Numerical Solution For The Forward And Inverse Problem Of Differential Equations By LS-SVM Method

The numerical solution of differential equations has always been an important research direction in Mathematics and related fields.In recent decades,the research of various calculation methods based on Finite Difference and Finite Element to solve differential equations has made new progress.In recent years,solving differential equations have gained more and more research results by Support Vector Machines,Artificial Neural Networks and other methods.Based on Least Squares Support Vector Machines(LS-SVM),this paper studies the calculation of forward and backward problems of differential equations.The main work of this paper is as follows:(1)The solution of high-order linear ordinary differential equations and nonlinear ordinary differential equations is discussed in the framework of LS-SVM.For linear ordinary differential equation problems,it can be converted into solving corresponding optimization problems.Lagrange multiplier method and Optimality conditions are used to obtain algebraic equations.For nonlinear ordinary differential equations,the constraint is made linearization by adding unknown variables,and the original problem is transformed into an optimization problem.Finally,the nonlinear discrete equations are obtained.The solution of the nonlinear discrete equations is realized by the Newton-iteration method.(2)Studying the LS-SVM numerical solution of two-dimensional and threedimensional development partial differential equations.For two-dimensional development partial differential equations,the cases of boundary regular regions,irregular regions and nonlinear partial differential equations are discussed respectively.The solution of partial differential equations are transformed into the solution of corresponding optimization problems by using LS-SVM method.When dealing with the irregular region problem,the method of discretizing the region is different from the regular region problem,and different optimization problems and different discrete equations are obtained.When solving nonlinear partial differential equations,the constraint is linear by adding unknown variables,which is transformed into an optimization problem,and the specific format of discrete equations is derived.(3)LS-SVM method in studying the inverse problem of initial conditions for two-dimensional diffusion equation.Firstly,the expression of approximate solution of unknown function is constructed by using boundary conditions,and the inverse problem is transformed into an optimization problem.Secondly,the discrete equations are derived based on LS-SVM method.Then,the approximate solution of the inverse problem is obtained by solving the equations,and the numerical solution of initial conditions is obtained.Finally,the stability and the convergence of the algorithm are analyzed,and the effectiveness of the method is illustrated by numerical examples.