Space Discretization of Time-Fractional Telegraph Equation with Mamadu-Njoseh Basis Functions

Mamadu, Ebimene James; Njoseh, Ignatius N.; Ojarikre, Henrietta Ify

doi:10.4236/am.2022.139048

Cited by 7 publications

(11 citation statements)

References 18 publications

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“…Now, if we choose , i.e, if then , then (2.6) corresponds to the Ritz-Galenkin method. A special feature of the FEM as a particular Ritz-Galenkin method is that the simple functions in are piecewise polynomials [12]. To solve any integral or differential equation using FEM, the following steps are involved: (i) equivalent variational formulation of the stated problem.…”

Section: Finite Element Methodsmentioning

confidence: 99%

“…with usual definition of parameters as given in Mamadu et al [12]. Let a linear and continuous finite element space defined in [a, b] be given as .…”

Section: Finite Element Methods For Time Fractional Telegraph Equationmentioning

confidence: 99%

“…We study in this section some basic concepts and properties of fractional calculus. The properties of fractional derivatives and integrals of for , essential to this research are stated as follows [12] :…”

Section: Basic Preliminaries and Propertiesmentioning

confidence: 99%

“…From Mamadu et al [12], it was deduced that Choosing in (4.15) and using Lemma 3.2(i) and the Cauchy-Schwartz inequality (see [18], [19]), we obtain Using the Gronwall inequality (see [20], [21]) on (4.16), we get…”

Section: Convergence Analysismentioning

confidence: 99%

“…In Mamadu et al [12], a space discretization of time fractional telegraph equation with Mamadu-Njoseh basis functions was studied with detailed numerical illustrations for various values of the fractional order, where resulting numerical evidences were expressed in and maximum errors. The aim of this paper is to investigate the convergence of the space discretization of time fractional telegraph equation.…”

Section: Introductionmentioning

confidence: 99%

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Convergence Analysis of Space Discretization of Time Fractional Telegraph Equation

Mamadu¹,

Ojarikre²,

Njoseh³

2023

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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.

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Section: Finite Element Methodsmentioning

confidence: 99%

“…with usual definition of parameters as given in Mamadu et al [12]. Let a linear and continuous finite element space defined in [a, b] be given as .…”

Section: Finite Element Methods For Time Fractional Telegraph Equationmentioning

confidence: 99%

Section: Basic Preliminaries and Propertiesmentioning

confidence: 99%