Meshless Method For Solving Two Kinds Of Partial Differential Equations

Many practical problems in life can be modeled as partial differential equation models.Solving partial differential equations will help people better understand practical problems,but the analytical solution of partial differential equations is not easy to obtain,so it is very necessary to study the numerical solution of partial differential equations.In this paper,two kinds of partial differential equations are solved numerically.Firstly,a class of piecewise continuous delayed partial differential equations is studied.The time-discrete scheme of the equation is obtained by using the θ-weighted finite difference method,and then the spatial derivative is approximated by the meshless interpolation method based on the radial basis function,and the fully discrete numerical scheme is obtained.The basis function adopted is Multiquadric(MQ)radial basis function,which is superior to other radial basis functions in accuracy,stability and other aspects.According to the Fourier method,the stability condition of the discrete model of one-dimensional piecewise continuous delay partial differential equation is obtained,that is,the stability is only related to the time step,and the error between the numerical solution and the analytic solution is calculated by using MATLAB,and the numerical solution of the equation is calculated under different time step,θ value and initial value,which verifies the effectiveness of the method.Secondly,the one-dimensional piecewise continuous delayed partial differential equation is extended to n-dimensional,and the analytic solution of the equation in series form is given.The discrete model of the equation is established by using the θ-weighted finite difference method and the radial basis function method,and the stability condition similar to the one-dimensional equation is obtained.The error between the analytical solution and the numerical solution in the one-dimensional equation is calculated in the numerical example.The component diagram of the stability of the numerical solution in the two-dimensional equation is given.The results show that the numerical solution is asymptotically stable and tends to 0 with the increase of time.Finally,a predator-prey model with reaction-diffusion term is studied.Based on Crank-Nicolson format and radial basis point interpolation(RPIM),the discrete model used MQ radial basis function.The linearities of discrete models are eliminated by the predictor-corrector method,and the linear equations are obtained.The influences of different shape parameters on three radial basis functions are analyzed.Finally,Finally,the numerical simulation diagram of predation-prey model is given.