An application of the differential transformation method to the boundary value problems of the system of integro-differential equations

Selçuk J. Appl. Math. Selçuk Journal of

Vol. 6. No. 1. pp. 43-53, 2005 Applied Mathematics

An Application of the Di¤erential Transformation Method to the Boundary Value Problems of the System of Integro-Di¤erential Equa-tions

Ozan Özkan1 and Y¬ld¬ray Keskin2

1_{Department of Mathematics, Art and Science Faculty of Selçuk University, 42031}

Konya, Turkey;

e-mail:o ozkan@ selcuk.edu.tr

2_{Research Center of Applied Mathematics, Selçuk University,42031 Konya, Turkey;}

e-mail:ykeskin@ selcuk.edu.tr

Received: March 25, 2005

Summary. The di¤erential transformation is one of the numerical methods for solving di¤erential equations. An advantage of this method is transforming the di¤erential equations into algebraic ones. This article is presents an e¢ cient nu-merical method for the solution of the systems of integro-di¤erential equations with two point boundary conditions. The di¤erential transformation method is applied to construct the numerical scheme for the problem. The accuracy of the method is shown by the numerical examples. The results we obtained by using this technique is much more accurate than the existing ones..

Key words: Di¤erential Transformation, Integro-Di¤erential Equations, Nu-merical methods.

1. Introduction

The majority of the problem in physics and engineering fall naturally into many integro-di¤erential equations. Numerical techniques developed for the solution of integro-di¤erential equations consist of Wawelet-Galerkin method [6], Taylor polynomial methods [7] and the Adomian decomposition method[8]. Now we discuss an e¢ cient numerical scheme called the di¤erential transformation (DT) in order to solve the system of integro-di¤erential equations with two points boundary values.

The concept of the DT technique was …rst introduced by Zhou [1] for electrical circuits. This method, called the di¤erential transformation method, serves for

constructing an analytical solution in the form of a polynomial. Di¤erent aspects of DT method have been analyzed and developed in many researchers [1-5, 11, 12, 14]. Initial value problems were solved in [2,14], the DT approximation for the system of ordinary di¤erential equations was investigated in [11,14], solution of boundary value problems for integro-di¤erential equations by using DT were studied in [12] and etc. DT method to …nd a numeric solution of di¤erential equation in one- dimensional grid points were considered in [5] and using the DT and central …nite di¤erence methods simultaneously were studied in [9]. In this work more general approach to the notion of di¤erential transformation method are considered and using the DT for the numerical solution of integro-di¤erential equations on grid points is studied. The usefulness of this method is illustrated comparatively by some numerical examples.

2. Preliminaries

In this section we consider some notions and descriptions corresponding to the di¤erential transformation method.

Let I = (a; b) _{R be an arbitrary real interval and y : I ! R is an} analytic function on I.

Using the de…nition of analytic function [13], for an arbitrary x0 2 I we can

write the power series representation of the function y(x) as,

y(x) =

1

X

k=0

Ck(x x0)k

This series converge to the function y(x) in some neighborhood of the point x0.

Referring on properties of power series and analytic function we can a¢ rm that the function y(x) is in…nitely continuously di¤erentiable and

Ck= y(k)_(x 0) k! so y(x) = 1 X k=0 y(k)_(x 0) k! (x x0) k_:

The di¤erential transform of the function y(x) is de…ned as follows [1-5]:

(1) Y(k) = 1 k! dk_y(x) dxk x=0

where y(x)is the original function and Y (k)is the transformed function. The di¤erential inverse transform of Y (k) is de…ned as

(2) y(x) =

1

X

k=0

Y (k) xk

Combining equations (1) and (2) we obtain

(3) y(x) = 1 X k=0 1 k! dk_y(x) dxk x=0 xk

From these de…nitions it is easy to see that the concept of di¤erential transform is derived from Taylor series expansion. Similarly as in works [1-5] we can consider the weight DTs of analytic functions and get the same table of correspondence of the functions and their spectrum (see Table 1).

Original function Transformed function

y(x) = u1(x) u2(x) Y (k) = U1(k) U2(k)

y(x) = u(x) =const Y (k) = U (k)

y(x) = du(x) dx Y (k) = (k + 1)U k + 1) y(x) = d m_u(x) dxm Y (k) = (k + n)! k! U (k + n) y(x) = u1(x)u2(x) Y (k) = U1(k) U2(k) = k P l=0 U1(l)U2(k l) y(x) = 1 Y (k) = (k); (k) = 1 if k = 0 0 _{if k 6= 0} y(x) = xn _{Y (k) = (k n); (k n) =} 1 if k = n 0 _{if k 6= n} y(x) = x R x0 u(t)dt Y (k) = U (k 1) k ; k> 1 y(x) = x R x0 u1(t)u2(t)dt Y (k) =_k1 k_P1 k1=0 U1(k)U2(k k1 1) ! ; k> 1

3. Applying the di¤erential transformation to the Integro-di¤erential equations

Consider a boundary value problem for the following systems of the integro-di¤erential equations: (4) 8 > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > : df1(t) dt = h1(t; f1(t); :::; fn(t); t R 0 k1(t; s; f₁(s); :::; f_n(s))ds) f1(0) = 1; f1(a) = 1 df2(t) dt = h2(t; f1(t); :::; fn(t); t R 0 k2(t; s; f₁(s); :::; f_n(s))ds); f2(0) = ₂; f2(a) = 2 .. . dfn(t) dt = hn(t; f1(t); :::; fn(t); t R 0 kn(t; s; f₁(s); :::; f_n(s))ds); f_n(0) = _n; f_n(a) = _n

where each equation in system (4) represents the …rst derivative of one of the unknown functions as a mapping involving the independent variable t and n unknown functions f1; f2; :::; fn which have appeared partly in the integral sign.

For economy of writing let us consider this systems as the following

(5) dfi(t) dt = hi(t; f1(t); :::; fn(t); t Z 0 ki(t; s; f1(s); :::; fn(s))ds);

with the initial conditions

(6) fi(0) = i; i = 1; :::; n

and the boundary conditions

(7) fi(a) = i; i = 1; :::; n

Taking the di¤erential transformation of (4), we get

(k + 1)Fi(t0; k + 1) = Hi(k; F1(t0; k); :::; Fn(t0; k); 1

with the initial condition Fi(0) = i; i = 1; :::; n

where Fi( ); Hi( ) and Ki( ) denote the transformed function of

fi(t); hi(t; f1(t); :::; fn(t); t

Z

0

ki(t; s; f1(s); :::; fn(s))ds); ki(t; s; f1(s); :::; fn(s)):

Now, assume that we investigate the solution of (4) in the interval [0; a] and it is divided into N equally spaced subintervals with the grid points f0 = t0; t1; :::; tN = ag

such that ti= t0+ ih ; i = 0; 1; 2; :::; N 1 and h =

a N.

Therefore, the approximation of the function fi(i = 1; 2; :::; n) in each

subinter-val is denoted by fi;j; j = 0; 1; :::; N 1.From the initial condition (6), we can

write

(8) Fi(0) = i; Fi(1) = i

where the constants Fi(1) = i will be determined later by using the boundary

conditions at t = a.The approximation function for fi(t) in the …rst subinterval

can be denoted by fi;0(t) and after applying the di¤erential transform procedure

it can be obtained in terms of n th order Taylor polynomial at the point t0= 0

as

(9) fi;0(t) = Fi;0(0) + Fi;0(1)t + Fi;0(1)t2+ :::: + Fi;0(n)tn

Therefore the approximate value of the function fi(t) at the point t1 can be

obtained by (9), that is,

fi(t1) = fi;0(t1) = Fi;0(0) + Fi;0(1)t1+ Fi;0(2)t21+ :::: + Fi;0(n)tn1

= Fi;0(0) + Fi;0(1)h + Fi;0(2)h2+ :::: + Fi;0(n)hn

=

n

P

p=0

Fi;0(p)hp

Since the initial value of the second sub-domain is equal to the value of fi;0(t)

at the point t1 we can write

Therefore, applying the proposed transform method for the second subinterval gives

fi;1(t) = Fi;1(0) + Fi;1(1)(t t0) + Fi;1(2)(t t0)2+ :::: + Fi;1(n)(t t0)n

Again, using the equality fi;1(t2) = fi;2(t2) = Fi;2(0) determines the initial

value of the third interval, i.e.

Fi;2(0) = fi;1(t2) = n

X

p=0

Fi;1"(p)hp

After following the same procedure, we can …nd the solution of fi at the grid

point tj+1as

fi(tj+1) = fi;j(ti+1) = Fi;j(0) + Fi;j(1)(tj+1 tj) +

(10)

Fi;j(2)(tj+1 tj)2+ :::: + Fi;j(n)(tj+1 tj)n

= Fi;j(0) + Fi;j(2)h + Fi;j(2)h2+ :::: + Fi;j(n)hn

=

n

X

p=0

Fi;j(p)hp

Therefore the approximate values of the function fi at any grid point can be

obtained. But, the series solution fi(t) follows immediately with the constants i as yet undetermined. For this reason, we substitute the boundary conditions

(7) into equation (10) at t = a. This leads n equations in n unknowns i’s. Then

solving this system of equations enables us to specify i. Having determined i, the n-term approximant for the solution of the integral-di¤erential equations

follows immediately as fi(t) = N_X1 j=0 fi;j(t) [ti;ti+1](t); t 2 [0; a] where [xi; xi+1](t) = 1 _{t 2 [t}i ; ti+1] ; 0 t =_{2 [t}i+1 ; ti] ; [0; a] = N_[1 i=0 [xi; xi+1]:

Example 1. Consider the following linear system of integro- di¤erential equa-tion with two-point boundary condiequa-tions discussed in [10]

(11) y0 1= 1 + x + x2 y2 x R 0 (y₁(t) + y₂(t))dt y0 2= 1 x y1 x R 0 (y₁(t) y₂(t))dt 9 > > = > > ; with the initial conditions

(12) y1(0) = 1; y2(0) = 1

and the boundary conditions

(13) y1(1) = 1 + e; y2(1) = 1 e

The analytical solution of this problem is known

y1(x) = x + ex; y2(x) = x ex

Now, if we apply the di¤erential transformation to (11), we derive

(14) (k + 1)Y1(k + 1) = (k) + (k 1) + (k 2) Y2(k) 1_k(Y1(k 1) + Y2(k 1)) (k + 1)Y2(k + 1) = (k) (k 1) Y1(k) 1_k(Y1(k 1) + Y2(k 1)) 9 > > > > = > > > > ; From the initial conditions (12) , we obtainc

(15) Y1(0) = 2; Y2(0) = 0

Then combining equations (14) and (15) gives the approximation of y1(x), y2(x)

obtained in a series form with the constants A and B as yet undetermined, where the constants de…ned as

Y1(1) = A; Y2(2) = B

The approximated values of the solutions are given in Table 2. All calculations presented in this paper were carried out in the Mapple environment

x_j A n a l y t i c s o l u t i o n * [ D i ¤ e r e n t i a l T r a n s . o f o r d e r 4 ] * [ D i ¤ e r e n t i a l T r a n s . o f o r d e r 7 ] y₁( x_j) y₂( x_j) y₁( x_j) y₂( x_j) y₁( x_j) y₂( x_j) 0 . 0 1 . 0 0 0 0 0 0 - 1 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 1 1 . 2 0 5 1 7 0 - 1 . 0 0 5 1 7 0 1 . 8 E - 8 1 . 4 E - 7 5 . 5 E - 1 4 4 . 2 E - 1 3 0 . 2 1 . 4 2 1 4 0 2 - 1 . 0 2 1 4 0 2 4 . 2 E - 8 2 . 6 E - 7 1 . 2 E - 1 3 7 . 7 E - 1 3 0 . 3 1 . 6 4 9 8 5 8 - 1 . 0 4 9 8 5 8 7 . 0 E - 8 3 . 5 E - 7 2 . 0 E - 1 3 1 . 0 E - 1 2 0 . 4 1 . 8 9 1 2 4 6 - 1 . 0 9 1 8 2 4 9 . 7 E - 8 4 . 1 E - 7 2 . 8 E - 1 3 1 . 2 E - 1 2 0 . 5 2 . 1 4 8 7 2 1 - 1 . 1 4 8 7 2 1 1 . 2 E - 7 4 . 3 E - 7 3 . 5 E - 1 3 1 . 2 E - 1 2 0 . 6 2 . 4 2 2 1 1 8 - 1 . 2 2 2 1 1 8 1 . 3 E - 7 4 . 3 E - 7 4 . 0 E - 1 3 1 . 2 E - 1 2 0 . 7 2 . 7 1 3 7 5 2 - 1 . 3 1 3 7 5 2 1 . 3 E - 7 3 . 8 E - 7 4 . 0 E - 1 3 1 . 1 E - 1 2 0 . 8 3 . 0 2 5 5 4 0 - 1 . 4 2 5 5 4 0 1 . 2 E - 7 3 . 0 E - 7 3 . 5 E - 1 3 9 . 0 E - 1 3 0 . 9 3 . 3 5 9 6 0 3 - 1 . 5 5 9 6 0 3 7 , 6 E - 8 1 . 7 E - 7 2 . 3 E - 1 3 5 . 3 E - 1 3 1 . 0 3 . 7 1 8 2 8 1 - 1 . 7 1 8 2 8 1 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0

T able 2 N umerical results f or Example 1 Error = analytical solution numerical solution

Example 2. Consider the following linear system of integro- di¤erential equa-tion with two-point boundary condiequa-tions

(16) y0 1= 2y1 y2 x R 0 y3(t)dt y0 2= y1 x R 0 y3(t)dt y₃0= 1 y₂ x R 0 y1(t)dt 9 > > > > > > = > > > > > > ;

with the initial conditions

(17) y1(0) = 1; y2(0) = 1; y3(0) = 2

and the boundary conditions

(18) y1(1) = e; y2(1) =

1

e; y3(1) = e + 1 e

y1(x) = ex; y2(x) = e x; y3(x) = ex+ e x

Now, if we apply the di¤erential transformation to (16), we write

(19) (k + 1)Y1(k + 1)= 2Y1(k) Y2(k) 1_kY3(k 1) (k + 1)Y2(k + 1)= Y1(k)+1kY3(k 1) (k + 1)Y3(k + 1)= (k) Y2(k) 1kY1(k 1) 9 > = > ;

From the initial conditions (17), we obtain

(20) Y1(0) = 1; Y2(0) = 1; Y3(0) = 2

Then combining equations (19) and (20) gives the approximation of y1(x), y2(x),

y3(x) in a series form with the constants A; B and C as yet undetermined, where

the constants de…ned as

Y1(1) = A; Y2(1) = B; Y3(1) = C

The approximated values of the solution are given in Table 3.

x_j A n a l y t i c s o l u t i o n * [ D i ¤ e r e n t i a l T r a n s . o f o r d e r 4 ] * [ D i ¤ e r e n t i a l T r a n s . o f o r d e r 7 ] y₁( x_j) y₂( x_j) y₃( x_j) y₁( x_j) y₂( x_j) y₃( x_j) y₁( x_j) y₂( x_j) y₃( x_j) 0 . 0 1 . 0 0 0 0 0 0 1 . 0 0 0 0 0 0 2 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 1 1 . 1 0 5 1 7 0 0 . 9 0 4 8 3 7 2 . 0 1 0 0 0 8 5 . 5 E - 1 0 9 . 4 E - 8 1 . 4 E - 7 8 . 1 E - 1 4 3 . 0 E - 1 4 2 . 0 E - 1 3 0 . 2 1 . 2 2 1 4 0 2 0 . 8 1 8 7 3 0 2 . 0 4 0 1 3 3 2 . 8 E - 9 1 . 6 E - 7 2 . 5 E - 7 1 . 5 E - 1 3 4 . 1 E - 1 4 3 . 9 E - 1 3 0 . 3 1 . 3 4 9 8 5 8 0 . 7 4 0 8 1 8 2 . 0 9 0 6 7 7 6 . 9 E - 9 2 . 1 E - 7 3 . 3 E - 7 2 . 1 E - 1 3 3 . 9 E - 1 4 5 . 4 E - 1 3 0 . 4 1 . 4 9 1 8 2 4 0 . 6 7 0 3 2 0 2 . 1 6 2 1 4 4 1 . 2 E - 8 2 . 4 E - 7 3 . 7 E - 7 2 . 5 E - 1 3 2 . 7 E - 1 4 6 . 6 E - 1 3 0 . 5 1 . 6 4 8 7 2 1 0 . 6 0 6 5 3 0 2 . 2 5 5 2 5 1 1 . 9 E - 8 2 . 5 E - 7 3 . 9 E - 7 2 . 8 E - 1 3 9 . 8 E - 1 5 7 . 2 E - 1 3 0 . 6 1 . 8 2 2 1 1 8 0 . 5 4 8 8 1 1 2 . 3 7 0 9 3 0 2 . 5 E - 8 2 . 4 E - 7 3 . 8 E - 7 2 . 8 E - 1 3 9 . 1 E - 1 5 7 . 3 E - 1 3 0 . 7 2 . 0 1 3 7 5 2 0 . 4 9 6 5 8 5 2 . 5 1 0 3 3 8 3 . 0 E - 8 2 . 1 E - 7 3 . 3 E - 7 2 . 5 E - 1 3 2 . 5 E - 1 4 6 . 7 E - 1 3 0 . 8 2 . 2 2 5 5 4 0 0 . 4 9 6 5 8 5 2 . 6 7 4 8 6 9 3 . 0 E - 8 1 . 7 E - 7 2 . 5 E - 7 2 . 0 E - 1 3 3 . 2 E - 1 4 5 . 4 E - 1 3 0 . 9 2 . 4 5 9 6 0 3 0 . 4 0 6 5 6 9 2 . 8 6 6 1 7 2 2 . 1 E - 8 9 . 8 E - 8 1 . 4 E - 7 1 . 1 E - 1 3 2 . 6 E - 1 4 3 . 2 E - 1 3 1 . 0 2 . 7 1 8 2 8 1 0 . 3 6 7 3 7 9 3 . 0 8 6 1 6 1 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0

T able 3 N umerical results f or Example 2 Error = analytical solution numerical solution

4. Conclusion

The computations associated with the two examples discussed above were per-formed by using Maple 9 computer programme. Comparing the obtained results with the analytic solution, show that the di¤erential transformation technique were clearly reliable than the grid points techniques where the solution is de…ned at grid points only. Moreover, this approach we used would require considerably less computational e¤ort. It is also fortunate that the approach is implemented directly in a straightforward manner without using restrictive assumptions and requiring di¢ cult computations. E¤ective applications of the proposed method are easy to handle for such a type of integro-di¤erential equations.

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