Solution of nonlinear telegraph equation with discontinuities along the transmission line using meshless collocation method - Muhammad Asif; Tabassum Gul; Muhammad Bilal Riaz; Faisal Bilal (Partial Differential Equations in Applied Mathematics)
In this paper, we present a novel hybrid numerical technique combining radial basis functions (RBFs) and the finite difference method (FDM) for solving the one-dimensional telegraph interface model (TIM). The presence of interfaces or discontinuities along the transmission line requires additional boundary or interface conditions to accurately describe the voltage and current behaviors at these junctions. Our method incorporates these interface conditions, approximating spatial partial derivatives using RBFs and time derivatives using FDM. To validate our approach, we applied it to several benchmark linear and nonlinear TIMs. For linear models, the resulting algebraic system is solved using Gauss elimination, while for nonlinear models, we employed the quasi-Newton method to linearize the nonlinear terms. We evaluated the method’s performance by calculating the maximum absolute errors (MAEs) and root mean square errors (RMSEs) for different grid points (GPs). Additionally, we compared the true and estimate solutions through three dimensional (3D)graphs. The numerical results show that our technique provides simple implementation, quick convergence, and outstanding accuracy for both linear and nonlinear models. This hybrid approach proves to be a highly effective and reliable tool for solving telegraph interface problems.