Numerical methods for solving the multi-term time-fractional wave-diffusion equation

doi:10.2478/S13540-013-0002-2

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Numerical methods for solving the multi-term time-fractional wave-diffusion equation

Journal Article•DOI•

Numerical methods for solving the multi-term time-fractional wave-diffusion equation

Fawang Liu¹, Mark M. Meerschaert², Robert J. McGough², Pinghui Zhuang³, Qingxia Liu³ - Show less +1 more•Institutions (3)

Queensland University of Technology¹, Michigan State University², Xiamen University³

01 Mar 2013-Fractional Calculus and Applied Analysis (SP Versita)-Vol. 16, Iss: 1, pp 9-25

TL;DR: Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations and can be extended to other kinds of themulti-term fractional time-space models with fractional Laplacian.

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Abstract: In this paper, the multi-term time-fractional wave-diffusion equations are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.

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Book Chapter•DOI•

Fractional Differential Equations

[...]

Tadeusz Kaczorek¹, Krzysztof Rogowski¹•Institutions (1)

Bialystok University of Technology¹

01 Jan 2015

3,828 citations

Journal Article•DOI•

A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations

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Ali H. Bhrawy¹, Mahmoud A. Zaky•Institutions (1)

Beni-Suef University¹

15 Jan 2015-Journal of Computational Physics

TL;DR: This paper proposes and analyzes an efficient operational formulation of spectral tau method for multi-term time-space fractional differential equation with Dirichlet boundary conditions using shifted Jacobi operational matrices of Riemann-Liouville fractional integral, left-sided and right-sided Caputo fractional derivatives.

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277 citations

Cites background or methods or result from "Numerical methods for solving the m..."

...Consider the following power law wave equation [30]:...
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...To demonstrate the accuracy of the proposed method, in Table 3 we compare maximum absolute errors of the present method at two choices of Jacobi parameters by selecting a few terms of the shifted Jacobi polynomial expansion N = M = 4, 6, 8, 10 together with the results obtained by using fractional predictor–corrector method FPCM-1 [30] and FPCM-2 [30], for y = 1....
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...5) h−1 FPCM-2 [30] FPCM-1 [30] 4 2....
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...We also compare the results given from our scheme and those reported in the literature such as difference scheme with spline function [73], implicit difference approximation [18] and fractional predictor–corrector method [30]....
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...The power law wave equation [30] is used to model sound wave propagation in anisotropic media that exhibits frequency dependent attenuation α(ω)....
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Journal Article•DOI•

The Galerkin finite element method for a multi-term time-fractional diffusion equation

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Bangti Jin¹, Raytcho D. Lazarov², Yikan Liu³, Zhi Zhou²•Institutions (3)

University College London¹, Texas A&M University², University of Tokyo³

15 Jan 2015-Journal of Computational Physics

TL;DR: The initial/boundary value problem for a diffusion equation involving multiple time-fractional derivatives on a bounded convex polyhedral domain is considered and nearly optimal error estimates for both cases of initial data and inhomogeneous term are derived.

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235 citations

Cites methods from "Numerical methods for solving the m..."

...We also refer to [24] for a numerical scheme based on a fractional predictor–corrector method for the multi-term time fractional wave-diffusion equation....
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Journal Article•DOI•

Fractional spectral collocation method

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Mohsen Zayernouri¹, George Em Karniadakis¹•Institutions (1)

Brown University¹

14 Jan 2014-SIAM Journal on Scientific Computing

TL;DR: A new family of interpolants are introduced, called fractional Lagrange interpolants, which satisfy the Kronecker delta property at collocation points and are developed as an exponentially accurate fractional spectral collocation method for solving steady-state and time-dependent fractional PDEs (FPDEs).

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Abstract: We develop an exponentially accurate fractional spectral collocation method for solving steady-state and time-dependent fractional PDEs (FPDEs). We first introduce a new family of interpolants, called fractional Lagrange interpolants, which satisfy the Kronecker delta property at collocation points. We perform such a construction following a spectral theory recently developed in [M. Zayernouri and G. E. Karniadakis, J. Comput. Phys., 47 (2013), pp. 2108--2131] for fractional Sturm--Liouville eigenproblems. Subsequently, we obtain the corresponding fractional differentiation matrices, and we solve a number of linear FODEs in addition to linear and nonlinear FPDEs to investigate the numerical performance of the fractional collocation method. We first examine space-fractional advection-diffusion problem and generalized space-fractional multiterm FODEs. Next, we solve FPDEs, including the time- and space-fractional advection-diffusion equation, time- and space-fractional multiterm FPDEs, and finally the space...

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229 citations

Cites background from "Numerical methods for solving the m..."

...The notion of fractional derivatives has been rapidly extended to many fractional partial differential equations (FPDEs), such as the fractional Burgers equation [30], the fractional Fokker–Planck equation [1], the fractional advection-diffusion equation [10], and fractional-order multiterm equations [22]....
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Journal Article•DOI•

A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations

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Ali H. Bhrawy¹, Eid H. Doha², Dumitru Baleanu, S. S. Ezz-Eldien³•Institutions (3)

Beni-Suef University¹, Cairo University², Modern Academy In Maadi³

15 Jul 2015-Journal of Computational Physics

TL;DR: An efficient and accurate spectral numerical method is presented for solving second-, fourth-order fractional diffusion-wave equations and fractional wave equations with damping based on Jacobi tau spectral procedure together with the Jacobi operational matrix for fractional integrals, described in the Riemann-Liouville sense.

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186 citations

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References

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Open Access

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Book•

Fractional differential equations

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Igor Podlubny

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Journal Article•DOI•

The random walk's guide to anomalous diffusion: a fractional dynamics approach

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Ralf Metzler¹, Joseph Klafter¹•Institutions (1)

Tel Aviv University¹

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TL;DR: Fractional kinetic equations of the diffusion, diffusion-advection, and Fokker-Planck type are presented as a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns.

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7,412 citations

"Numerical methods for solving the m..." refers methods in this paper

...Generalized fractional partial differential equations have been used for describing important physical phenomena (see [1, 2, 10, 11, 14, 23, 24, 27])....
[...]

Book Chapter•DOI•

Fractional Differential Equations

[...]

Tadeusz Kaczorek¹, Krzysztof Rogowski¹•Institutions (1)

Bialystok University of Technology¹

01 Jan 2015

3,828 citations

"Numerical methods for solving the m..." refers background in this paper

...1) where x and t are the space and time variables, k is an arbitrary positive constant, f(x, t) is a sufficiently smooth function, 0 < α ≤ 2 and Dα t is a Caputo fractional derivative of order α defined as [28]...
[...]
...The Riemann-Liouville fractional derivative ∂ t p = 0D y t p of order y (0 ≤ m − 1 < y < m) is defined as (see [28]):...
[...]
...The relationship between the Caputo fractional derivative and the Riemann-Liouville fractional derivative (see [28]) is...
[...]

Journal Article•DOI•

Analysis of Fractional Differential Equations

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Kai Diethelm¹, Neville J. Ford²•Institutions (2)

Braunschweig University of Technology¹, Chester College of New England²

15 Jan 2002-Journal of Mathematical Analysis and Applications

TL;DR: In this paper, the authors discuss existence, uniqueness, and structural stability of solutions of nonlinear differential equations of fractional order, and investigate the dependence of the solution on the order of the differential equation and on the initial condition.

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3,047 citations