The purpose of this study is to develop an efficient and accurate numerical technique based on cubic B-spline collocation for solving the one-dimensional hyperbolic telegraph equation, which arises in the modeling of damped wave propagation phenomena in various physical applications.
The proposed method transforms the hyperbolic telegraph equation into a system of equations. Cubic B-spline basis functions are employed to approximate the spatial variable and its derivatives, while the collocation technique is used to obtain a system of differential equations with a tridiagonal coefficient matrix. The temporal discretization is carried out using the Crank–Nicolson finite difference scheme. The convergence and stability properties of the method are analyzed through theoretical investigations, including Von Neumann stability analysis. Error norms, such as (L2), (L∞) and root mean square errors, are computed to assess the accuracy of the numerical solutions.
The numerical experiments demonstrate that the proposed cubic B-spline collocation method provides highly accurate and reliable solutions for the hyperbolic telegraph equation. The method achieves a convergence order of O(h2 + ∆t) and exhibits unconditional stability. The obtained numerical results show excellent agreement with exact solutions and compare favorably with previously reported results in the literature.
This study focuses on the one-dimensional hyperbolic telegraph equation. Although the method is computationally efficient and robust, its applicability to higher-dimensional and more complex nonlinear telegraph-type equations requires further investigation. The study provides a foundation for extending B-spline-based collocation techniques to a broader class of partial differential equations.
This study presents an effective combination of cubic B-spline collocation and Crank–Nicolson time discretization for solving the hyperbolic telegraph equation. The method offers simplicity in implementation, reduced computational effort and high numerical accuracy. Its unconditional stability and favorable convergence properties make it a valuable numerical tool for solving challenging wave propagation problems.
