A Numerical Approach Based on Taylor Polynomials for Solving a Class of Nonlinear Differential Equations

C Guler,S O Kaya

doi:10.1155/2018/8256932

Abstract

In this study, a matrix method based on Taylor polynomials and collocation points is presented for the approximate solution of a class of nonlinear differential equations, which have many applications in mathematics, physics and engineering. By means of matrix forms of the Taylor polynomials and their derivatives, the technique we have used reduces the solution of the nonlinear equation with mixed conditions to the solution of a matrix equation which corresponds to a system of nonlinear algebraic equations with the unknown Taylor coefficients. On the other hand, to illustrate the validity and applicability of the method, some numerical examples together with residual error analysis are performed and the obtained results are compared with the existing results in literature.

Highlights

IntroductionWe consider the high order differential equation with nonlinear terms in the form m
They implement a relatively new analytic iterative technique to get approximate solutions of differential algebraic equations system based on generalized Taylor series formula and the analysis proposes an analytical-numerical approach for providing solutions of a class of nonlinear fractional Klein-Gordan equation subjected to appropriate initial conditions in Caputo sense by using the Fractional Reduced Differential Transform Method (FRDTM) in [13, 14], respectively
Nonlinear differential equations used in engineering, physics, mathematics or in many modelling problems are usually difficult to solve analytically

Summary

Introduction

We consider the high order differential equation with nonlinear terms in the form m. In [11], they present a new analytical technique for obtaining the analytical approximate solutions for system of Fredholm-Integral equations based on the use of residual power series method (RPSM). A new general form fractional power series is introduced in the sense of the Caputo fractional derivative in [12] They implement a relatively new analytic iterative technique to get approximate solutions of differential algebraic equations system based on generalized Taylor series formula and the analysis proposes an analytical-numerical approach for providing solutions of a class of nonlinear fractional Klein-Gordan equation subjected to appropriate initial conditions in Caputo sense by using the Fractional Reduced Differential Transform Method (FRDTM) in [13, 14], respectively.

Operational Matrix Relations

Accuracy of Solutions and Residual Error Estimation

Numerical Examples

Conclusion