Numerical Method For Solving A Class Of Nonlinear Singular Differential Equations

Singular differential equations appear in many applied models.Due to the sin-gular of differential equation,its solution is affected by various boundary value condi-tions,which brings some difficulties to the study of the problem.Therefore,it's a great value to study the solutions of such equations.The numerical solutions of nonlinear singular differential equations are studied by combining the improved reproducing kernel method with the least square method and the Quasi-Newton method in this paper.In this paper,the least squares-simplified reproducing kernel method is used to solve nonlinear singular differential equations.Firstly,the corresponding reproduc-ing kernel space is established according to the characteristics of the model and corresponding basis function is obtained on the established space.Then we com-bine the least square method and find an approximate solution to the equation.The method avoids the process of Smith orthogonalization.Compared with the previ-ous methods,the method greatly saves the computation time and the computation quantity.Finally,numerical examples are given to demonstrate the effectiveness of the proposed method.Newton's method is often used to solve nonlinear problems.The advantage of Newton's method is fast convergence.Because the method requires that the first derivative of function exist and at each step needs to compute the value of the deriva-tive function.it's not convenient to implement on computer.Quasi-Newton method is proposed to address this shortcoming.This method is used to solve nonlinear sin-gular differential equations.Firstly,the Quasi-Newton method is used to linearize the equation.According to the characteristics of the model,the corresponding reproduc-ing kernel space is established.Then the system of equations is established according to the projection operator constructed.Next,we write out the matrix form of the system and finally we find the coefficients.Finally,the effectiveness of the proposed method is demonstrated by comparison with the corresponding examples.