Analysis of the Fractional-Order Delay Differential Equations by the Numerical Method

Saadia Masood,Roman Ullah,Muhammad Naeem,Abdul Bariq,Saima Mustafa,Fathalla A Rihan

doi:10.1155/2022/3218213

Abstract

In this study, we implemented a new numerical method known as the Chebyshev Pseudospectral method for solving nonlinear delay differential equations having fractional order. The fractional derivative is defined in Caputo manner. The proposed method is simple, effective, and straightforward as compared to other numerical techniques. To check the validity and accuracy of the proposed method, some illustrative examples are solved by using the present scenario. The obtained results have confirmed the greater accuracy than the modified Laguerre wavelet method, the Chebyshev wavelet method, and the modified wavelet-based algorithm. Moreover, based on the novelty and scientific importance, the present method can be extended to solve other nonlinear fractional-order delay differential equations.

Highlights

Fractional calculus is used in various branches of mathematics due to its numerous applications in modeling different physical phenomena in engineering and science. e concept of fractional calculus has been derived from the fact Dα(f(x)), where alpha is noninteger
DEs are used to develop a different number of physical problems
A new technique has been used by the researchers known as fractional differential equations (FDEs)

Summary

Introduction

Fractional calculus is used in various branches of mathematics due to its numerous applications in modeling different physical phenomena in engineering and science. e concept of fractional calculus has been derived from the fact Dα(f(x)), where alpha is noninteger. Fractional calculus is used in various branches of mathematics due to its numerous applications in modeling different physical phenomena in engineering and science. Some are more complex and cannot be modeled with the help of simple differential equations For these complex problems, a new technique has been used by the researchers known as fractional differential equations (FDEs). Researchers pay more attention to FDDEs as compared to DEs because a slight delay has a large effect In this regard, numerous papers have been dedicated to the study of the numerical solution of FDDEs. FDDs have been widespread in mathematical modelings, such as population.

Basic Definitions of Fractional Derivatives

Chebyshev Series Expansion Is Used to Approximate a Caputo Derivative

Chebyshev

Chebyshev Collocation Method

Numerical Representation

Conclusion