TY - JOUR
T1 - A Reliable Approach for Solving Delay Fractional Differential Equations
AU - Hashim, Ishaq
AU - Sharadga, Mwaffag
AU - Syam, Muhammed I.
AU - Alrefai, Moh'd A.
N1 - Funding Information:
Acknowledgments: The first author acknowledges the financial support received from the UKM’s under Grant No. DIP-2021-018.
Publisher Copyright:
© 2022 by the authors. Licensee MDPI, Basel, Switzerland.
PY - 2022/2
Y1 - 2022/2
N2 - 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.
AB - 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.
KW - Operational matrix method
KW - Second-order fractional delay differential equation
KW - Shifted Chebyshev polynomials of the second kind
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U2 - 10.3390/fractalfract6020124
DO - 10.3390/fractalfract6020124
M3 - Article
AN - SCOPUS:85125175430
SN - 2504-3110
VL - 6
JO - Fractal and Fractional
JF - Fractal and Fractional
IS - 2
M1 - 124
ER -