Numerical Solutions Of Differential Equation Based On Reproducing Kernel Theory

In this paper,modified reproducing kernel Hilbert space method is employed to solve Fredholm type linear integro-differential equations.First a new reproducing kernel Hilbert space is constructed and its reproducing kernel is obtained.Then,traditional reproducing kernel Hilbert space method is introduced.Next,we propose a modified reproducing kernel Hilbert method.Finally,two numerical examples are presented.To test the method with CAS wavelet method,differential transformation method and improved homotopy perturbation method in accuracy and traditional reproducing kernel Hilbert space method in calculation speed,we find modified reproducing kernel Hilbert space method is of higher accuracy and faster.Also we demonstrate that the algorithm can be applied to other linear ordinary equations or partial differential equations with initial boundary value conditions and non-local boundary value conditions.Then we present a reliable algorithm to determine the new approximate solutions of nonlinear singular initial value problems of Emden-Fowler type.The method is designed to overcome the difficulty of the singular point.The approximate solution is obtained and is proved to converge to the exact solution.We have compared analytical and numerical solutions of linear and nonlinear Emden-Fowler equations respectively with the approximate solutions obtained by Differential Transformation Method(DTM),Chebyshev Neural Network Method(ChNN),traditional Artificial Neural Network(ANN),a method which is proposed on the series representation of the solution via suitable base functions and reproducing kernel Hilbert space method.The numerical results show the high precision and efficiency of the present method.