Study On Some Numerical Methods For Solving Nonlinear Evolution Equations

With the development of nonlinear science, it has become a hot problem toobtain the solutions of nonlinear evolution equations for a great number of scienceworkers who study physics, mechanics, geosciences, life sciences, appliedmathematics and engineering techniques. At the present, a number of methods areproposed and developed to look for the exact solutions of nonlinear evolutionequations, but, in most cases, we can only obtain their exact solutions when somespecial conditions are given. Then, it is more important to solve nonlinear evolutionequations numerically. In this dissertation, we have studied modified Bernsteinpolynomials Galerkin's approximation method, Quartic B-spline Galerkin finiteelement method and differential quadrature method for solving some nonlinearevolution equations numerically, and some works as follow.1. Numerical solutions of Burgers'equation and KdV-Burgers'equation areobtained by modified Bernstein polynomials Galerkin,s approximation method. Theresults show that the present algorithm uses only a few basic functions but can obtainhighly accurate numerical solutions with less computer efforts, and is effective for along time evolution behavior.2. Numerical solutions of Kuramoto-Sivashinsky equation are obtained byquartic B-spline Galerkin finite element method. Numerical experiments show that thenumerical solution is highly accurate and the present scheme is highly adaptable .3. Differential quadrature method based on Cosine is derived for solvingimproved Boussinesq equation. The results show that the highly accurate numericalsolutions can be acquired by the present algorithm with less amounts of grid points,small calculations and better time complexity.4. Numerical solutions of coupled Burgers'equations are derived separatly bycubic B-spline Galerkin finite element method, modified Bernstein polynomialsGalerkin's approximation method and differential quadrature method. The numericalresults show that the cubic B-spline Galerkin finite element method is highly accurateand adaptable, but has complicated theory, difficult in programming, large calculationsand long time-consuming. In contrast, the modified Bernstein polynomial-Galerkin and differential quadrature methods have simpled theory, easy programming, smallcalculations and high efficience, but are poorly adaptable and highly dependent oninitial conditions.