Study On Numerical Methods For Several Classes Of SDEs And PDEs

In this paper,several kinds of stochastic differential equations and partial differential equations are studied.Euler method is an important method for solving stochastic differential equations.Based on this method,a new numerical method is constructed,and the convergence of the improvement Euler method for solving stochastic differential equations is studied.For autonomous scalar stochastic differential equations,the local convergence orders of the improvement Euler method in mean and mean square are 2 and 1,the order of strong convergence in mean square is 1.The numerical solution is closer to the analytical solution by introducing parameters and .Finally,numerical examples show that the numerical solution obtained by this method is closer to the analytical solution than that obtained by trapezoidal Euler-Maruyama method.The Regular long wave equation and KdV equation play an important role in many fields of Applied Science,and there are many kinds of numerical methods to solve these kinds of equations.In order to obtain higher numerical accuracy,the barycentric interpolation collocation method is introduced in this paper to solve this kind of equation.In order to better solve the equation,get higher convergence accuracy,faster running speed and fewer iterations,direct linearization and Newton Raphson iteration are applied to the calculation.In this paper,a direct linearization iterative method of barycentric interpolation for solving the coupled KdV equation is proposed.In order to make the numerical solution more accurate,a new method for solving the coupled KdV equation and the generalized Hirota Satsuma coupled KdV equation is introduced and its convergence is studied.Different parameters and exact solutions are discussed in the numerical examples.The accuracy and effectiveness of the method are proved by comparing with the previous work Numerical experiments show that the method is simple,fast and accurate.