Mathematics and Statistics Vol. 11(2), pp. 245 - 251
DOI: 10.13189/ms.2023.110202
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Convergence Analysis of Space Discretization of Time Fractional Telegraph Equation

Ebimene James Mamadu ^*, Henrietta Ify Ojarikre , Ignatius Nkonyeasua Njoseh
Department of Mathematics, Delta State University, Abraka, Nigeria

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, H¹ – 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 [0,T] with a finite element space in Mamadu-Njoseh polynomial space, φ_m-1, of degree ≤m-1. 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 L₂ - norm between them to avoid a difficult Ritz operator construction to achieve the convergence of the method.

KEYWORDS
Time-Fractional Telegraph Equation, Caputo Derivative, Mamadu-Njoseh Basis Functions, Finite Element Method, Ritz Projection, Convergence

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Ebimene James Mamadu , Henrietta Ify Ojarikre , Ignatius Nkonyeasua Njoseh , "Convergence Analysis of Space Discretization of Time Fractional Telegraph Equation," Mathematics and Statistics, Vol. 11, No. 2, pp. 245 - 251, 2023. DOI: 10.13189/ms.2023.110202.

(b). APA Format:
Ebimene James Mamadu , Henrietta Ify Ojarikre , Ignatius Nkonyeasua Njoseh (2023). Convergence Analysis of Space Discretization of Time Fractional Telegraph Equation. Mathematics and Statistics, 11(2), 245 - 251. DOI: 10.13189/ms.2023.110202.