A Novel Representation of the Exact Solution for Differential Algebraic Equations System Using Residual Power-Series Method

Khaled Moaddy,Mohammed Al-Smadi,Ishak Hashim

doi:10.1155/2015/205207

Abstract

We implement a relatively new analytic iterative technique to get approximate solutions of differential algebraic equations system based on generalized Taylor series formula. The solution methodology is based on generating the residual power series expansion solution in the form of a rapidly convergent series with easily computable components. The residual power series method (RPSM) can be used as an alternative scheme to obtain analytical approximate solution of different types of differential algebraic equations system applied in mathematics. Simulations and test problems were analyzed to demonstrate the procedure and confirm the performance of the proposed method, as well as to show its potentiality, generality, viability, and simplicity. The results reveal that the proposed method is very effective, straightforward, and convenient for solving different forms of such systems.

Highlights

Differential algebraic equations (DAEs) have gained in the recent years considerable importance and popularity partly due to its powerful potential applications
The DAEs are type of differential equations, in which the unknown functions are satisfying additional algebraic equations, whereas the derivatives are not in general expressed explicitly and typically derivatives of some of the dependent variables may not appear in the equations at all
Many explicit solutions have been found to the linear DAEs, but there exists no method that yields an explicit solution for the nonlinear DAEs

Summary

Introduction

Differential algebraic equations (DAEs) have gained in the recent years considerable importance and popularity partly due to its powerful potential applications. There are many powerful numerical methods in literature that can be used to approximate solutions to the DAEs system. The RPSM has been developed as an efficient numerical as well as analytical method to determine the coefficients of power series solutions for a class of fuzzy differential equation by Abu Arqub [27]. The analytical approximate solution should be constructed in the form of a polynomial which does not exhibit the real behaviors of the problem but gives a good approximation to the true solution in the given interval This approach is different from the traditional higher order Taylor series method. An excellent account of the study of error analysis, which includes its definitions, varieties, applications, and method of derivations, can be found in [34]

Simulations and Test Problems

Conclusion