A new spectral Galerkin method for solving the two dimensional hyperbolic telegraph equation

Telegraph equation is more suitable than ordinary diffusion equation in modeling reaction-diffusion for several branches of sciences and engineering. In this paper, a new numerical technique is proposed for solving the second order two dimensional hyperbolic telegraph equation subject to initial and Dirichlet boundary conditions. Firstly, a time discrete scheme based on the finite difference method is obtained. Unconditional stability and convergence of this semi-discrete scheme are established. Secondly, a fully discrete scheme is obtained by the Sinc-Galerkin method and the problem is converted into a Sylvester matrix equation. Especially, when a symmetric Sinc-Galerkin method is used, the resulting matrix equation is a discrete ADI model problem. Then, the alternating-direction Sinc-Galerkin (ADSG) method is applied for solving this matrix equation. Also, the exponential convergence rate of Sinc-Galerkin method is proved. Finally, some examples are given to illustrate the accuracy and efficiency of proposed method for solving such types of differential equations compared to some other well-known methods.