, (4)

[β1∂

∂zT − β2T ]|z=0 = −g0(r, t), [β3

∂

∂zT + β4T ]|z=` = g1(r, t), (5)

k(r) = k0rm (k0 − const, 0 ≤ m ≤ 1), ρ − const, c − const,

α21 + α2

2 > 0, β21 + β22 > 0, β23 + β24 > 0, β22 + β24 > 0,

where a is the radius of the cylinder.T, k(r), ρ and c are the temperature, ther-mal conductivity, density and specific heat of the cylinder, and α1, α2, β1, β2,β3, β4 are the coefficient of surface heat transfer are constants, and ϕ(r, z), f0(z, t),f1(z, t), g0(r, t), g1(r, t)are given functions.

3. Solving Method

The boundary condition (3) is discontinuous,we can write its as follows:

[α1∂

∂rT + α2T ]|r=a = f1(z, t)− f2(z, t)η(t− S−1(z)) ≡ f(z, t); (6)

where

η(τ) =

1, if τ > 0,

0, if τ < 0,

S−1(z) is inverse function to z = S(t), f2(z, t) = f1(z, t)− f0(z, t).

1130 G.R. Gasimov, Z.A. Abusutash

We choose a solution of this problem in the form:

T (r, z, t) = T0(r, z, t) + T1(r, t) + T2(r, t)z, (7)

where T0(r, z, t) satisfies homogeneous boundary conditions as follows:

[β1∂

∂zT0 − β2T0]|z=0 = 0, [β3

∂

∂zT0 + β4T0]|z=` = 0. (8)

Using (7) and boundary conditions (5) we get:

[β1(∂∂zT0 + T2)− β2(T0 + T1 + T2z)]|z=0 = −g0(r, t),

[β3(∂∂zT0 + T2) + β4(T0 + T1 + T2z]|z=` = g1(r, t).

(9)

This system (9) we rewrite as following

[β1∂∂zT0 − β2T0]|z=0 = (β2T1 − β1T2 − g0),

[β3∂∂zT0 + β4T0]|z=` = (−β4T1 − (β4`+ β3)T2 + g1).

(10)

In order to satisfies conditions of (8), the most T1, T2 satisfies the followingsystem:

β2T1 − β1T2 = g0,

β4T1 + (β4`+ β3)T2 = g1.(11)

The determinant this system

∆1 =

∣∣∣∣ β2 −β1β4 β4`+ β3

∣∣∣∣ ;also ∆1 = β1β4 + β2 (β4`+ β3) 6= 0, we find that the system (11) with respectto T1, T2 there are non- trivial solution:

T1(r, t) =1

∆1[g0(r, t)(β4`+ β3) + β1g1(r, t)],

T2(r, t) =1

∆1[β2g1(r, t)− β4g0(r, t)].

Now we use integral transform with respect to z; on the basis of the theoryof integral transformations [5], the kernel of this transformation assignment assolution of Sturm- Liouville problem in the following form:

d2

dz2K(z) + γ2K = 0, 0 < z < `, (12)

SOLUTION OF HEAT CONDUCTION PROBLEM WITH... 1131

[β1d

dzK − β2K]|z=0 = 0, [β3

d

dzK + β4K]|z=` = 0. (13)

The general solution of (12) is

K(z) = C1 cos γz + C2 sin γz,

where C1 and C2 are arbitrary constants, from the condition (2), we can befound C1 and C2 asfollows:

− β2C1 + β1γC2 = 0,

(−β3γ sin γ`+ β4 cos γ`)C1 + (β3γ cos γ`+ β4 sin γ`)C2 = 0.(14)

For the existence non trivial solution to this system a necessary and sufficientcondition the determinant of this system equal to zero:∣∣∣∣ −β2 β1γ

−β3γ sin γ`+ β4 cos γ` β3γ cos γ`+ β4 sin γ`

∣∣∣∣ = 0.

For parameter γ we get the equation as follows:

(β1β4 + β2β3) γ cos γ` =(β1β3γ

2 − β2β4)

sin γ`

or

tgγ` =γ (β1β4 + β2β3)

γ2β1β3 − β2β4. (15)

The roots of (15) are all real, simple, and have an arithmetic numbers: γn, n =1, 2, ... are the eigenvalues of the problem (12)-(13), and the corresponding so-lutions are eigenfunctions for this problem.

Assign the eigenfunctions of the first equation for the system (14)

−β2C1 + β1γnC2 = 0;

from here we choose C1 = β1γn, C2 = β2.Then

Kn(z) = β1γn cos γnz + β2 sin γnz;n = 1, 2, ...

and

Kn(z) =1

cnKn(z), cn =

∫ `

0K2n(z)dz,

1132 G.R. Gasimov, Z.A. Abusutash

are the kernel of transform with respect to z. This transformation is integraltransform Fourier. We apply Fourier transform by kernel Kn(z) for (1), andwith respect to Fourier representation, we get on the following equation:

ρc∂

∂tT0,n(r, t) =

∂

∂r(k(r)

∂

∂rT0,n)− γ2nT0,n + T ∗n(r, t), n = 1, 2, ... (16)

where

T0,n(r, t) =

∫ `

0T0(r, z, t)Kn(z)dz,

T ∗n(r, t) = [∂

∂r(k(r)

∂

∂rT1(r, t))− ρc

∂

∂tT1(r, t)]

∫ `

0Kn(z)dz

+[∂

∂r(k(r)

∂

∂rT2(r, t))− ρc

∂

∂tT2(r, t)]

∫ `

0zKn(z)dz.

For T0,n(r, t) we find initial and boundary conditions as follows:

T0,n(r, 0) = ϕn(r)− T1(r, 0)

∫ `

0Kn(z)dz − T2(r, 0)

∫ `

0zKn(z)dz, (17)

[α1∂

∂r+ α2]T0,n(r, t)|r=a = fn(t)− [α1

∂

∂r+ α2]T1(r, t)|r=a

∫ `

0Kn(z)dz

− [α1∂

∂r+ α2]T2(r, t)|r=a

∫ `

0zKn(z)dz ≡ f0,n(t), (18)

where

ϕn(r) =

∫ `

0ϕ(r, z)Kn(z)dz,

fn(t) =

∫ `

0f(z, t)Kn(z)dz.

In order to be converges uniformly with respect to r, we write the solution(16)-(18) on the following formula:

T0,n(r, t) = T ∗0,n(r, t) +1

α2fn(t),

SOLUTION OF HEAT CONDUCTION PROBLEM WITH... 1133

thus T ∗0,n(r, t) satisfies boundary condition:

[α1∂

∂r+ α2]T

∗0,n(r, t)|r=a = [α1

∂

∂r+ α2]T0,n(r, t)|r=a︸ ︷︷ ︸

f0,n(t)

− α2 ·1

α2f0,n(t) = 0.

With respect to T ∗0,n(r, t) we get the following equation:

ρc∂

∂tT ∗0,n(r, t) =

∂

∂r(k(r)

∂

∂rT ∗0,n(r, t))− γ2nT ∗0,n(r, t) + T ∗n(r, t)

− 1

α2

[ρcd

dt+ γ2n

]f0,n(t), (19)

T ∗0,n(r, 0) = ϕ∗0,n(r), (20)

T ∗0,n(r, t)|r=0 = T0,n(r, t)|r=0 −1

α2f0,n(t) <∞, (21)

[α1∂

∂r+ α2]T

∗0,n(r, t)|r=a = 0, (22)

where

ϕ∗0,n(r) = ϕn(r)− T1(r, 0)

∫ `

0Kn(z)dz − T2(r, 0)

∫ `

0zKn(z)dz.

For problem (19)-(22), we use integral transform with respect to r.So that the kernel and weight of this integral be found by using eigenfunctionsfor Sturm-Liouvill problem following:(

k(r)Q′(r))′

+ λ2Q(r) = 0, 0 < r < a, (23)

Q(0) <∞, [α1d

dr+ α2]Q(r)|r=a = 0, (24)

the solution of this problem (23) and (24) are the kernel of integral transformwith respect to r.

Using permutation variables:

r → y : y =2

2−mr

2−m2 ; r = (

2−m2

y)2

2−m ,

and when the equation (23) takes the following form

1134 G.R. Gasimov, Z.A. Abusutash

∂2

∂y2Q+

m

y

∂

∂yQ+ (

λ√k0

)2Q = 0, 0 < r < a.

The solution that bounded when r = 0, appointed in the following form:

Jν(λ

β√k0rβ) · r(1−m)/2;β =

2−m2

; ν = (1−m)/(2−m).

Using a second condition of (24), we get

α1√k0λJ ′ν(

λ

β√k0aβ) + a

m2−1((1−m)

α1

2+ aα2)Jν(

λ

β√k0aβ) = 0. (25)

The roots of (25) are all real, simple and have an arithmetic numbers,λη , η = 1, 2, ... are the roots of the eigenvalues for Sturm-Liouville problem(23) and (24).

Using eigenfunctions corresponding, we find kernel of integral transformwith respect to r.

K∗η(r) =1

NηJν(

λη

β√k0rβ)r(1−m)/2,

where

Nη =

∫ a

0(Jν(

λη

β√k0rβ)r(1−m)/2)2dr.

Now using integeral transform with respect to r,by kernel K∗η(r) and weight

1, we get on problem for representation the function T ∗0,n(r, t)

ρcd

dtT ∗0,n,η(t) = −µ2n,ηT ∗0,n,η(t) + T ∗n,η(t)− F0,n,η(t), (26)

T ∗0,n,η(t)|t=0 = ϕ∗0,n,η, (27)

where

µ2n,η = (λ2η + γ2n),

T ∗n,η(t) =

∫ a

0T ∗n(r, t)K∗η(r)dr,

F0,n,η(t) =1

α2[ρc

d

dt+ γ2n]f0,n(t) ·

∫ a

0K∗η(r)dr,

SOLUTION OF HEAT CONDUCTION PROBLEM WITH... 1135

ϕ∗0,n,η =

∫ a

0ϕ∗0,n(r)K∗η(r)dr.

The solution of (26) and (27) as:

T ∗0,n,η(t) = e−µ2n,η/(ρc)·t[ϕ∗0,n,η +

1

ρc

∫ t

0(T ∗n,η(τ)− F0,n,η(τ))eµ

2n,η/(ρc)·τdτ ]. (28)

Using the inverse Hankel transform with respect to r , we obtain:

T ∗0,n(r, t) =∞∑η=1

e−µ2n,η/(ρc)·t[ϕ∗0,n,η +

1

ρc

∫ t

0(T ∗n,η(τ)− F0,n,η(τ))

·eµ2n,η/(ρc)·τdτ ]Jν(λη

β√k0rβ)r(1−m)/2.

Then

T0,n(r, t) =1

α2f0,n(t) +

∞∑η=1

e−µ2n,η/(ρc)·t[ϕ∗0,n,η +

1

ρc

∫ t

0(T ∗n,η(τ)

−F0,n,η(τ))eµ2n,η/(ρc)·τdτ ]Jν(

λη

β√k0rβ)r(1−m)/2.

Finally, applying the inverse Fourier transform with respect to z ,we findthe relation to T (r, z, t) :

T (r, z, t) = T1(r, t) + zT2(r, t) +∞∑n=1

{ 1

α2f0,n(t) +

∞∑η=1

e−µ2n,η/(ρc)·t

·[ϕ∗0,n,η +1

ρc

∫ t

0(T ∗n,η(τ)− F0,n,η(τ))eµ

2n,η/(ρc)·τdτ ]

Jν(λη

β√k0rβ)r(1−m)/2}Kn(z).

4. Steps Solution using Program Maple 18

To consider that the outer surface of cylinder is silver: k0 = 1, ρ = 10.49,c = 0.0556,(using the units are c.g.s, caloric, and 0c).

SOLUTION OF HEAT CONDUCTION PROBLEM WITH... 1149

5. Conclusion

In this paper, the spread heat in nonhomogeneous moving entire cylinder isstudied, the solution behavior with discontinuous boundary conditions of theboundary problem is given as an infinite series, using two sequences of integraltransformations, Bessel functions theory, we get the solution to the problem ofthe spread heat in the form of the series,and with writing special programmingin Maple program as for a special cases,we constructed to a spread heat as afunction of r as a function of z.A numerical solutions were obtained and tablesas curves defined in 2D,3D. It can be noticed that the inner surface of thecylinder under the effect of two waves be a convex curve and direction fromthe top to the bottom,as temperatures regularly drop.As for the cylinder whichbe under the effect one wave, be a convex curve and direction to top. Andalso through the comparison can be seen that the curves are the different asstructure is different.This problem has numerous engineering applications, suchas,machining, welding, grinding,are all others practical examples.

Acknowledgements

The authors are very much grateful to Prof. Ahmed M. K. Tarabia and Dr. R.Tantawy members of mathematics department at Damietta university, Egyptfor their valuable comments to bring the paper in the present form.

References

[1] G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd edition, CambridgeUniversity Press, Cambridge (1966).

[2] H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd edition, Oxford Uni-versity Press, London, England, UK, (1959).

[3] M. N. Ozis¯ik, Heat Conduction, 2nd edition, John Wiley and Sons, New York (1993).

[4] G. A. Articolo, Partial Differential Equations and Boundary Value Problems with Maple,2nd edition, Academic Press, San Diego (2009).

[5] N. S. Koshlyakov,et al, Equation in Partial Mathematical Physics, Visshaya Shkola,Moscow (1970).

[6] G. R. Gasimov, Z. A. Abusutash, Heat Conduction Equation in Physically Inhomo-geneous Moving Composite Solids, International Journal of Differential Equations andApplications, AP, LTD, V.14, N.3, (2015),252-270.

[7] A. M. Shahout, G. R. Gasimov and E. A. Rzayev, Solution problem of heat conductionin the circular hollow cylinder using the program Maple11 ,The Second Conference onMathematical Sciences , Zarqa Private University, Jordan , (October 22-23,2008), 63-73.

1150 G.R. Gasimov, Z.A. Abusutash

[8] V. Y. Lotarev, Temperature filed of finite continuous moving cylinder, J. New of TulaState University, V.5, N.2, (1999), 82-86.

[9] S. Y. Kholodovskii, Solving Boundary Value Problems in Cylinders with Multilayer Filminclusion, J, Scientific Notes of Zabaikalskii State University, N. 3, (2012), 145-151.