Spectral framework using modified shifted Chebyshev polynomials of the third-kind for numerical solutions of one- and two-dimensional hyperbolic telegraph equations

This investigation discusses a numerical approach to solving the hyperbolic telegraph equation in both one and two dimensions. Applying the Galerkin method is the basis of this approach. We use appropriate combinations of third-kind modified shifted Chebyshev polynomials (3KMSCPs) as basis functions to transform the governing partial differential equations into a collection of algebraic equations. Through spectral Galerkin techniques, we establish the convergence error to demonstrate that our algorithm is more effective and efficient. Five examples are examined to verify the effectiveness and resilience of the applied method by comparing errors and illustrating the results. Our results show that the current numerical solutions align closely with exact solutions. The current algorithm is simple to set up and is better suited to solving certain difficult partial differential equations.