Spectral solutions for the time-fractional heat differential equation through a novel unified sequence of Chebyshev polynomials

Waleed Mohamed Abd-Elhameed; Hany Mostafa Ahmed; Waleed Mohamed Abd-Elhameed; Hany Mostafa Ahmed

doi:10.3934/math.2024107

AIMS Mathematics

2024, Volume 9, Issue 1: 2137-2166. doi: 10.3934/math.2024107

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Spectral solutions for the time-fractional heat differential equation through a novel unified sequence of Chebyshev polynomials

Waleed Mohamed Abd-Elhameed ^{1
,
,},
Hany Mostafa Ahmed ²

1.
Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah, Saudi Arabia
2.
Department of Mathematics, Faculty of Technology and Education, Helwan University, Cairo 11281, Egypt; Email: hanyahmed@techedu.helwan.edu.eg

Received: 08 October 2023 Revised: 04 December 2023 Accepted: 13 December 2023 Published: 20 December 2023
MSC : 65XX, 65Mxx, 33C45

In this article, we propose two numerical schemes for solving the time-fractional heat equation (TFHE). The proposed methods are based on applying the collocation and tau spectral methods. We introduce and employ a new set of basis functions: The unified Chebyshev polynomials (UCPs) of the first and second kinds. We establish some new theoretical results regarding the new UCPs. We employ these results to derive the proposed algorithms and analyze the convergence of the proposed double expansion. Furthermore, we compute specific integer and fractional derivatives of the UCPs in terms of their original UCPs. The derivation of these derivatives will be the fundamental key to deriving the proposed algorithms. We present some examples to verify the efficiency and applicability of the proposed algorithms.
- Chebyshev polynomials,
- recurrence relation,
- spectral methods,
- fractional differential equations,
- convergence analysis
Citation: Waleed Mohamed Abd-Elhameed, Hany Mostafa Ahmed. Spectral solutions for the time-fractional heat differential equation through a novel unified sequence of Chebyshev polynomials[J]. AIMS Mathematics, 2024, 9(1): 2137-2166. doi: 10.3934/math.2024107

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Abstract

In this article, we propose two numerical schemes for solving the time-fractional heat equation (TFHE). The proposed methods are based on applying the collocation and tau spectral methods. We introduce and employ a new set of basis functions: The unified Chebyshev polynomials (UCPs) of the first and second kinds. We establish some new theoretical results regarding the new UCPs. We employ these results to derive the proposed algorithms and analyze the convergence of the proposed double expansion. Furthermore, we compute specific integer and fractional derivatives of the UCPs in terms of their original UCPs. The derivation of these derivatives will be the fundamental key to deriving the proposed algorithms. We present some examples to verify the efficiency and applicability of the proposed algorithms.

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