Numerical investigation of unconditionally stable spline function for three-dimensional time-fractional telegraph equations

Telegraph equations are hyperbolic partial differential equations that may be used to represent reaction-diffusion processes in a variety of engineering and biological disciplines. The development of numerical techniques for telegraph-type equations has received a lot of interest in the literature in recent years. The main objective of this article is to introduce and evaluate a new approach for approximating the time-fractional telegraph equation utilizing spline functions. Initially, an operational matrix method based on the consolidation of block pulse functions and Fibonacci wavelets is presented to derive the solutions to Time-Fractional Telegraph Equations (TFTs). The suggested approach converts the fractional model into an algebraic equation system that can be solved utilizing the Newton iteration method. The Crank Nicolson approach is also offered for the solution of three-dimensional time-fractional telegraph equations using the Trigonometric Quintic B-spline (TQBS). The rationale behind using the collocation method is to select specific collocation spots where the differential equation is fulfilled exactly. The suggested technique combats nonlinearity by employing a quasi-linearization procedure. The time-fractional derivative's discretization is done using the Caputo fractional derivative formula. The calculated solutions are obtained using a combination of a trigonometric Quintic B-spline and the Caputo fractional derivative. The primary goal is to verify the well-posedness and produce a numerical solution for an initial-boundary value issue for a hyperbolic equation using finite-difference methods. Accordingly, the research developed the exponentially fitted approach for solving initial boundary value problems utilizing finite difference formulae and temporal frequencies. The scheme's convergence is demonstrated using normal analytical approaches, demonstrating that the method is unconditionally stable and has an order of convergence. MATLAB software is used to run the numerical simulations. Two model examples with boundary layer behaviour are investigated to support the theoretical conclusion. Furthermore, the graphs show that exact and numerical solutions are near together, demonstrating the method's precision.