A High Order Numerical Scheme for Time-Fractional Telegraph Equation Via Cubic Spline in Tension

In this work, we examine the numerical solution of the time-fractional telegraph equation by applying a higher-order numerical scheme via the tension spline method. The fractional order derivative of two different orders is analyzed using Caputo's definition. Moreover, the numerical scheme formation was carried out using spline functions incorporating the tension parameter in the spatial direction and the discretization technique based on a finite difference approach in the temporal direction. The proposed scheme includes some parameters, and by a suitable choice of parameters, its order can be enhanced from two to four in the spatial direction. The present technique is proven unconditionally stable through meticulous analysis. The convergence analysis is also demonstrated using the Fourier series. Numerical results of two test examples are presented through tables and plots that show the proficiency of the proposed numerical scheme and validate our theoretical findings.