An Explicit and Implicit Discretized Forward Difference Schemes with Cubic B-Spline Functions for the Time Fractional Telegraph Equation with a Damping Coefficient

The motivation behind this work is the recent advances in literature for seeking numerical techniques for time fractional telegraph equations. A coupled technique combining the finite difference method, <img src=image/12640095_01.gif>-weighted method and cubic B-splines is underexplored in literature specific for time fractional telegraph equations. Also, there are limited comparative studies between cubic B-splines-based numerical schemes and other hybrid numerical methods for time fractional telegraph equation. Most studies focus on deploying standard cubic B-splines as basis functions and not orthogonal polynomials. This paper aims to find an approximate solution to the time fractional telegraph equation using discretized explicit and implicit forward difference schemes combined with the <img src=image/12640095_01.gif>-weighted method. To achieve this, we employed cubic B-splines as basis functions to enhance the efficiency and accuracy of the solutions. The orthogonal collocation method is then employed to generate a linear system of equations, which are solved using numerical methods for the approximate solution. Two numerical examples illustrate the efficiency of this method, showing how the schemes rapidly converge when the step size is reduced. Interestingly, the results show that the implicit system outperforms the explicit design. Furthermore, the paper advances the area by proposing a new framework for solving time fractional telegraph equations that involves the application of orthogonal polynomials as basis functions, providing better accuracy and convergence.