On Convergence of Higher Order Differential-Difference Schemes for Telegraph Equation

The purpose of this article is to introduce an approximation method for solving the Dirichlet initial boundary value problem (IBVP) for the telegraph equation. The approach involves the idea of discretizing a spatial variable and utilizing the integral-interpolation method with specific fundamental functions. The original equation is multiplied by auxiliary functions, and subsequently interpolation and integration methods are applying on the spatial variable to generate system of ordinary differential equations (ODEs). Flexible applications of the Newton–Stirling and Hermite–Birkhoff interpolations are made to internal and close-to-boundary nodes. Additionally, the boundary conditions are automatically satisfied without the need for a separate approximation as in classical numerical methods like the grid method or the method of lines. As a result, the suggested schemes have a higher level of approximation. To prove the convergence of differential-difference schemes of high degree of accuracy, the logarithmic norm of the matrix is used.