| The design and implementation of numerical solutions to differential equations has become an important branch of computational mathematics and control science.Due to the high computational complexity and the discrete solution form of the traditional numerical solution,researchers try to use the Support Vector Machines(SVM)method with better generalization ability as a new exploration direction for solving differential equations.This paper proposes to solve three classes of ordinary differential equations through an effectively improved Proximal Support Vector Machines(P-SVM)method,giving full play to its strong generalization ability,fast training speed and high degree of interpretability in nonlinear regression problems to make up for the shortcomings of traditional methods.First,this paper converts the ordinary differential equation solving problem into an objective optimization problem with equality constraints based on the P-SVM,and adds a bias term in the objective function to convert the optimization problem into a strict convex quadratic programming to simplify the calculation and solution process,in order to obtain an approximate solution with simple form and fixed structure.Secondly,using the above ideas,the initial value problem of the first-order linear ordinary differential equation,the initial-boundary value problem of the second-order linear ordinary differential equation,and the periodic solution problem of the delayed differential equation are respectively carried out.The theoretical derivation of the solution process and the approximate solution expression is verified.Its feasibility and effectiveness are proved by numerical experiments,and its accuracy is compared with other methods.Finally,the experimental results show that the obtained closed-form approximate solution can effectively improve the calculation speed while ensuring the accuracy,and the simple and fixed expression is also convenient for the qualitative analysis and application of the approximate solution,which proves that the approximate solution of three classes ordinary differential equations based on P-SVM proposed in this paper has important theoretical significance and application value in the field of numerical solution of differential equations. |
