Numerical Solution of second order hyperbolic telegraph equation via new Cubic Trigonometric B-Splines Approach

TL;DR

This paper introduces a new cubic trigonometric B-spline collocation method for solving second order hyperbolic telegraph equations, demonstrating stability, accuracy, and computational efficiency through various test problems.

Contribution

The paper develops a novel cubic trigonometric B-spline collocation scheme for hyperbolic telegraph equations, combining finite difference time discretization with spline-based spatial interpolation.

Findings

The scheme is unconditionally stable for a range of parameters.

Numerical results agree well with exact solutions and previous studies.

The method is computationally economical and effective for complex problems.

Abstract

This paper presents a new approach and methodology to solve the second order one dimensional hyperbolic telegraph equation with Dirichlet and Neumann boundary conditions using the cubic trigonometric B-spline collocation method. The usual finite difference scheme is used to discretize the time derivative. The cubic trigonometric B-spline basis functions are utilized as an interpolating function in the space dimension, with a weighted scheme. The scheme is shown to be unconditionally stable for a range of values using the von Neumann (Fourier) method. Several test problems are presented to confirm the accuracy of the new scheme and to show the performance of trigonometric basis functions. The proposed scheme is also computationally economical and can be used to solve complex problems. The numerical results are found to be in good agreement with known exact solutions and also with earlier studies.