Galerkin-based solution for the time-fractional diffusion-wave equation

Waleed Mohamed Abd-Elhameed; Mohamed A. Abdelkawy; Naher Mohammed A. Alsafri; Ahmed Gamal Atta; Waleed Mohamed Abd-Elhameed; Mohamed A. Abdelkawy; Naher Mohammed A. Alsafri; Ahmed Gamal Atta

doi:10.3934/era.2025232

Electronic Research Archive

2025, Volume 33, Issue 9: 5179-5206. doi: 10.3934/era.2025232

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Galerkin-based solution for the time-fractional diffusion-wave equation

1.
Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
2.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia
3.
Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23831, Saudi Arabia
4.
Department of Mathematics, Faculty of Education, Ain Shams University, Roxy 11341, Cairo, Egypt

Received: 30 June 2025 Revised: 12 August 2025 Accepted: 19 August 2025 Published: 02 September 2025

In this work, a new spectral Galerkin approach to solving the time-fractional diffusion-wave equation (TFDWE) with non-homogeneous initial and boundary conditions is presented. A suitable transformation is used to convert the TFDWE governed by non-homogeneous conditions into a modified one governed by homogeneous conditions. New basis functions in terms of specific shifted Horadam polynomials are used. Some new definite integral formulas that are crucial to the numerical implementation are developed, and the Galerkin scheme is analyzed in detail to obtain the approximate solutions. A thorough convergence and error analysis of the proposed expansion is established. A number of numerical experiments are carried out to show the scheme's applicability and accuracy when compared to other methods.
- Horadam polynomials,
- diffusion equations,
- spectral methods,
- homogeneous conditions,
- convergence analysis
Citation: Waleed Mohamed Abd-Elhameed, Mohamed A. Abdelkawy, Naher Mohammed A. Alsafri, Ahmed Gamal Atta. Galerkin-based solution for the time-fractional diffusion-wave equation[J]. Electronic Research Archive, 2025, 33(9): 5179-5206. doi: 10.3934/era.2025232

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Abstract

In this work, a new spectral Galerkin approach to solving the time-fractional diffusion-wave equation (TFDWE) with non-homogeneous initial and boundary conditions is presented. A suitable transformation is used to convert the TFDWE governed by non-homogeneous conditions into a modified one governed by homogeneous conditions. New basis functions in terms of specific shifted Horadam polynomials are used. Some new definite integral formulas that are crucial to the numerical implementation are developed, and the Galerkin scheme is analyzed in detail to obtain the approximate solutions. A thorough convergence and error analysis of the proposed expansion is established. A number of numerical experiments are carried out to show the scheme's applicability and accuracy when compared to other methods.

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