Spectral Galerkin treatment of linear one-dimensional telegraph type problem via the generalized Lucas polynomials

The major goal of this research is to develop and test a numerical technique for solving a linear one-dimensional telegraph problem. The generalized polynomials, namely, the generalized Lucas polynomials are selected as basis functions. To solve the linear one-dimensional telegraph type equation, we solve instead its corresponding integral equation via the application of the spectral Galerkin method that serves to convert the equation with its underlying conditions into a system of linear algebraic equations that may be solved by a suitable numerical solver. The convergence and error analysis of the generalized Lucas expansion are discussed in depth. The current analysis is based on the assumption that the problem’s solution is separable. Finally, some explanatory numerical examples are displayed together with comparisons to some other articles, to demonstrate the suggested method’s validity, applicability, and accuracy.