New Iterative Method Based on Laplace Decomposition Algorithm

Since 2001, Laplace decomposition algorithm (LDA) has been one of the reliable mathematical methods for obtaining exact or numerical approximation solutions for a wide range of nonlinear problems. The Laplace decomposition algorithm was developed by Khuri in [2] to solve a class of nonlinear differential equations. The basic idea in Laplace decomposition algorithm, which is a combined form of the Laplace transform method with the Adomian decomposition method, was developed to solve nonlinear problems. The disadvantage of the Laplace decomposition algorithm is that the solution procedure for calculation of Adomian polynomials is complex and difficult and takes a lot of computational time for higher-order approximations as pointed out by many researchers [3–5]. The Laplace decomposition algorithm plays an important role in modern scientific research for solving various kinds of nonlinear models; for example, Laplace decomposition algorithm was used in [6] to solve a model for HIV infection of CD4+T cells; LDA was employed in [7] to solve Abel’s second kind singular integral equations. In [8] it was used to solve boundary Layer equation. Even though there has been some developments in the LDA [8–11], the use of Adomian polynomials has not been abandoned. The main purpose of this paper is to introduce a new iterative method based on Laplace decomposition algorithm procedure without the need to compute Adomian polynomials and thus reduce the size of calculations needed. The scheme is tested for some classes of pantograph equations, and the results demonstrate reliability and efficiency of the proposed method.