An Approximate Solution of the Time-Fractional Fisher Equation with Small Delay by Residual Power Series Method

Ali Demir,Mine Aylin Bayrak,Ebru Ozbilge

doi:10.1155/2018/9471910

Ali Demir, Mine Aylin Bayrak + Show 1 more

Open Access

https://doi.org/10.1155/2018/9471910

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Abstract

An analytical solution of the time-fractional Fisher equation with small delay is established by means of residual the residual power series method (RPSM) where the fractional derivative is taken in the Caputo sense. Taking advantage of small delay, the time-fractional Fisher equation is expanded in powers series of delay term ϵ. By using RPSM analytical solution of time-fractional of Fisher equation is constructed. The final results and graphical consequences illustrate that the proposed method in this study is very efficient, effective, and reliable for the solution of the time-fractional Fisher equation with small delay.

Highlights

In last few decades, fractional calculus growing considerable interest is used in bioengineering, thermodynamics, viscoelasticity, control theory, aerodynamics, electromagnetics, signal processing, chemistry, and finance [1,2,3,4,5,6,7]
Delay differential equations (DDEs) can be considered as the generalization of the ordinary differential equations which are appropriate for modelling physical systems with memory
The residual power series method (RPSM) was established as a powerful method for fuzzy differential equations [13]

Summary

Introduction

Fractional calculus growing considerable interest is used in bioengineering, thermodynamics, viscoelasticity, control theory, aerodynamics, electromagnetics, signal processing, chemistry, and finance [1,2,3,4,5,6,7]. Construction of multidimensional and multiple solutions for fractional differential equations in the form of power series is an important advantage of RPSM. We extend the application of the RPSM in order to establish an approximate solution to time-fractional Fisher equation with small delay. By making use of the residual power series method (RPSM), we find series solution for the time-fractional Fisher equation with small delay. Applying RPSM we determine the solution of equations which are the obtained from the Mathematical Problems in Engineering coefficients of ε. Replacing these solutions in the Taylor series the solution of the original problem is obtained

Preliminaries

Numerical Results

Conclusion