Modified wavelets–based algorithm for nonlinear delay differential equations of fractional order

Muhammad Asad Iqbal,Muhammad Shakeel,Syed Tauseef Mohyud-Din,Muhammad Rafiq

doi:10.1177/1687814017696223

Abstract

Most of the physical phenomena located around us are nonlinear in nature and their solutions are of great significance for scientists and engineers. In order to have a better representation of these physical phenomena, fractional calculus is developed. Some of these nonlinear physical models can be represented in the form of delay differential equations of fractional order. In this article, a new method named Gegenbauer Wavelets Steps Method is proposed using Gegenbauer polynomials and method of steps for solving nonlinear fractional delay differential equations. Method of steps is used to convert the fractional nonlinear fractional delay differential equation into a fractional nonlinear differential equation and then Gegenbauer wavelet method is applied at each iteration of fractional differential equation to find the solution. To check the accuracy and efficiency of the proposed method, the proposed method is implemented on different nonlinear fractional delay differential equations including singular-type problems also.

Highlights

In functional differential equations, the rate of change of unknown function depends upon the values of unknown functions at present time and on previous time values
Over the past few decades, several researchers had devoted their study in finding numerical solutions of fractional delay differential equations due to nonavailability of exact solutions in most of the cases
We propose a new method called Gegenbauer Wavelet Steps Method for solving nonlinear fractional delay differential equations and show that it is strongly reliable method for such nonlinear problems than the other existing methods

Summary

Introduction

The rate of change of unknown function depends upon the values of unknown functions at present time and on previous time values. In Falbo,[14] the method of steps is utilized to solve linear and nonlinear discrete delay differential equations with different types of delay. Rawashdeh[17] implemented Legendre wavelets method to obtain solutions of fractional integro-differential equations. E Babolian and FF Zadeh[20] obtained numerical solutions of differential equations using Chebyshev wavelet operational matrix of integration. Since these methods are newly developed, these methods had few shortcomings while dealing with nonlinear differential equations of fractional order. We propose a new method called Gegenbauer Wavelet Steps Method for solving nonlinear fractional delay differential equations and show that it is strongly reliable method for such nonlinear problems than the other existing methods. DaR;ayðxÞ and DaC;ayðxÞ become zero for x 1⁄4 a

Ji-Huan He’s fractal derivative

Procedure of implementation

Conclusion