Fractional Difference Approximations for Time-Fractional Telegraph Equation

The exact solution is calculated for fractional telegraph partial differential equation depend on initial boundary value problem. Stability estimates are obtained for this equation. Crank-Nicholson difference schemes are constructed for this problem. The stability of difference schemes for this problem is presented. This technique has been applied to deal with fractional telegraph differential equation defined by Caputo fractional derivative for fractional orders α = 1.1, 1.5, 1.9. Numerical results confirm the accuracy and effectiveness of the technique.

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“…Remark 3. Applying the method in [16,17], we can get the convergence of the method from stability and consistency of the proposed method.…”

Section: Crank-nicolson Difference Scheme and Its Stabiltymentioning

confidence: 99%

“…In [15], the time-fractional advection dispersion equations have been presented. In [16], Liu has studied fractional difference approximations for time-fractional telegraph equation. Modanli and Akgül [12] have worked the second-order partial differential equations by two accurate methods.…”

Section: Introductionmentioning

confidence: 99%

On Solutions of Fractional order Telegraph Partial Differential Equation by Crank-Nicholson Finite Difference Method

Modanlı

Akgül

2020

Applied Mathematics and Nonlinear Sciences

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“…Wang and Mei [21] proposed a method for solving TFTE via the Legendre spectral Galerkin method and generalized finite difference scheme. Liu [22] discussed the Caputo fractional difference formula and Grünwald difference formula for the solution of TFTE.…”

Section: Application and Literature Reviewmentioning

confidence: 99%

Extended cubic B-splines in the numerical solution of time fractional telegraph equation

et al. 2019

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A finite difference scheme based on extended cubic B-spline method for the solution of time fractional telegraph equation is presented and discussed. The Caputo fractional formula is used in the discretization of the time fractional derivative. A combination of the Caputo fractional derivative together with an extended cubic B-spline is utilized to obtain the computed solutions. The proposed scheme is shown to possess the unconditional stability property with second order convergence. Numerical results demonstrate the applicability, simplicity and the strength of the scheme in solving the time fractional telegraph equation with accuracies very close to the exact solutions.and ∂ α u(x,t) ∂t α denote the Caputo fractional derivative of order 2α and α, respectively, which will be defined in Sect. 2.This model has been developed to overcome the shortcomings of the classical telegraph equation which might not adequately model the abnormal diffusion phenomena during a long transmission process in a transmission line [2].

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“…A fairly large number of works on approximate solutions, for the above linear problem, can be found in the literature using Legendre polynomials, the Chebyshev Tau method, decomposition method, homotopy perturbation method, He's variational iteration method, Rothe-Wavelet-Galerkin method, Haar wavelets, numerical spectral Legendre-Galerkin algorithm, finite difference scheme approximation, Sinc-collocation techniques, Bernstein polynomials operational matrices, the Caputo fractional difference formula and Grünwald difference, and the natural transform decomposition method [32][33][34][35][36][37][38].…”

Section: Introductionmentioning

confidence: 99%

Well‐posedness and Mittag–Leffler stability for a nonlinear fractional telegraph problem

Othmani

Tatar

2023

Asian Journal of Control

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The ordinary diffusion models in continuous media are derived from basic principles: Conservation of energy and momentum. The classical telegraph equation is one of the earliest and most commonly used equation to describe the transmission of electromagnetic waves. In fractal and disordered media, to capture the complex diffusion feature, we consider rather fractional models. Indeed, the mean square displacement in anomalous media is no longer proportional to the time but rather proportional to a power of time whose exponent may be between zero and one or between one and two. This is the case for the present fractional telegraph equation in the presence of non‐negligible voltage wave. Here, the well‐posedness and stability of a telegraph problem with two time‐fractional derivatives is discussed. In addition to being a noninteger problem, the equation in the model is subject to a nonlinear source as well as a nonlinear lower order fractional derivative term. First, the well‐posedness in an appropriate underlying space is established by the use of resolvent operators. Then, it is proved that, despite the presence of these nonlinearities, solutions are Mittag–Leffler stable. This confirms (and extends) the fact that the lower order term plays the role of a damper in the fractional case. To this end, we combine the energy method with some properties of the Caputo fractional derivative.

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Fractional Difference Approximations for Time-Fractional Telegraph Equation

Abstract: In this paper, we approximate the solution to time-fractional telegraph equation by two kinds of difference methods: the Grünwald formula and Caputo fractional difference.

Cited by 5 publications

References 19 publications

On Solutions of Fractional order Telegraph Partial Differential Equation by Crank-Nicholson Finite Difference Method

On Solutions of Fractional order Telegraph Partial Differential Equation by Crank-Nicholson Finite Difference Method

Extended cubic B-splines in the numerical solution of time fractional telegraph equation

Well‐posedness and Mittag–Leffler stability for a nonlinear fractional telegraph problem

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