Bernstein Collocation Method for Solving Nonlinear Differential Equations

Aysegul Dascioglu,Nese Isler

doi:10.3390/mca18030293

Abstract

In this study, a collocation method based on Bernstein polynomials is developed for solution of the nonlinear ordinary differential equations with variable coefficients, under the mixed conditions. These equations are expressed as linear ordinary differential equations via quasilinearization method iteratively. By using the Bernstein collocation method, solutions of these linear equations are approximated. Combining the quasilinearization and the Bernstein collocation methods, the approximation solution of nonlinear differential equations is obtained. Moreover, some numerical solutions are given to illustrate the accuracy and implementation of the method.

Highlights

The quasilinearization method [2, 4, 12] based on the Newton-Raphson method is an effective approximation technique for solution of the nonlinear differential equations and partial differential equations
This method provides a sequence of functions which converges rather rapidly to the solutions of the original nonlinear equations. This method has been applied to a variety of problems involving different equations like nonlinear initial and boundary value problems involving functional differential equations [1], functional differential equations with retardation and anticipation [5], singular boundary value problems [11], nonlinear Volterra integral equations [8,10], mix integral equations [3], integro-differential equation [13]
The numerical results show that the proposed method can be applicable to the nonlinear differential equations and have effective results for increasing n

Summary

Introduction

The quasilinearization method [2, 4, 12] based on the Newton-Raphson method is an effective approximation technique for solution of the nonlinear differential equations and partial differential equations. Aim of this method is to solve a nonlinear nth order ordinary or partial differential equation in N dimensions as a limit of a sequence of linear differential equations. It is a powerful tool that nonlinear differential equations are expressed as a sequence of linear differential equations This method provides a sequence of functions which converges rather rapidly to the solutions of the original nonlinear equations. This method has been applied to a variety of problems involving different equations like nonlinear initial and boundary value problems involving functional differential equations [1], functional differential equations with retardation and anticipation [5], singular boundary value problems [11], nonlinear Volterra integral equations [8,10], mix integral equations [3], integro-differential equation [13].

Methods

Results

Conclusion