Open Access

ARTICLE

Analysis of a Laplace Spectral Method for Time-Fractional Advection-Diffusion Equations Incorporating the Atangana-Baleanu Derivative

Kamran^1,*, Farman Ali Shah¹, Kallekh Afef ², J. F. Gómez-Aguilar ³, Salma Aljawi⁴, Ioan-Lucian Popa^5,6,*

1 Department of Mathematics, Islamia College Peshawar, Peshawar, Khyber Pakhtoon Khwa, 25120, Pakistan
2 Department of Mathematics, College of Science, King Khalid University, Abha, 61413, Saudi Arabia
3 Centro de Investigación en Ingenierìa y Ciencias Aplicadas (CIICAp-IICBA)/UAEM, Universidad Autónoma del Estado de Morelos, Av. Universidad 1001. Col. Chamilpa, Cuernavaca, 62209, Morelos, México
4 Department of Mathematical Sciences, Princess Nourah Bint Abdulrahman University, Riyadh, 11671, Saudi Arabia
5 Department of Computing, Mathematics and Electronics, “1 Decembrie 1918” University of Alba Iulia, Alba Iulia, 510009, Romania
6 Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Iuliu Maniu Street 50, Brasov, 500091, Romania

* Corresponding Authors: Kamran. Email: ; Ioan-Lucian Popa. Email:

(This article belongs to the Special Issue: Analytical and Numerical Solution of the Fractional Differential Equation)

Computer Modeling in Engineering & Sciences 2025, 143(3), 3433-3462. https://doi.org/10.32604/cmes.2025.064815

Received 25 February 2025; Accepted 07 May 2025; Issue published 30 June 2025

Abstract

In this article, we develop the Laplace transform (LT) based Chebyshev spectral collocation method (CSCM) to approximate the time fractional advection-diffusion equation, incorporating the Atangana-Baleanu Caputo (ABC) derivative. The advection-diffusion equation, which governs the transport of mass, heat, or energy through combined advection and diffusion processes, is central to modeling physical systems with nonlocal behavior. Our numerical scheme employs the LT to transform the time-dependent time-fractional PDEs into a time-independent PDE in LT domain, eliminating the need for classical time-stepping methods that often suffer from stability constraints. For spatial discretization, we employ the CSCM, where the solution is approximated using Lagrange interpolation polynomial based on the Chebyshev collocation nodes, achieving exponential convergence that outperforms the algebraic convergence rates of finite difference and finite element methods. Finally, the solution is reverted to the time domain using contour integration technique. We also establish the existence and uniqueness of the solution for the proposed problem. The performance, efficiency, and accuracy of the proposed method are validated through various fractional advection-diffusion problems. The computed results demonstrate that the proposed method has less computational cost and is highly accurate.

---

Keywords

---