Shifted Gegenbauer–Galerkin algorithm for hyperbolic telegraph type equation

This paper is concerned with a numerical spectral solution to a one-dimensional linear telegraph type equation with constant coefficients. An efficient Galerkin algorithm is implemented and analyzed for treating this type of equations. The philosophy of utilization of the Galerkin method is built on picking basis functions that are consistent with the corresponding boundary conditions of the telegraph type equation. A suitable combination of the orthogonal shifted Gegenbauer polynomials is utilized. The proposed method produces systems of especially inverted matrices. Furthermore, the convergence and error analysis of the proposed expansion are investigated. This study was built on assuming that the solution to the problem is separable. The paper ends by checking the applicability and effectiveness of the proposed algorithm by solving some numerical examples.