Sinc Collocation Method For Solving Several Classes Of Differential Equations

In many important natural science and engineering technology problems,it is important to establish mathematical models of differential equations as to describe the evolution process of some characteristics over time or space.This method can be used in many important fields,such as fluid movement,optical fiber propagation,population genetics,population development,nuclear reaction,etc.Therefore,differential equations are widely used in physics,biology,engineering and other fields,and the study of differential equations has always been one of the hot spots in the field of mathematics.It is of great value to obtain the solutions of differential equations for the problems described by the equations.Generally speaking,it is very difficult to obtain the exact solutions for most differential equations.However,for many practical problems,only the results of some discrete points are needed.At the same time,by constructing an efficient numerical method,we can quickly find the numerical solution to meet the needs of practical application.Therefore,it is very meaningful to study the numerical solutions of differential equations.At present,the commonly used numerical methods mainly include spectral method,finite difference method,collocation method,spline interpolation method,finite element method,etc.In this paper,Burgers equation,coupled Burgers equation,coupled KdV equation,the modified Kawahara equation and Burgers-KdV equation are solved by using sinc collocation method.Firstly,using ?-weighted scheme to discretize the time derivative and Taylor expansion to process the nonlinear term,we obtain the semi-discrete scheme of the equation.Subsequently,sinc collocation method is used to discretize the spatial terms,and then a fully discrete scheme is obtained.Moreover,the stability condition is given by expressing the fully discrete scheme in matrix form.Finally,numerical solution of the original equation can be obtained by solving the fully discrete scheme.The article details are as follows:Chapter ? introduces research background and current situation of Burgers equation,coupled Burgers equation,coupled KdV equation,the modified Kawahara equation,Burgers-KdV equation and sinc collocation method.The main content of this paper is also briefly summarized.Chapter II gives the definitions and properties of sinc collocation method and Von Neumann condition is also stated.In Chapter ?,sinc collocation method is used to solve Burgers equation and coupled Burgers equation.The stability analysis is also given.At the same time,numerical solutions obtained in this paper are compared with those of other methods.The results show that numerical solutions obtained by present method has higher accuracy.Chapter IV uses sinc collocation method to solve coupled KdV equation.The feasibility of present method is illustrated through stability analysis and numerical experiments.In Chapter ?,the modified Kawahara equation is solved by using sinc collocation method.In the numerical experiments,in addition to comparing the errors with those of other methods,the relative changes of two invariants are also analyzed,which shows the high-precision of present method.In Chapter ?,sinc collocation method is employed to solve Burgers-KdV equation.The analysis of absolute errors at some points shows that the numerical results obtained by present method are reliable.Chapter ? summarizes the content of full text and concludes that sinc collocation method is effective for solving these classes of differential equations.