Numerical solution and error analysis of the fractional nonlinear hyperbolic wave equation using the finite difference method

Hazal Yüksekkaya; Hazal Yüksekkaya

doi:10.3934/math.20261361

AIMS Mathematics

2026, Volume 11, Issue 10: 34963-34974. doi: 10.3934/math.20261361

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Numerical solution and error analysis of the fractional nonlinear hyperbolic wave equation using the finite difference method

Hazal Yüksekkaya ^,

Department of Mathematics, Hakkari University, Hakkari, Türkiye

Received: 06 August 2026 Revised: 12 September 2026 Accepted: 23 September 2026 Published: 26 October 2026
MSC : 35R30, 35R11, 65N06, 35Q55

In this work, we examined the numerical results for a nonlinear fractional hyperbolic-type wave equation governed by a time-fractional derivative of order $ \alpha $. For the fractional term, an L2-1$ \sigma $ memory discretization approach was integrated with standard central difference approximations of spatial second-order derivatives. Explicitly, the study formulates the discrete algebraic relations for the governing system, proves the comprehensive numerical scheme, and evaluates computational efficiency and accuracy across diverse spatial grid refinements.
- fractional nonlinear wave equation,
- finite difference method,
- L2-1$ \sigma $ memory coefficients,
- numerical analysis,
- error estimation
Citation: Hazal Yüksekkaya. Numerical solution and error analysis of the fractional nonlinear hyperbolic wave equation using the finite difference method[J]. AIMS Mathematics, 2026, 11(10): 34963-34974. doi: 10.3934/math.20261361

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Abstract

In this work, we examined the numerical results for a nonlinear fractional hyperbolic-type wave equation governed by a time-fractional derivative of order $ \alpha $. For the fractional term, an L2-1$ \sigma $ memory discretization approach was integrated with standard central difference approximations of spatial second-order derivatives. Explicitly, the study formulates the discrete algebraic relations for the governing system, proves the comprehensive numerical scheme, and evaluates computational efficiency and accuracy across diverse spatial grid refinements.

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