The New Methods For Solving Linear And Nonlinear Weakly-singular Volterra Integro-differential Equations

This paper studies the numerical solution algorithm for the nth-order linear andnonlinear weakly-singular Volterra integro-diferential equations. the study of thetheory of singular integro-diferential equations, only a small part of the equationcould write a closed form solution, only for qualitative research, such as Noether’stheorem, it is very important therefore seek the approximation solution of singularintegro-diferential equations.The aim of this paper is to use Taylor’s approximation and then transform thegiven nth-order linear and nonlinear weakly-singular Volterra integro-diferentialequations into an ordinary linear and nonlinear diferential equations, Using theRKM to solve ordinary nth-order linear diferential equation and using the RKMand ADM to solve ordinary nonlinear diferential equation. Reproducing kerneltheory is based on reproducing kernel space W_n+1[a, b], the approximate solution ofthe equation in the form of a series reproducing kernel spaces W_n+1[a, b]. Using theRKM and ADM to solve ordinary nonlinear diferential equation. Some examplesand compared to existing examples to prove this efectiveness of the algorithm.