Numerical solutions of hyperbolic telegraph equation by using the Bessel functions of first kind and residual correction

In this study, a collocation method is presented to solve one-dimensional hyperbolic telegraph equation. The problem is given by hyperbolic telegraph equation under initial and boundary conditions. The method is based on the Bessel functions of the first kind. Using the collocation points and the operational matrices of derivatives, we reduce the problem to a set of linear algebraic equations. The determined coefficients from this system give the coefficients of the approximate solution. Also, an error estimation method is presented for the considered problem and the method. By using the residual function and the original problem, an error problem is constructed and thus the error function is estimated. By aid of the estimated function, the approximated solution is improved. Numerical examples are given to demonstrate the validity and applicability of the proposed method and also, the comparisons are made with the known results.