Solution of boundary value problems for integro-differential equations by using differential transform method

doi:10.1016/j.amc.2004.10.009

Applied Mathematics and Computation

Volume 168, Issue 2, 15 September 2005, Pages 1145-1158

https://doi.org/10.1016/j.amc.2004.10.009 Get rights and content

Abstract

In this study, we extended the differential transform method (DTM) to solve the integro-differential equations. New theorems for the transformation of integrals are introduced and prooved. The commonly encountered linear and nonlinear integro-differential equations that appear in literature are solved as an illustration for the efficiency of the method. Some numerical results are also given to demonstrate the superiority of the method to other common techniques. The results we obtained by using this technique is much more accurate compared to the existing ones.

Introduction

Nonlinear integro-differential equations are usually hard to solve analytically and exact solutions are scarce. Therefore, they have been of great interest by several authors. In literature, numerical techniques such as Wavelet–Galerkin method (WGM) [1], Lagrange interpolation method [2] and Tau method [3] and semi analytical–numerical techniques such as Adomian’s decomposition method [4], Taylor polynomials [5] and rationalized Haar functions method [6] do exist. However, none of them propose a methodical way to solve these equations. Moreover, previous studies require more effort to achieve the results, they are not accurate and usually they are developed for special types of integro-differential equations.

The technique that we used is the differential transform method (DTM), which is based on Taylor series expansion. It is introduced by Zhou [7] in a study about electrical circuits. It gives exact values of the nth derivative of an analytical function at a point in terms of known and unknown boundary conditions in a fast manner.

In this study, DTM is applied to integro-differential equations and new theorems are introduced. We gave the theorems in general forms to be able to apply to any kind of integro-differential equation in any order. We also gave three numerical examples with comparison to the literature to show how accurate and fast the results can be obtained. In Example 1, Example 2, we obtained closed form exact series solutions and in Examples 3, we gave a high order series solution with error analysis.

Section snippets

Differential transform method

The transformation of the nth derivative of a function in one variable is as follows: $F (k) = \frac{1}{k!} {[\frac{d^{k} f}{d x^{k}} (x)]}_{x = x_{0}}$ and the inverse transformation is defined as $f (x) = \sum_{k = 0}^{\infty} F (k) (x - x_{0})^{k} .$ The following theorems that can be deduced from Eqs. (1), (2) are given below:

Theorem 1

If f(x) = g(x) ± h(x), then F(k) = G(k) ± H(k).

Theorem 2

If f(x) = cg(x), then F(k) = cG(k), where c is a constant.

Theorem 3

If $f (x) = \frac{d^{n} g (x)}{d x^{n}}$ , then $F (k) = \frac{(k + n)!}{k!} G (k + n)$ .

Theorem 4

If f(x) = g(x)h(x), then $F (k) = \sum_{k_{1} = 0}^{k} G (k_{1}) H (k - k_{1})$ .

Theorem 5

If f(x) = xⁿ, then F(k) = δ(k−n) where, $δ (k - n) = \{\begin{matrix} 1 & k = n \\ 0 & k \neq n . \end{matrix}$

Theorem 6

If f(x) = g₁(x)g

Numerical results

Example 1

We first consider the following linear integro-differential equation, which is also solved by ADM in the study of Wazwaz [8]. $y^{(iv)} (x) = x (1 + e^{x}) + 3 e^{x} + y (x) - \int_{0}^{x} y (t) d t$ with the B.C.’s $\begin{matrix} y (0) = 1, y^{'} (0) = 1, \\ y (1) = 1 + e, y^{'} (1) = 2 e, \end{matrix}$ and at x = 0 Eq. (3) gives out another B.C. as $y^{(iv)} (0) = 4 .$ Knowing that the differential transform of e^x is 1/k! and applying Theorem 3, Theorem 5, Theorem 6, Theorem 7 to Eq. (3), the following recurrence relation is obtained $Y (k + 4) = \frac{δ (k - 1) + \sum_{k_{1} = 0}^{k} δ (k_{1} - 1) \frac{1}{(k - k_{1})!} + \frac{3}{k!} + Y (k) - \frac{1}{k} Y (k - 1)}{(k + 1) (k + 2) (k + 3) (k}$

Conclusion

In this study, we introduced new theorems for DTM to solve the integro-differential equations. We first gave their proofs and then applied to linear and nonlinear integro-differential equations. It is shown that DTM is a very fast convergent, precise and cost efficient tool for solving integro-differential equations in the bounded domains.

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