Numerical Solution Of Nonlinear Partial Differential Equations By Jacobi Collocation Method

Spectral methods are becoming more and more widely used in many fields,such as marine engineering,hydrodynamics,atmospheric science,quantum mechanics and other science and engineering.After many years of development,the spectral method is not only improved in theoretical analysis,but also a lot of important achievements in numerical simulation.The rapid development of spectral methods in recent years is due to its spectral accuracy.Its convergence is only related to the smoothness of the approximation problem.If the solution of the problem is infinite smooth,its convergence rate is exponential.This paper will mainly discuss the numerical solution of the 3-coupled nonlinear Schrodinger equation(3-CNLS)and the generalized Hirota Satsuma coupled KdV equation by the Jacobi collocation method and analyse the error of numerical solutions.The 3-coupled nonlinear Schrodinger equation can describe different scientific phenomena,such as nonlinear optics,optical communication,multicomponent Bose-Einstein condensates at zero temperature,and plasma physics.In the process of solution,we first change complex function into real function and then use the Jacobi collocation method to change the 3-coupled nonlinear Schrodinger equations into ordinary differential equations in the interval-]1,1[.Lastly,we use the software Mathematica with the implicit Euler method for error estimation of the ordinary differential equation,which proves the effectiveness and accuracy of this method.The generalized Hirota Satsuma coupled KdV equation is used to describe the interaction of long waves with different dispersion relations in mathematics.Mathematicians used these equations to construct a set of important nonlinear evolution systems in history.Because of their wide application,mathematicians have gained wide attention in mathematics.In the process,the H-S coupled KdV equation is transformed into the ordinary differential equations on the interval-]1,1[ by using the Jacobi collocation method.Then the error of the numerical solution of the ordinary differential equations is estimated by software Matlab with fourth-order Runge Kutta method,which proves the validity and accuracy of the method.