Iterative Methods For Solving Nonlinear Pantograph Equations

This thesis presents a comparative study between two powerful methods that weredeveloped recently. The first method is the diferential transform method (DTM), whichdeveloped by the Chinese scientist Zhou in1986and the second one, is the Laplace de-composition algorithm (LDA), which introduced by the Arabian researcher Khuri in2001.Both methods are used heavily in the literature. The efectiveness of these two method-s when used for certain categories of nonlinear pantograph diferential equation is alsoevaluated in this thesis.This thesis is divided into five chapters. The first Chapter is devoted to clarify thefundamental properties of the diferential transform method (DTM) and it introduces thestandard Laplace decomposition algorithm (SLDA). The second chapter introduces a newform of SLDA called modified Laplace decomposition algorithm (MLDA), which ledto an increase in the speed of convergence. In this chapter, SLDA, MLDA and DTM areadapted and applied to solve Functional diferential equations with proportional delays. Inthe third chapter, the solution of the higher-order nonlinear pantograph integro-diferentialequation is presented. The fourth chapter is devoted to apply SLDA, MLDA and DTM tosolve neutral functional–diferential equations. Finally, a relation between the variationaliteration method (VIM) and the modified Laplace decomposition algorithm is presented.The application of SLDA, MLDA and DTM for solving partial diferential equation withproportional delays is presented in the fifth Chapter. In addition, a brief introduction aboutthe variational iteration method (VIM) is given in the appendix.The conclusion that is pointed out in this study is that the diferential transformmethod is easier to use, time and efort saving more than Laplace decomposition algorith-m. However, the rate of convergence of Laplace decomposition algorithm is higher thanthat of the diferential transform method.