• Uday Singh

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 primary goal of this article is to introduce and evaluate a new approach for approximating the time-fractional telegraph equation using spline functions. Initially, an operational matrix technique based on the consolidation of Fibonacci wavelets and block pulse functions 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 using 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 quasilinearization linearization procedure. The discretization of the time-fractional derivative is done using the Caputo fractional derivative formula. The calculated solutions are obtained using a combination of the Caputo fractional derivative and a trigonometric Quintic B-spline. The main objective 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 using 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 numerical and exact solutions are near together, demonstrating the method's precision.

Time Fractional Telegraph Equation, Three-Dimensional, Quintic B-spline, Fibonacci Wavelets, Crank Nicholson

"NUMERICAL INVESTIGATION OF UNCONDITIONALLY STABLE SPLINE FUNCTION FOR THREE-DIMENSIONAL TIME-FRACTIONAL TELEGRAPH EQUATIONS", International Journal of Emerging Technologies and Innovative Research (www.jetir.org), ISSN:2349-5162, Vol.9, Issue 5, page no.g51-g71, May-2022, Available :http://www.jetir.org/papers/JETIR2205705.pdf

"NUMERICAL INVESTIGATION OF UNCONDITIONALLY STABLE SPLINE FUNCTION FOR THREE-DIMENSIONAL TIME-FRACTIONAL TELEGRAPH EQUATIONS", International Journal of Emerging Technologies and Innovative Research (www.jetir.org | UGC and issn Approved), ISSN:2349-5162, Vol.9, Issue 5, page no. ppg51-g71, May-2022, Available at : http://www.jetir.org/papers/JETIR2205705.pdf