Capturing Discontinuities with Precision: A Numerical Exploration of 3D Telegraph Interface Models via Multi-Resolution Technique

Khawaja Shams Ul Haq, Muhammad Asif (), Muhammad Faheem and Ioan-Lucian Popa ()
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Khawaja Shams Ul Haq: Department of Mathematics, University of Peshawar, Peshawar 25120, Pakistan
Muhammad Asif: Department of Mathematics, University of Peshawar, Peshawar 25120, Pakistan
Muhammad Faheem: Higher Education Department, Govt. Degree College Badaber, Peshawar 25000, Pakistan
Ioan-Lucian Popa: Department of Computing, Mathematics and Electronics, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, Romania

Mathematics, 2025, vol. 13, issue 15, 1-27

Abstract: This study presents a hyperbolic three-dimensional telegraph interface model with regular interfaces, numerically solved using a hybrid scheme that integrates Haar wavelets and the finite difference method. Spatial derivatives are approximated via a truncated Haar wavelet series, while temporal derivatives are discretized using the finite difference method. For linear problems, the resulting algebraic system is solved using Gauss elimination; for nonlinear problems, Newton’s quasi-linearization technique is applied. The method’s accuracy and stability are evaluated through key performance metrics, including the maximum absolute error, root mean square error, and the computational convergence rate R c ( M ) , across various collocation point configurations. The numerical results confirm the proposed method’s efficiency, robustness, and capability to resolve sharp gradients and discontinuities with high precision.

Keywords: Haar wavelet; telegraph interface model; partial differential equations; hyperbolic equations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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