Two Functional Expansion Methods And Homotopy Perturbation Method And Their Applications To The Differential Equations

In this thesis, based on two new general ansatzes, the extended (?)-expansion method and compound (?)-expansion method is proposed to seek exact solutions of nonlinear evo-lution equations. Also we propose the variational homotopy perturbation iteration method to solve the inverse problem in mathematical physics.In chapter 2, we use the extended (?)-expansion method to construct travelling wave solutions of Whitham-Broer-Kaup-Like equations and coupled Hirota-Satsuma KdV equa-tions. By using this method, new exact solutions involving parameters, expressed by three types of functions which are the hyperbolic functions, the trigonometric functions and the rational functions, are obtained. When the parameters are taken as special values, some solitary wave solutions are derived from the hyperbolic function solutions.In chapter 3, we use the compound (?)-expansion method to construct non-travelling wave solutions of (2+1)-dimensional Painleve integrable Burgers equations and the (2+1)-dimensional breaking soliton equations with the aid of symbolic computation. As a result, hyperbolic function solutions, trigonometric function solutions, rational solutions and a series of complexiton solutions with parameters of the two equations are obtained. When the parameters are taken as special values, some double solitary wave solutions are derived from hyperbolic function solutions.In chapter 4, by combining the variational iteration method and homotopy perturba-tion method, a new method named variational homotopy perturbation iteration method is proposed for solving an inverse problem of determining an unknown parameter in a linear parabolic equation. Using this method, a rapid convergent sequence tending to the exact solution of the inverse problem can be obtained.