A Rothe-Chebyshev collocation algorithm for the hyperbolic telegraphic type equations with variable coefficients

We construct a semi-discretized spectral approach for the second-order telegraphic-type equations with Dirichlet or Neumann boundary conditions. The successive method of Rothe is first employed for the temporal discretization procedure to transform the model equations into a system of boundary value problems. Subsequently, the spectral matrix procedure utilizing the shifted modified Chebyshev polynomials (SMCPs) is formulated for the spatial variable. The family of discrete solutions obtained by the hybrid Rothe-SMCPs algorithm is demonstrated to exhibit uniform convergence to the continuous solution of order $O(\Delta \tau +R^{-3})$. In this context, Δτ signifies the time step, while R represents the number of SMCPs employed in the approximation procedure. Simulation experiments are carried out to highlight the strong agreement between the numerical results and theoretical predictions. The numerical results utilizing a larger time-step size exhibit greater accuracy compared to the computational values available in existing research works.