Solutions of hyperbolic telegraph equation using radial basis function-finite difference method

This study introduces a novel Radial Basis Function–Finite Difference (RBF-FD) numerical framework for the solution of the hyperbolic telegraph equation, a fundamental model arising in reaction–diffusion phenomena and wave–diffusion processes. The proposed scheme employs a fully meshless formulation based on polyharmonic spline radial basis functions augmented with polynomials, thereby eliminating the need for shape-parameter selection that commonly hampers classical RBF methods. This design ensures enhanced numerical stability, and high-order accuracy while retaining the inherent flexibility of meshless discretization for handling irregular and complex geometries. Unlike traditional grid-based approaches, the presented RBF-FD method seamlessly accommodates scattered node distributions and time-dependent dynamics. The effectiveness and robustness of the scheme are demonstrated through four benchmark test problems, where numerical solutions exhibit excellent agreement with available analytical solutions. Detailed error analyses confirm consistently negligible errors across all examples. These results establish the RBF-FD approach as an efficient and versatile computational tool for solving hyperbolic-type partial differential equations on complex domains.