Finite Difference Method Solves Two-Dimensional Nonlinear Fractional Telegraph Equations

In a research paper published in the journal Boundary Value Problems, mathematicians Mahmut Modanli and Mousa J. Huntul investigated an inverse source problem for a two-dimensional nonlinear fractional-order telegraph equation. The investigated model incorporates mixed fractional Caputo derivatives alongside an unknown time-dependent source term that must be accurately determined. The model under consideration features a nonlinear diffusion operator formulated in divergence form, coupled with an unknown time-dependent source term.

The authors explain that such mathematical models are essential for accurately describing complex physical phenomena where standard classical diffusion equations prove fundamentally insufficient to capture the underlying dynamics. Specifically, the two-dimensional nonlinear fractional telegraph model provides an appropriate mathematical description for wave propagation through complex, heterogeneous media. Furthermore, the formulation is necessary for characterizing stress relaxation processes in viscoelastic materials and for investigating various anomalous transport phenomena that cannot be captured by standard integer-order diffusion models.

To address the challenges posed by this inverse source problem, Modanli and Huntul constructed a numerical finite difference scheme based on the L2-1 approximation and central difference discretizations. The finite difference scheme is based on the L2-1 approximation for the fractional Caputo derivatives. Central difference approximations are employed for the spatial discretization.

Stability estimates of the proposed scheme are established in the discrete L2 norm. The convergence of the method is rigorously proven under appropriate mesh constraints.

To substantiate their theoretical derivations, the authors carried out several numerical experiments that also served to illustrate the underlying physical behavior of the solution. These computational simulations examined multiple physical characteristics of the modeled system, including the evolution of energy, the distribution of flux density, and the specific influences of nonlinear diffusion effects over time. Finally, the numerical results obtained from the scheme were directly compared with the corresponding exact solution, demonstrating the strong accuracy and practical efficiency of the proposed finite difference method for solving this two-dimensional inverse fractional telegraph problem.

Subject of Research: Numerical solutions for two-dimensional fractional telegraph inverse problems

Article Title: Inverse source problem for 2D nonlinear mixed fractional-order telegraph equations using finite difference method

Article References: Modanli, M., & Huntul, M. J. (2026). Inverse source problem for 2D nonlinear mixed fractional-order telegraph equations using finite difference method. Boundary Value Problems. https://doi.org/10.1186/s13661-026-02371-0

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DOI: 10.1186/s13661-026-02371-0

Keywords: inverse source problems, fractional telegraph equations, finite difference method, Caputo derivatives, nonlinear diffusion, mesh convergence, viscoelastic materials, wave propagation, Inverse, source, problem, nonlinear