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Article

A Reliable Approach for Solving Delay Fractional Differential Equations

1
Department of Mathematical Sciences, Universiti Kebangsaan Malaysia, Bangi Selangor 43600, Malaysia
2
Academic Support Department, Abu Dhabi Polytechnic, Abu Dhabi P.O. Box 111499, United Arab Emirates
3
Department of Mathematical Sciences, College of Science, UAE University, Al-Ain P.O. Box 15551, United Arab Emirates
4
Department of Mathematics, Yarmouk University, Irbid 21163, Jordan
*
Author to whom correspondence should be addressed.
Fractal Fract. 2022, 6(2), 124; https://doi.org/10.3390/fractalfract6020124
Submission received: 17 January 2022 / Revised: 7 February 2022 / Accepted: 18 February 2022 / Published: 21 February 2022

Abstract

In this paper, we study a class of second-order delay fractional differential equations with a variable-order Caputo derivative. This type of equation is an extension to ordinary delay equations which are used in the modeling of several biological systems such as population dynamics, epidemiology, and immunology. Usually, fractional differential equations are difficult to solve analytically, and with fractional derivatives of variable-order, they become more challenging. Therefore, the need for reliable numerical techniques is worth investigating. To solve this type of equation, we derive a new approach based on the operational matrix. We use the shifted Chebyshev polynomials of the second kind as the basis for the approximate solutions. A convergence analysis is discussed and the uniform convergence of the approximate solutions is proven. Several examples are discussed to illustrate the efficiency of the presented approach. The computed errors, figures, and tables show that the approximate solutions converge to the exact ones by considering only a few terms in the expansion, and illustrate the novelty of the presented approach.
Keywords: second-order fractional delay differential equation; operational matrix method; shifted Chebyshev polynomials of the second kind second-order fractional delay differential equation; operational matrix method; shifted Chebyshev polynomials of the second kind

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MDPI and ACS Style

Hashim, I.; Sharadga, M.; Syam, M.I.; Al-Refai, M. A Reliable Approach for Solving Delay Fractional Differential Equations. Fractal Fract. 2022, 6, 124. https://doi.org/10.3390/fractalfract6020124

AMA Style

Hashim I, Sharadga M, Syam MI, Al-Refai M. A Reliable Approach for Solving Delay Fractional Differential Equations. Fractal and Fractional. 2022; 6(2):124. https://doi.org/10.3390/fractalfract6020124

Chicago/Turabian Style

Hashim, Ishak, Mwaffag Sharadga, Muhammed I. Syam, and Mohammed Al-Refai. 2022. "A Reliable Approach for Solving Delay Fractional Differential Equations" Fractal and Fractional 6, no. 2: 124. https://doi.org/10.3390/fractalfract6020124

APA Style

Hashim, I., Sharadga, M., Syam, M. I., & Al-Refai, M. (2022). A Reliable Approach for Solving Delay Fractional Differential Equations. Fractal and Fractional, 6(2), 124. https://doi.org/10.3390/fractalfract6020124

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