Convergence Analysis of Space Discretization of Time Fractional Telegraph Equation

Abstract

The role of fractional differential equations in the advancement of science and technology cannot be overemphasized. The time fractional telegraph equation (TFTE) is a hyperbolic partial differential equation (HPDE) with applications in frequency transmission lines such as the telegraph wire, radio frequency, wire radio antenna, telephone lines, and among others. Consequently, numerical procedures (such as finite element method, H 1 -Galerkin mixed finite element method, finite difference method, and among others) have become essential tools for obtaining approximate solutions for these HPDEs. It is also essential for these numerical techniques to converge to a given analytic solution to certain rate. The Ritz projection is often used in the analysis of stability, error estimation, convergence and superconvergence of many mathematical procedures. Hence, this paper offers a rigorous and comprehensive analysis of convergence of the space discretized time-fractional telegraph equation. To this effect, we define a temporal mesh on with a finite element space in Mamadu-Njoseh polynomial space, , of degree An interpolation operator (also of a polynomial space) was introduced along the fractional Ritz projection to prove the convergence theorem. Basically, we have employed both the fractional Ritz projection and interpolation technique as superclose estimate in -norm between them to avoid a difficult Ritz operator construction to achieve the convergence of the method.

An Error Analysis of Implicit Finite Difference Method with Mamadu-Njoseh Basis Functions for Time Fractional Telegraph Equation

“…Tables 1 and 2 shows the maximum error of the proposed method at . In comparison with the finite difference method in [30][31][32] show the superiority of the proposed method with maximum errors of order and , respectively. Results are also presented in graphs showing the comparison of results as shown in the Figs.…”

An Orthogonal Polynomial Based Iterative Procedure for Finding the Root of the Equation f (x) = 0

“…This paper offers to derive and implement new iterative scheme that performs better than other methods, especially the Newton-Raphson method. The new scheme is orthogonal polynomials based, that is, applying Mamadu-Njoseh polynomials [10][11][12][13][14][15][16][17] as basis functions as series solutions subjects to three relevant conditions. The class of algebraic and transcendental equations and possibly any nonlinear equations are employed to test the rate of convergence, accuracy and effectiveness of the new scheme.…”