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Fractal Fract
Volume 7
Issue 11
10.3390/fractalfract7110780
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Article

A Spectral Collocation Method for Solving the Non-Linear Distributed-Order Fractional Bagley–Torvik Differential Equation

by
Ahmed Z. Amin
1,
Mohamed A. Abdelkawy
2,3,*,
Emad Solouma
2,3 and
Ibrahim Al-Dayel
2
1
Department of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia UKM, Bangi 43600, Malaysia
2
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11564, Saudi Arabia
3
Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef 2722165, Egypt
*
Author to whom correspondence should be addressed.
Fractal Fract. 2023, 7(11), 780; https://doi.org/10.3390/fractalfract7110780
Submission received: 29 August 2023 / Revised: 9 October 2023 / Accepted: 20 October 2023 / Published: 26 October 2023
(This article belongs to the Special Issue Recent Developments on Mathematical Models of Deadly Disease)
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Abstract

One of the issues in numerical solution analysis is the non-linear distributed-order fractional Bagley–Torvik differential equation (DO-FBTE) with boundary and initial conditions. We solve the problem by proposing a numerical solution based on the shifted Legendre Gauss–Lobatto (SL-GL) collocation technique. The solution of the DO-FBTE is approximated by a truncated series of shifted Legendre polynomials, and the SL-GL collocation points are employed as interpolation nodes. At the SL-GL quadrature points, the residuals are computed. The DO-FBTE is transformed into a system of algebraic equations that can be solved using any conventional method. A set of numerical examples is used to verify the proposed scheme’s accuracy and compare it to existing findings.
Keywords: spectral collocation method; fractional Bagley–Torvik differential equation; Caputo fractional derivative; shifted Legendre polynomials spectral collocation method; fractional Bagley–Torvik differential equation; Caputo fractional derivative; shifted Legendre polynomials

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MDPI and ACS Style

Amin, A.Z.; Abdelkawy, M.A.; Solouma, E.; Al-Dayel, I. A Spectral Collocation Method for Solving the Non-Linear Distributed-Order Fractional Bagley–Torvik Differential Equation. Fractal Fract. 2023, 7, 780. https://doi.org/10.3390/fractalfract7110780

AMA Style

Amin AZ, Abdelkawy MA, Solouma E, Al-Dayel I. A Spectral Collocation Method for Solving the Non-Linear Distributed-Order Fractional Bagley–Torvik Differential Equation. Fractal and Fractional. 2023; 7(11):780. https://doi.org/10.3390/fractalfract7110780

Chicago/Turabian Style

Amin, Ahmed Z., Mohamed A. Abdelkawy, Emad Solouma, and Ibrahim Al-Dayel. 2023. "A Spectral Collocation Method for Solving the Non-Linear Distributed-Order Fractional Bagley–Torvik Differential Equation" Fractal and Fractional 7, no. 11: 780. https://doi.org/10.3390/fractalfract7110780

APA Style

Amin, A. Z., Abdelkawy, M. A., Solouma, E., & Al-Dayel, I. (2023). A Spectral Collocation Method for Solving the Non-Linear Distributed-Order Fractional Bagley–Torvik Differential Equation. Fractal and Fractional, 7(11), 780. https://doi.org/10.3390/fractalfract7110780

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