Two-parameter Homotopy Analysis Method For Solving Nonlinear Boundary Value Problems

Many problems in science and engineering can be reduced to the problems of solving non-linear boundary value problems. Homotopy analysis method(namely,HAM) is a popular analy-sis method for solving linear and nonlinear boundary value problems. HAM was proposed by Liao in 1992, since then, it has been improved by many scientists. It can be concluded that there are mainly two optimization ways. The first is converting q or c0 in the zero-order deformation equation to general expressions, which provides us more freedom for solving nonlinear prob-lems. The other is optimizing convergence-control parameter c, which makes series solution more accurate.Another frequently used method for solving nonlinear differential equations is the Adomian decomposition method(namely,ADM) proposed by Georgie Adomian. Duan modified the tradi-tional ADM by add an constant c called Duan's convergence parameter in 2011. Experiments show that the modified ADM can give more accurate series solutions to the given problems by choosing proper c.A modified HAM with an additional parameter p is proposed for solving nonlinear bound-ary value problems. The two-parameter method reduces to the traditional one-parameter HAM when p= 0. It is demonstrated by examples that solutions given by the two-parameter method are more accurate than solutions given by the one-parameter HAM. It is further demonstrated that solutions given by the ADM and the Duan-Rach modified ADM, which are usually not opti-mal, can be recovered from the solutions given by the two-parameter method and more accurate solutions are obtained by choosing proper values of c0 and p.This paper is organized as follows. In chapter 1, the thesis elaborated the background and significance of the research, the HAM and part of optimized HAM are introduced. In chapter 2, the ADM and Duan-Rach modified ADM are described. In chapter 3, the two-parameter ?AM is developed. In chapter 4, some given nonlinear boundary problems are solved with this new method. In the last section, some concluding remarks are given.