The derivation of convergence analysis for the multistage adomian decomposition method for solving the autonomous van der pol system

الملخص EN

The autonomous Van der Pol system was solved accurately by MADM [4].

The method has the advantage of giving the form of the numerical solution within each time interval which is not possible in purely numerical techniques like RK45 and classical ADM.

In this paper we will derive the convergence of A Multistage Adomian Decomposition Method (MADM) by deriving the Homotopy Analysis Method for this system.

We will show that if the series        2 1 0 0 1 ( 0) ( ) j n t n i ij in in x t a x dt x t dt where     2 1 0 0 i2 2 1 0 0 , 1 ) , , ( R ~ ! 1 ! ~ ( , ) ! ~ ( ) ( ) p t n p q n q ipq n i n i in dt n d d x x n a n d d x t n x t x t      is convergent and it must be a solution of Van der Pol system, then the two sequences ( ) , ( ) 1 1 R t v h t k n k in k n k in   are converges to zero.