A method based on meshless approach for the numerical solution of the two‐space dimensional hyperbolic telegraph equation

Abstract

Communicated by S. GeorgievIn the current article, we investigate the RBF solution of second-order two-space dimensional linear hyperbolic telegraph equation. For this purpose, we use a combination of boundary knot method (BKM) and analog equation method (AEM). The BKM is a meshfree, boundary-only and integration-free technique. The BKM is an alternative to the method of fundamental solution to avoid the fictitious boundary and to deal with low accuracy, singular integration and mesh generation. Also, on the basis of the AEM, the governing operator is substituted by an equivalent nonhomogeneous linear one with known fundamental solution under the same boundary conditions. Finally, several numerical results and discussions are demonstrated to show the accuracy and efficiency of the proposed method.where @ is the boundary of , which is a closed subset of R 2 and T is a positive number.When˛andˇare constants and˛>ˇ> 0, A D B D 1, Eqn (1) represents the two-dimensional telegraph equation which will be studied in the current investigation.Equation (1), referred as the second-order telegraph equation with constant coefficients, models the mixture between diffusion and wave propagation by introducing a term that accounts for the effects of finite velocity to standard heat or mass transport equation [1]. Problem (1) is commonly used in signal analysis for transmission and propagation of the electrical signals [2].Authors of [33,34] introduced the BKM, which eliminates the disadvantage of applying fictitious boundary and preserves all benefits of the MFS. The BKM employs the non-singular general solutions in place of the singular fundamental solutions. This method is truly meshfree, integration-free, mathematically simple, boundary-type and easy to implement. The BKM has been employed for a wide range of physical and engineering problems. In [35], the BKM is extended to solve the 2D Helmholtz and convection-diffusion problems under rather complex-shaped interior and exterior contours, which shows that unlike the MFS, the BKM is insensitive to geometric irregularity. This method is used in [36] for solving the Poisson equation . In that article, authors used the high-order general solutions of the Helmholtz and modified Helmholtz equations to evaluate the particular solution. In [37], the BKM is employed for solving the Cauchy problem associated with the inhomogeneous Helmholtz equation. For problems with noisy boundary data, the standard methods for solving matrix equations yield unstable results; thus, these authors [37] used truncated singular value decomposition to solve the resulting matrix equation and obtained a stable and accurate numerical solution. Authors of [38] investigated the regularization techniques for BKM, and their experiments show that Thikhonov regularization with generalized cross-validation approach is more efficient than other regularization techniques for the BKM solution of Helmholtz and modified Helmholtz equations. Authors of [39] used the geodesic distance instead of the standard isotropic Euc...

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