Differential Quadrature Method For Two Types Of Coupled Nonlinear Partial Differential Equations

In this thesis,we mainly study two types of equations: the generalized Zakharov equations and the Klein-Gordon-Zakharov equations,these two equations are coupled nonlinear partial differential equations.We study the numerical solutions of the Dirichlet initial-boundary value problem for these two systems.Here,we adopt high precision differential quadrature method to solve the above two kinds of partial differential equations.The modified cubic B-spline basis functions applied into the differential quadrature method are to determine the weighting coefficients.Then the partial differential equations are reduced into the ordinary differential equations system,and finally we use the optimal four-stage,order three strong stability preserving time-stepping Runge-Kutta scheme to solve this system.Introduction briefly introduces the background of the generalized Zakharov equations and Klein-Gordon-Zakharov equations,research contents and some preliminaries.In this paper,we combine the differential quadrature method and the modified cubic B-spline basis functions into two dimensional,three dimensional generalized Zakharov equations to solve the numerical solutions of the equations.Also we continue to use the modified cubic B-spline differential quadrature method to solve the Klein-Gordon-Zakharov equations.Then we give the numerical simulation of these two equations,and compare the numerical solutions obtained by our method and some finite difference methods with the exact solutions.Compared with the finite difference methods,it can be seen that the results obtained by the modified cubic B-spline differential quadrature method are more close to the exact solutions and the errors are relatively smaller.Also the wave grapes of the numerical and exact solutions are shown.We can see the numerical results obtained by our method are in good agreements with the exact solutions.Especially for the Klein-Gordon-Zakharov equations,we also simulate the propagation process of a single wave.In three dimensional case,we draw the graphs of the numerical solutions and the exact solutions.The validity of our method can be seen from the results of numerical experiments,and the numerical solutions from our method are more accurate than the difference methods' s.Finally,we gives a summary of this paper.