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Article

Stability Analysis and Finite Difference Approximations for a Damped Wave Equation with Distributed Delay

1
Department of Mathematics, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia
2
Center for Integrative Petroleum Research (CIPR), King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia
Mathematics 2025, 13(17), 2714; https://doi.org/10.3390/math13172714 (registering DOI)
Submission received: 14 July 2025 / Revised: 13 August 2025 / Accepted: 21 August 2025 / Published: 23 August 2025
(This article belongs to the Special Issue Advances in Numerical Analysis of Partial Differential Equations)

Abstract

This paper presents a fully implicit finite difference scheme for the numerical approximation of a wave equation featuring strong damping and a distributed delay term. The discretization employs second-order accurate approximations in both time and space. Although implicit, the scheme does not ensure unconditional stability due to the nonlocal nature of the delayed damping. To address this, we perform a stability analysis based on Rouché’s theorem from complex analysis and derive a sufficient condition for asymptotic stability of the discrete system. The resulting criterion highlights the interplay among the discretization parameters, the damping coefficient, and the delay kernel. Two quadrature techniques, the composite trapezoidal rule (CTR) and the Gaussian quadrature rule (GQR), are employed to approximate the convolution integral. Numerical experiments validate the theoretical predictions and illustrate both stable and unstable dynamics across different parameter regimes.
Keywords: wave equation with delay; finite-difference method; Rouché’s theorem; time-distributed delay; asymptotic stability wave equation with delay; finite-difference method; Rouché’s theorem; time-distributed delay; asymptotic stability

Share and Cite

MDPI and ACS Style

Alotaibi, M. Stability Analysis and Finite Difference Approximations for a Damped Wave Equation with Distributed Delay. Mathematics 2025, 13, 2714. https://doi.org/10.3390/math13172714

AMA Style

Alotaibi M. Stability Analysis and Finite Difference Approximations for a Damped Wave Equation with Distributed Delay. Mathematics. 2025; 13(17):2714. https://doi.org/10.3390/math13172714

Chicago/Turabian Style

Alotaibi, Manal. 2025. "Stability Analysis and Finite Difference Approximations for a Damped Wave Equation with Distributed Delay" Mathematics 13, no. 17: 2714. https://doi.org/10.3390/math13172714

APA Style

Alotaibi, M. (2025). Stability Analysis and Finite Difference Approximations for a Damped Wave Equation with Distributed Delay. Mathematics, 13(17), 2714. https://doi.org/10.3390/math13172714

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