The Wavelet Homotopy Analysis Method For Nonlinear Boundary Value Problems And Its Applications

Nonlinear phenomenons widely arise in science,engineering and even social life,and lots of them are described by nonlinear differential equations.It is well-known that nonlinear problems are much more difficult to solve than linear ones.Since the homotopy analysis method(HAM)was proposed by Liao in 1992,this powerful and easy-touse analytical tool has been widely applied to solve many highly nonlinear problems in different fields.Based on the homotopy analysis method and modern wavelet theories,a new method for nonlinear boundary value problems,namely,the wavelet homotopy analysis method(wHAM)is proposed in this thesis.The main contributions of this thesis are as follows:(1)The homotopy analysis method and the generalized Coiflet wavelet are success-fully combined,and the wavelet homotopy analysis method is proposed.Thebasic ideas and main algorithms for nonlinear ordinary and partial differentialequations are described by one-dimensional and two-dimensional Bratu equa-tions,respectively.(2)An approach to solve multi-solution problems by the wavelet homotopy analysismethod is introduced.The main idea of this approach is to transform the originalnonlinear problem into a new one depending on parameters.To solve the problemdepending on parameters,zeroth-order deformation equations for the governingequation as well as the boundary conditions are constructed,respectively.Thevalidity of this approach is verified by numerical results.(3)In the frame of the wavelet homotopy analysis method,a generalized waveletapproach is developed to deal with different kinds of non-homogeneous bound-ary conditions.This approach is widely adaptive and highly efficient for variouskinds of boundary conditions varying from simple boundary conditions to com-plicated mixed ones.Besides,due to the simple and standardized procedure,thisapproach is very easy to use.(4)The generalized wavelet-Galerkin method is proposed to solve differential equa-tions with non-constant coefficients.Compared with the traditional wavelet-Galerkin method,the generalized wavelet-Galerkin method avoids computingthe complicated connection coefficients and is computationally very efficient tosolve various types of differential equations with non-constant coefficients.In general,the wavelet homotopy analysis method possesses the following advantages:a)Based on the homotopy analysis method,the convergence-control parameter pro-vides us a convenient way to control the convergence of the solution;b)The accuracy and the efficiency can be conveniently balanced by adjusting theresolution level;c)Compared with the traditional homotopy analysis method,the wavelet homotopyanalysis method possesses larger freedom to choose the auxiliary linear opera-tors,because the solution expression and the computational efficiency are notsensitive to the choice of the auxiliary linear operators;d)The wavelet homotopy analysis method possesses very high computational ef-ficiency,especially,the CPU time just increases linearly with respect to the ap-proximation order;e)The iterative wavelet homototy analysis method can significantly accelerate theconvergence of the solution.In summary,the wavelet homotopy analysis method provides us a convenient and efficient way to solve nonlinear boundary value problems.