Numerical analysis of a two-parameter fractional telegraph equation

Ford, Neville J.; Rodrigues, M. M.; Xiao, Jingyu; Yan, Yubin

doi:10.1016/j.cam.2013.02.009

Cited by 29 publications

(22 citation statements)

References 20 publications

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“…Actually, some authors have already studied the numerical solutions to some kinds of time or space fractional telegraph equations, such as C. Li [14], Z. Zhao [15], N. J. Ford [16], A. Sevimlican [17], and M. Dehghan [18]. The fractional telegraph equation we consider here is different from all of which they discussed in their papers.…”

Section: Introductionmentioning

confidence: 99%

Fractional Difference Approximations for Time-Fractional Telegraph Equation

Liu¹

2018

JAMP

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Section: Introductionmentioning

confidence: 99%

Fractional Difference Approximations for Time-Fractional Telegraph Equation

Liu¹

2018

JAMP

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“…Considering the particular case of β = 2 (see [14]), the wave solution is given by 5) and the corresponding quantum mechanical probability density is given as…”

Section: )mentioning

confidence: 99%

“…For example, the one-dimensioc 2013 Diogenes Co., Sofia pp. 454-468 , DOI: 10.2478/s13540-013-0028-5 nal time-fractional diffusion-wave equation [1,13], the fractional telegraph equation [5] and the multi-dimensional fractional Schrödinger equation [14] have all provided effective models. We draw attention also to the important work of Francesco Mainardi (see for example the recent works [6,11]) to whom this paper is dedicated.…”

Section: Introductionmentioning

confidence: 99%

A numerical method for the fractional Schrödinger type equation of spatial dimension two

Ford

Rodrigues

Vieira

2013

fcaa

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This work focuses on an investigation of the (n + 1)−dimensional timedependent fractional Schrödinger type equation. In the early part of the paper, the wave function is obtained using Laplace and Fourier transform methods and a symbolic operational form of the solutions in terms of Mittag-Leffler functions is provided. We present an expression for the wave function and for the quantum mechanical probability density. We introduce a numerical method to solve the case where the space component has dimension two. Stability conditions for the numerical scheme are obtained.MSC 2010 : Primary 35R11; Secondary 42A38, 33E12, 65M06, 47H10

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“…In this paper, we will use the idea in Diethelm [6] to define a finite difference method for solving (1)-(3), see also our recent works in [13], [14]. We first express the fractional derivative by using the Hadamard finite-part integral, i.e., with 1 < α < 2,…”

Section: Introductionmentioning

confidence: 99%

“…There is considerable interest in developing various numerical methods for solving space-fractional partial differential equations in literature: the finite difference methods [2], [13], [14] - [15], [18] - [19], [21] - [25], [31] - [35], the finite element methods, [3] - [4], [9] - [12] and the spectral methods [16]- [17]. By using shifted Grünwald -Letnikov formulae (6) and (7), Meerschaert and Tadjeran [24] introduced a finite difference method for solving two-sided space-fractional partial differential equations (1)- (3) and proved that the convergence order of spatial discretisation is O(∆x).…”

Section: Introductionmentioning

confidence: 99%

An Algorithm for the Numerical Solution of Two-Sided Space-Fractional Partial Differential Equations

Ford

Pal²,

Yan

2015

Computational Methods in Applied Mathematics

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We introduce an algorithm for solving two-sided space-fractional partial differential equations. The space-fractional derivatives we consider here are left-handed and right-handed Riemann–Liouville fractional derivatives which are expressed by using Hadamard finite-part integrals. We approximate the Hadamard finite-part integrals by using piecewise quadratic interpolation polynomials and obtain a numerical approximation of the space-fractional derivative with convergence order ${O(\Delta x^{3- \alpha })}$, ${1<\alpha <2}$. A shifted implicit finite difference method is applied for solving the two-sided space-fractional partial differential equation and we prove that the order of convergence of the finite difference method is ${O (\Delta t + \Delta x^{\min (3- \alpha , \beta )})}$, ${1< \alpha <2}$, ${\beta >0}$, where ${\Delta t,\Delta x}$ denote the time and space stepsizes, respectively. Numerical examples where the solutions have varying degrees of smoothness are presented and compared with the exact analytical solution to compare the practical performance of the method with the theoretical order of convergence.

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Numerical analysis of a two-parameter fractional telegraph equation

Cited by 29 publications

References 20 publications

Fractional Difference Approximations for Time-Fractional Telegraph Equation

Fractional Difference Approximations for Time-Fractional Telegraph Equation

A numerical method for the fractional Schrödinger type equation of spatial dimension two

An Algorithm for the Numerical Solution of Two-Sided Space-Fractional Partial Differential Equations

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