A uniform hyperbolic polynomial B-spline approach for solving the fractional diffusion-wave equations in the Caputo-Fabrizio sense

Muhammad Umar Manzoor; Muhammad Yaseen; Muath Awadalla; Hajer Zaway; Muhammad Umar Manzoor; Muhammad Yaseen; Muath Awadalla; Hajer Zaway

doi:10.3934/math.2025765

AIMS Mathematics

2025, Volume 10, Issue 7: 17049-17081. doi: 10.3934/math.2025765

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A uniform hyperbolic polynomial B-spline approach for solving the fractional diffusion-wave equations in the Caputo-Fabrizio sense

1.
Department of Mathematics, University of Sargodha, Sargodha, 40100, Pakistan
2.
Department of Mathematics and Statistics, College of Science, King Faisal University, Al Ahsa 31982, Saudi Arabia

Received: 23 May 2025 Revised: 19 July 2025 Accepted: 21 July 2025 Published: 30 July 2025
MSC : 26A33, 35R11, 65D07, 65M70

Piecewise polynomial functions serve as powerful tools for function approximation and the numerical solution of differential equations. In this study, we presented a robust numerical method for solving the time-fractional diffusion-wave equation involving the Caputo-Fabrizio fractional derivative. The proposed scheme combines the uniform hyperbolic polynomial B-spline basis for spatial discretization with a $ \theta $-weighted finite difference approach for temporal integration. The uniform hyperbolic polynomial B-spline, an advanced generalization of B-splines, integrates hyperbolic functions to enhance smoothness and flexibility, making it especially well-suited for problems exhibiting hyperbolic behavior. Rigorous stability and convergence analyses were carried out to ensure the reliability of the method. To demonstrate its effectiveness, the scheme was applied to several benchmark problems. Numerical results reveal that the proposed approach is highly accurate and computationally efficient.
Citation: Muhammad Umar Manzoor, Muhammad Yaseen, Muath Awadalla, Hajer Zaway. A uniform hyperbolic polynomial B-spline approach for solving the fractional diffusion-wave equations in the Caputo-Fabrizio sense[J]. AIMS Mathematics, 2025, 10(7): 17049-17081. doi: 10.3934/math.2025765

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Abstract

Piecewise polynomial functions serve as powerful tools for function approximation and the numerical solution of differential equations. In this study, we presented a robust numerical method for solving the time-fractional diffusion-wave equation involving the Caputo-Fabrizio fractional derivative. The proposed scheme combines the uniform hyperbolic polynomial B-spline basis for spatial discretization with a $ \theta $-weighted finite difference approach for temporal integration. The uniform hyperbolic polynomial B-spline, an advanced generalization of B-splines, integrates hyperbolic functions to enhance smoothness and flexibility, making it especially well-suited for problems exhibiting hyperbolic behavior. Rigorous stability and convergence analyses were carried out to ensure the reliability of the method. To demonstrate its effectiveness, the scheme was applied to several benchmark problems. Numerical results reveal that the proposed approach is highly accurate and computationally efficient.

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