A Strong Method for Solving Systems of Integro-Differential … · 2013. 12. 24. · E-mail:...

Applied Mathematics, 2011, 2, 1105-1113 doi:10.4236/am.2011.29152 Published Online September 2011 (http://www.SciRP.org/journal/am)

Copyright © 2011 SciRes. AM

A Strong Method for Solving Systems of Integro-Differential Equations

Jafar Biazar, Hamideh Ebrahimi Department of Applied Mathematics, Faculty of Mathematical Science, University of Guilan, Rasht, Iran

E-mail: [email protected], [email protected] Received June 5, 2011; revised June 28, 2011; accepted July 6, 2011

Abstract The introduced method in this paper consists of reducing a system of integro-differential equations into a system of algebraic equations, by expanding the unknown functions, as a series in terms of Chebyshev wavelets with unknown coefficients. Extension of Chebyshev wavelets method for solving these systems is the novelty of this paper. Some examples to illustrate the simplicity and the effectiveness of the proposed method have been presented. Keywords: Systems of Integro-Differential Equations, Chebyshev Wavelets Method, Mother Wavelet, Op-

erational Matrix

1. Introduction In recent years, many different orthogonal functions and polynomials have been used to approximate the solution of various functional equations. The main goal of using or-thogonal basis is that the equation under study reduces to a system of linear or non-linear algebraic equations. This can be done by truncating series of functions with orthogonal basis for the solution of equations and using the operational matrices. In this paper, Chebyshev wavelets basis, on the interval [0, 1], have been considered for solving systems of integro-differential equations. There are some applications

of Chebyshev wavelets method in the literature [1-3]. Systems of integro-differential equations arise in ma-

thematical modeling of many phenomena. Some tech- niques have been used for solving these systems such as, Adomian decomposition method (ADM) [4], He’s homotopy perturbation method (HPM) [5,6], variational iteration method (VIM) [7], The Tau method [8,9], differential transform method (DTM ) [10], power series method, ra-tionalized Haar functions method and Galerkin method for linear systems [11-13]. The general form of these systems are considered as follows

1) Systems of Volterra integro-differential equations

1

2

1 1 11

, , 1 101

, , , , , ,

, , , , ,

0 1, 1,2, , ,

mm m

i i i k nk

mx m m

i j i j nj

u x f x F x u x u x u x u x

k x t G u t u t u t t

x i n

d ,

m

d ,

m

(1)

2) Systems of Fredholm integro-differential equations

1

2

1 1 11

1

, , 1 101

( ) , , , , ,

, , , , ,

0 1, 1, 2, , ,

mm m

i i i k nk

mm m

i j i j nj

u x f x F x u x u x u x u x

k x t G u t u t u t t

x i n

(2)

where , and are positive integers, m 1m 2m 2, 0,1 0,ijk x t L 1 are the kernels,

J. BIAZAR ET AL. 1106

, , 1, 2, ,if x i n are known functions, i kF and are linear or non-linear functions and

are unknown functions. ijG , 1, 2, ,iu x i n ,This paper is organized as follows: in Section 2, Che-

byshev wavelets method is explained. In Section 3, the application of the method for introduced systems (1) and (2) is studied. Some numerical examples are presented in Section 4, Conclusions are presented in Section 5. 2. Wavelets and Chebyshev Wavelets Wavelets constitute a family of functions constructed from dilation and translation of a single function called the mother wavelet [14,15]. When the dilation parameter a and the translation parameter b vary continuously we have the following family of continuous wavelets

1

2, , , ,a b

x bx a a b a

a

0. (3)

Chebyshev wavelets , , , ,n m x k n m x 1, , 2 k k

have four arguments, , is an arbitrary positive integer and m is the order of Chebyshev polynomials of the first kind. They are defined on the interval [0,1], as fo

1, 2n

llows:

21 1

, , ,

12 ( 2 2 1) ,

2 20, otherwise

nm

kk

m k k

x k n m x

n nT x n x

, (4)

where

1, 0

π

2, 0

π

m

m

m

T x

T x m

,

.

1,

(5)

For and .

m are the famous Chebyshev polynomials of the first kind of degree m, which construct an orthogonal system with respect to the weight function

0,1, , 1m M , 0 ,1, ,T x m M

11, 2, , 2 kn

2

1

1W x

x

, on the interval 1,1 , and satisfy the

following recursive formula:

0

1

1 1

1,

,

2 , m m m

T x

T x x

T x xT x T x m

1,2,

.

(6)

The set of Chebyshev wavelets is an orthogonal set with

respect to the weight function 2 2 1knW x W x n

A function f x defined on the interval [0, 1] may

be presented as

1 0

,nm nmn m

f x c

x (7)

where ,

nnm nm W x

c f x x . Series presentation

(7), can be approximated by the following truncated se-ries.

12 1

1 0

,k M

Tnm nm

n m

f x c x C

x (8)

where C and x are 12k 1M matrices given by

1 1

1

10 11 1 1 20 21 2 1

T

2 0 2 1

T

1 2 1 2

, , , , , , , ,

, , ,

, , , , , ,

C

k k

k

M M

M

M M M

c c c c c c

c c

c c c c c

,

(9)

and

1 1

1

10 11 1, 1 20 21

T

2, 1 2 0 2 , 1

T

1 2 1 2

, , , , , ,

, , , , ,

, , , , , ,

.

k k

k

M

M M

M M M

x

x x x x x

x x x

x x x x x

(10) Also a function ,f x y defined on 0,1 0,1

can be approximated as the following

, . T f x y x K y (11)

Here the entries of matrix 1 12 2k kij M MK k

will be obtained by

,

1

, , ,

, 1, , 2 .

nn

i j i j W y W x

k

k x f x y y

i j M

, (12)

The integration of the vector x , defined in (10), can be achieved as

0

dx

t t P x . (13)

where P is the 1 12 2k kM M operational matrix of integration [1,2]. This matrix is determined as follows.

,

L F F F

O L F

P O O L F

F

O O O L

(14)


J. BIAZAR ET AL.


1107

where F, and O L are M M matrices given by

1 1

2

2 0

0 0

2 20 0

3

2 1 1 1 120 0

2 1 1

1 1 1 120 0

2 2

F

k r r

M M

r r

M M

0

0

,

3. Application of Chebyshev Wavelets Method

In this section the introduced method will be applied to solve systems (1) and (2). 3.1. Application to Solve System (1)

Consider system of integro-differential Equations (1), with the following conditions

(15)

0 0 0

0 0 0,

0 0 0

O

(16)

The property of the product of two Chebyshev wave-lets vector functions will be as follows

.T x x Y Y x (18)

where is a given vector and is a Y Y 1 12 2k kM M matrix. This matrix is called the operational matrix of product. The integration of the product of two Cheby-shev wavelets vector functions with respect to the weight function is derived as ,nW x

1 0

d ITnW x x x x ,

1.

(19)

0 , 1, , , 0, , si i su a i n s m

n

.

(20)

First we assume the unknown functions are approximated in the follow-

ing forms

, 1, 2, ,miu x i

, 1, 2, , m Ti iu x C x i n (21)

Therefore we have

1

0

,!

1, , , 0, , 1.

jm ss T m s

i i i sj

xu x C P x a

j

i n s m

(22)

Using (20) and (21) other terms will be considered as the following general expansions

1 1 1

1 1

1 2

,

, , , , , , , ,

,

, , , , , , ,

, ,

1, 2, , , 1, 2, , , 1,2, , .

Ti i

m mik n n

Tik

m m Tij n n ij

Tij ij

f x F x

F x u x u x u x u x u x

X x

G u t u t u t u t Y t

k x t x K t

i n k m j m

where is an identity matrix. I

1 1

2

11 0 0 0

2

2 10 0 0 0

4 4

2 1 10 0 0

3 2 6

2 ,

1 12 10 0 0

2 1 1 2 1 2 1

1 12 10 0 0 0

2 2 2 2

L

k

r r

M M

r r r r

M M M

0

1

(17)

J. BIAZAR ET AL.


1108

where iF and ijK are known matrices, ijX and are column vectors of elements of the vectors

.

ijY

, 1,,iC i nSubstituting these approximations into system (1),

leads to

1

2

1

2

1 2

1

01

1

01

1 1

1 2

( ) d

d

,

1, 2, , , 1, 2, , , 1, 2, , .

T

mT T Ti i i k

k

mx T T

ij ijj

mT T

i i kk

mxT T

i j i jj

m mT T T

i i k ij ijk j

C x F x X x

x K t Y t t

F x X x

x K t Y t t

F x X x x K P x

i n k m j m

1 1k k

(23)

where areTij 2 2M M TW x x

matrices. Multiplying , in to both sides of the n

1system (23) and applying

0. d ,x a linear or a non-

linear system, in terms of the entries of , will be obtained and the elements of vectors

can be obtained by solving this sys-tem.

, 1,2, ,iC i n

, 1,2, ,iC i n

3.2. Solving the System (2) Consider the system of integro-differential Equations (2) subject to the conditions (20). To solve this system, let’s considered the unknown functions as a linear combination of Chebyshev wavelets, by the following

, 1, 2, ,miu x i n

. , 1, 2, , m Ti iu x x C i n (24)

and we have

1

0

,!

1, , , 0, , 1

jm sm ss T Ti i

j

xu x x P C a

j

i n s m

is (25)

Therefore the following general expansions are achieved.

1 1 1

1 1

1 2

,

, , , , , , , ,

,

, , , , , , ,

, ,

1, 2, , , 1, 2, , , 1, 2, , .

Ti i

m mi k n n

Ti k

m m Tij n n ij

Ti j ij

f x x F

F x u x u x u x u x u x

x X

G u t u t u t u t x Y

k x t x K t

i n k m j m

Substitution of these approximations into the system (2), would be obtained results in

1

2

1

2

1 2

1

1

01

1

1

01

1 1

( ) d

d

,

1, 2,..., ,

mT T T

i i i kk

mT T

i j ijj

mT T

i i kk

mT T

i j i jj

m mT T T

i i kk j

x C x F x X

x K t t Y t

x F x X

x K t t t Y

ij ijx F x X x K

i n

DY

(26)

where is a D 1 12 2k kM M matrix. Multiplying both sides of the system (26) by nW x x and applying

0 1. d ,x the following

linear or non-linear system will be obtained

1 2

1 1

, 1,2,..., ,m m

i i i k i j ijk j

C F X K DY i n

(27)

Solving system (27) the entries of , will be obtained.

, 1,2, ,iC i n

Also one can check the accuracy of the method. Since the truncated Chebyshev wavelets series are approximate the solutions of the systems (1) and (2), so the error function ie u x is constructed as follows

12 1

1 0

.k M

ii i n m n m

n m

e u x u x c x

(28)

If we set jx x where 0,1jx , the error values can be obtained.

4. Numerical Results In this section some systems of integro-differential equa-tions are considered and solved by the introduced method. Parameters and k M are considered to be 1 and 6 respectively.

Example 1: Consider the following linear system of Fredholm integro-differential equations

1

0

2

1 2

0

2 3 d

33 8,

10

3 2 2 d

421 , 0 1.

5

u x v x x t u t v t t

x x

v x u x x t u t v t t

x x

(29)


1

Subject to initial conditions and .The exact

solutions are and [8].

0 1, 0 0, 0 1u u v 23u x x

0 2v v x 3 2 1x x

Let’s

1 2

1

2 2

2

1

2

1 0

2

2

2

2 0

21 2

21 2

, ,

,

2 ,

1

,

2 1

,

2 , 3 2

3 43 8 , 21 ,

10 5

T T

T T

T T T T T

T T

T T T

T T

T T T

T T

T T

u x x C v x x C

u x x P C

v x x P C x P C x V

u x x P C

x P C x U

v x x P C x

x P C x V

1

,x t x K t x t x K t

x x x F x x F

Substituting into (28), the following linear system will be obtained

1 2 1

2 2

1 1 0 2 0

1 2

2 2

2 1 0 2 0

3 3

2( 2 .

T

T T

T

T T

C P C V

K D P C U P C V F

P C C

1

2

,

K D P C U P C V F

.

Solving this system; the solutions would be achieved as follows

2 21 1 3 1,T Tu x x P C x

2 32 2 1 2 1T Tv x x P C x x x

This is the exact solution. Example 2: Consider the following non-linear system

of Fredholm integro-differential equations with the exact solutions and 2 1u x x e xv x and boundary conditions 0 1, 0 0,u u and 0 1v 0 1v .

1 2

0

1 3

0

2 3 3

4 2 d

2 134e 2e ,

15 61

3 d2

7 1 1 19e 3 e e e ,

6 6 9 180 1,

x

x

u x v x x t u t x t v t t

x x

v x u x x t u t v t x t v t t

x x x

x

(30)

In this example, let’s take

1

2

1

2 2

2 2

1 1

,

,

,

1 ,

1

T

T

T T

T T T T T

T T T T T

u x x C

v x x C

u x x P C

v x x P C x P C x V

u x x P C x P C x V

1

1

2

2

2

2 0

21 2

33

1 2

3

1

2 3 32

1

,

, ,

,

, 2 ,

,

2 134e 2e ,

15 67 1 1 19

e 3 e e e6 6 9 18

T T

T T T

T T

T

T T

T

x T

x T

v x x P C x

x P C x V

u x x Y u x v x x Y

v x x Y

x t x K t x t x K t

x t x K t

x x x F

.x x x x F

Applying the Chebyshev wavelets method, the fol-lowing non-linear system will be obtained

2

1 2 0 1 1 2 2 1

2

2 1 1 3 2 1 3

4 4

1 13 3

2 2

T T

T

C P C V K DY K D P C V F

C P C V K DY K DY F

1

2

,

.

Solving this system the following results would be achieved.

1,0 1,1

71,2 1,3

7 71, 4 1,5

2,0 2,1

2,2 2,3

2.506705578, 0.00009696248441,

0.00001788458767, 5.080260046 10 ,

1.449170655 10 , 2.373895017 10 ,

2.197451850, 0.7536322881,

0.09324678580, 0.0077327666

c c

c c

c c

c c

c c

2,4 2,5

17,

0.0004818914245, 0.00002403741177.c c

Therefore, the following approximate solutions will be achieved.

5 4

3 2

8

0.000001345778125 0.00001026370827

0.00001182646441 0.9999858747

1.336715265 10 1,

u x

x

x x

x x

5 4

3 2

0.01391739110 0.03477166926

0.1704167644 0.4990585121 1.

x x

x x

v

x

x

Plots of the exact and approximate solutions are pre-



sented in Figure 1 and also plots of error functions are shown in Figure 2.

Example 3: consider non-linear system of Volterra integro-differential equations with conditions 0 0, 0 1, 0 0,u u u . 0 1, 0 0v v and

0 1.v

2 2

0

2

0

d ,

1sin sin d , 0 1.

2

x

x

u x x u x u t v t t

v x x x u t v t t x

(31)

The exact solutions are and [7].

sinu x x

C cosv x xEntries of vectors 1C and 2 are computed by

solving a system of nonlinear equations with six un-knowns and six equations as follows

1,0 1,1

1,2 1,3

1,4 1,5

2,0 2,1

2, 2 2,3

2,

1.032210259 .2058700709

0.04760379553 0.002178545560

0.0002500199685 0.000006823287551

.5638994174 .3768424359

0.0260060615

, ,

, ,

, ,

7 0.00398782176

, 0 ,

, ,4

0

c c

c c

c c

c c

c c

c

4 2,50.0001365887993 0.0000141989 5, .04 9c

Therefore, one gets the following approximate solutions

3

3 2

7

1

5 40.007222202515 0.001672032401

0.1676491115 0.0002483913114

0.3210839427 10 ,

T

x x

x x

u x C P x x

x

2

3

4

3

2

5

2

0.003945506187 0.04597489600

10.002189489246

20.00004265668610

12

1.

T

x x

x x

xx C

x

v P x

Approximate and exact solutions are plotted in Figure 3. Error functions are plotted in Figure 4.

Example 4: Consider the following non-linear system of Volterra integro-differential equations with the exact solutions , 2u x x v x x and on 23w x x0 1x [5].

3 2 2

0

2

0

3 2 2

2 2 3

0

2 2 d

3 d

,

42 2

3

d ,

x

x

x

u x x x v x v t u t w t t

v x x xu t x t v t u t w t t

w x x u t u x

x v t u t t w t t

,

(32)

(a)

(b)

Figure 1. (a) and (b) plots of exact and approximate solu-tions of Example 2.

Figure 2. Plots of error functions of Examples 2.



(a)

(b)

Figure 3. (a) and (b) plots of exact and approximate solu-tions of Example 3.

Figure 4. Plots of error functions of Examples 3.

(a)

(b)

(c)

Figure 5. (a), (b) and (c) plots of exact and approximate solutions of Example 4.



Figure 6. Plots of error functions of Examples 4. Initial conditions are 0 0, 0 0u u ,

0 0 0w

and . 0 0, 0 1v v , w 0The following approximate solutions would be achie-

ved by using Chebyshev wavelets method.

11 5 10 4

11 3

2

2

5

1

3

1

8.527181902 10 2.509321144 10

4.653979568 10 3.641136902 10

6.551942965 0 ,1

Tu x

x

x

P

x

x

C x

1 x

11 5 10 4

1

22

0 3 10 2

14

7.880203686 10 1.637978948 10

1.014393404 10 3.511874249 10

1.461792049 10 ,

T

x x

x x

v x C P x x

x

14

10 5 9 4

10 3

2

2

1

3

2

4.043702026 10 1.228923763 10

7.249989732 10 3.000000001

1.753892960 10 2.955310652 0 .1

Tw x C P

x x

x x

x

x

Plots of the exact solution, approximate solution and error functions are shown in Figures 5 and 6. 5. Conclusions This paper proposes a powerful technique for solving systems of integro-differential equations using Cheby-shev wavelets method. Comparison of the approximate solutions and the exact solutions shows that the proposed method is more efficient tool and more practical for solving linear and non-linear systems of integro-difierential equations, and plots confirm. Researches for finding

more applications of this method and other orthogonal basis functions are one of the goals in our research group. The package Maple 13 has been used to carry the com-putations associated with these examples. 6. Acknowledgements Authors are grateful to the anonymous reviewers for his (her) influential comments which have improved the quality of the paper. 7. References [1] E. Babolian and F. Fattahzadeh, “Numerical Computation

Method in Solving Integral Equations by Using Cheby-shev Wavelet Operational Matrix of Integration,” Applied Mathematics and Computations, Vol. 188, No. 1, 2007, pp. 1016-1022. doi:10.1016/j.amc.2006.10.073

[2] E. Babolian and F. Fattahzadeh, “Numerical Solution of Differential Equations by Using Chebyshev Wavelet Op-erational Matrix of Integration,” Applied Mathematics and Computations, Vol. 188, No. 1, 2007, pp. 417-426.

[3] Y. Li, “Solving a Nonlinear Fractional Differential Equa-tion Using Chebyshev Wavelets,” Communications in Nonlinear Science and Numerical Simulation, Vol. 15, No. 9, 2010, pp. 2284-2292. doi:10.1016/j.cnsns.2009.09.020

[4] J. Biazar, “Solution of Systems of Integral-Differential Equations by Adomian Decomposition Method,” Applied Mathematics and Computation, Vol. 168, No. 2, 2005, pp. 1232-1238. doi:10.1016/j.amc.2004.10.015

[5] J. Biazar, H. Ghazvini and M. Eslami, “He’s Homotopy Perturbation Method for Systems of Integro-Differential Equations,” Chaos, Solitions and Fractals, Vol. 39, No. 3, 2009, pp. 1253-1258. doi:10.1016/j.chaos.2007.06.001

[6] E. Yusufoglu (Agadjanov), “An Efficient Algorithm for Solving Integro-Differential Equations System,” Applied Mathematics and Computation, Vol. 192, No. 1, 2007, pp. 51-55. doi:10.1016/j.amc.2007.02.134

[7] J. Biazar and H. Aminikhah, “A New Technique for Solving Integro-Differential Equations,” Computers and Mathematics with Applications, Vol. 58, No. 11-12, 2009, pp. 2084-2090. doi:10.1016/j.camwa.2009.03.042

[8] J. Pour-Mahmoud and M. Y. Rahimi-Ardabili and S. Sha- morad, “Numerical Solution of the System of Fredholm Integro-Differential Equations by the Tau Method,” Ap-plied Mathematics and Computation, Vol. 168, 2005, pp. 465-478.

[9] S. Abbasbandy and A. Taati, “Numerical Solution of the System of Nonlinear Volterra Integro-Differential Equa-tions with Nonlinear Differential Part by the Operational Tau Method and Error Estimation,” Journal of Computa-tional and Applied Mathematics, Vol. 231, No. 1, 2009, pp. 106-113. doi:10.1016/j.cam.2009.02.014

[10] A. Arikoglu and I. Ozkol, “Solutions of Integral and In-tegro-Differential Equation Systems by Using Differen-tial Transform Method,” Computers and Mathematics


http://dx.doi.org/10.1016/j.amc.2006.10.073

http://dx.doi.org/10.1016/j.cnsns.2009.09.020


http://dx.doi.org/10.1016/j.chaos.2007.06.001


http://dx.doi.org/10.1016/j.camwa.2009.03.042

http://dx.doi.org/10.1016/j.cam.2009.02.014

J. BIAZAR ET AL.


1113

with Applications, Vol. 56, No. 9, 2008, pp. 2411-2417. doi:10.1016/j.camwa.2008.05.017

[11] M. Gachpazan, “Numerical Scheme to Solve Integro- Difierential Equations System,” Journal of Advanced Re-search in Scientific Computing, Vol. 1, No. 1, 2009, pp. 11-21.

[12] K. Maleknejad, F. Mirzaee and S. Abbasbandy, “Solving Linear Integro-Differential Equations System by Using Rationalized Haar Functions Method,” Applied Mathe-matics and Computation, Vol. 155, No. 2, 2004, pp. 317- 328. doi:10.1016/S0096-3003(03)00778-1

[13] K. Maleknejad and M. Tavassoli Kajani, “Solving Linear Integro-Differential Equation System by Galerkin Meth-ods with Hybrid Functions,” Applied Mathematics and Computation, Vol. 159, No. 3, 2004, pp. 603-612. doi:10.1016/j.amc.2003.10.046

[14] I. Daubeches, “Ten Lectures on Wavelets,” SIAM, Philadelphia, 1992.

[15] Ole Christensen and K. L. Christensen, “Approximation Theory: From Taylor Polynomial to Wavelets,” Birk-hauser, Boston, 2004.

http://dx.doi.org/10.1016/S0096-3003(03)00778-1


A Strong Method for Solving Systems of Integro-Differential … · 2013. 12. 24. · E-mail:...

Documents

Transcript of A Strong Method for Solving Systems of Integro-Differential … · 2013. 12. 24. · E-mail:...