The Study On The Approximate Solution Of Linear Differential Equation By LS-SVM Algorithm

The problem of solving differential equations and the properties of solutions have always been an important content of differential equation the study.In practical application and scientific research,it is difficult to obtain analytical solutions for most differential equations.In recent years,with the development of computer technology,some new intelligent algorithms have been used to solve differential equations.They overcomes the drawback of traditional methods and provide the approximate solution in closed form(i.e.,continuous and differentiable).Least Squares Support Vector Machine(LS-SVM)shows nice performances in solving differential equations.This method adds the original differential equation as a constraint to the LS-SVM regression model,which greatly improves the accuracy and generalization ability of the algorithm.In this paper,we study the problems encountered by LS-SVM algorithm for solving ordinary differential equations,and a new approach based on LS-SVM is proposed for solving linear pantograph delay differential equation.The main contents are listed as following:Firstly,for linear ordinary differential equations,in order to further improve the accuracy of approximate solutions,a novel method based on numerical methods and LS-SVM is presented to solve linear ordinary differential equations.In this method,a high-precision numerical solution is added as a constraint to the nonlinear LS-SVM regression model,and the optimal parameters of the model are adjusted to minimize an appropriate error function.Then,a high-precision approximate solution is obtained.A large number of numerical experiments prove that our proposed method can improve the accuracy of approximate solutions.Secondly,concerning high-order ordinary differential equations,in order to avoid solving high-order derivatives of the kernel function,it is transformed into a system of first-order differential,and then the nonlinear LS-SVM regression model is constructed containing the first-order derivative forms.In this model,these model parameters are obtained by minimizing the error function and a high-precision approximate solution is obtained by solving three linear equations.Experimental results verify the effectiveness of the proposed method.Finally,the LS-SVM algorithm is used to solve the proportional delay differential equation.In this method,equality constraints replace inequality constraints,therefore,it avoids solving complex quadratic programming problems,and then an approximate solution is obtained by solving a linear system of equations.Numerical experiments demonstrate the effectiveness of the proposed method.