Absorbing boundary conditions for the fractional wave equation

Dea, John R.

doi:10.1016/j.amc.2013.03.113

Cited by 8 publications

(6 citation statements)

References 36 publications

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“…where the initial data u 0 and the source term f (x, t) are assumed to be compactly supported in the interval Ω i := {x|x l < x < x r }. To solve this problem using a finite difference scheme, one needs to truncate the computational domain to a finite interval and impose some boundary conditions at the end points, see [3,4,8,13,24,17,18,21]. The exact nonreflecting boundary conditions for the above problem have been derived in [17] via standard Laplace transform method and it is shown in [17] that the above problem is equivalent to the following initial-boundary value problem…”

Section: Application I: Linear Fractional Diffusion Equationmentioning

confidence: 99%

Fast Evaluation of the Caputo Fractional Derivative and its Applications to Fractional Diffusion Equations

Jiang

Zhang

et al. 2017

Commun. Comput. Phys.

399

235

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We present an efficient algorithm for the evaluation of the Caputo fractional derivative C 0 D α t f (t) of order α ∈ (0, 1), which can be expressed as a convolution of f ′ (t) with the kernel t −α . The algorithm is based on an efficient sum-of-exponentials approximation for the kernel t −1−α on the interval [∆t, T ] with a uniform absolute error ε, where the number of exponentials Nexp needed is ofAs compared with the direct method, the resulting algorithm reduces the storage requirement from O(N T ) to O(Nexp) and the overall computational cost from O(N 2 T ) to O(N T Nexp) with N T the total number of time steps. Furthermore, when the fast evaluation scheme of the Caputo derivative is applied to solve the fractional diffusion equations, the resulting algorithm requires only O(N S Nexp) storage and O(N S N T Nexp) work with N S the total number of points in space; whereas the direct methods require O(N S N T ) storage and O(N S N 2 T ) work. The complexity of both algorithms is nearly optimal since Nexp is of the order O(log N T ) for T ≫ 1 or O(log 2 N T ) for T ≈ 1 for fixed accuracy ε. We also present a detailed stability and error analysis of the new scheme for solving linear fractional diffusion equations. The performance of the new algorithm is illustrated via several numerical examples. Finally, the algorithm can be parallelized in a straightforward manner.

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Section: Application I: Linear Fractional Diffusion Equationmentioning

confidence: 99%

Fast Evaluation of the Caputo Fractional Derivative and its Applications to Fractional Diffusion Equations

Jiang

Zhang

et al. 2017

Commun. Comput. Phys.

399

235

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“…where we used the Laplace transform of the Caputo fractional derivative ( 14) and the initial value condition (5). If take ( 24)-( 25) as artificial boundary conditions, then the previous problem ( 4)-( 8) on unbounded domain is reduced to the following initial-boundary value problem on bounded domain…”

Section: The Exact Artificial Boundary Conditionsmentioning

confidence: 99%

“…Awotunde et al [3] obtained ABCs for a modified fraction diffusion problem. Using the similar approach given by Engquist and Majda, Dea obtains a ABCs for the two-dimensional time-fractional wave equation in his recent work [5]. It is important that the boundaries are designed such that the reduced problem is well-posed.…”

Section: Introductionmentioning

confidence: 99%

On the stability of exact ABCs for the reaction-subdiffusion equation on unbounded domain

2015

Preprint

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In this note we propose the exact artificial boundary conditions formula to the fractional reaction-subdiffusion equation on an unbounded domain. With the application of Laplace transformation, the exact artificial boundary conditions (ABCs) are derived to reformulate the original problem on the unbounded domain to an initial-boundary-value problem on the bounded computational domain. By the Kreiss theory, we prove that the reduced initialboundary value problem is stability. Based on the properties of tempered fractional calculus, we obtain that the reduced initial-boundary value problem is long-time stability.

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“…The basic idea is to build a suitable ABC to eliminate the waves striking a fictitious boundary introduced to truncate the computational domain. In the literature, a lot of attention has been directed towards the construction of ABCs for time-fractional linear PDEs [7,12,14,18] as well as Schrödingertype equations [2,4,5]. However, the question of designing accurate and stable ABCs for the TFNSE remains open, mainly due to the presence of the nonlinear term f .…”

Section: Introductionmentioning

confidence: 99%

Efficient Numerical Computation of Time-Fractional Nonlinear Schrödinger Equations in Unbounded Domain

Zhang¹,

Li²,

Antoine³

2019

CiCP

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The aim of this paper is to derive a stable and efficient scheme for solving the one-dimensional time-fractional nonlinear Schrödinger equation set in an unbounded domain. We first derive absorbing boundary conditions for the fractional system by using the unified approach introduced in [47, 48] and a linearization procedure. Then, the initial boundary-value problem for the fractional system with ABCs is discretized, a stability analysis is developed and the error estimate O(h 2 +τ) is stated. To accelerate the L1-scheme in time, a sum-of-exponentials approximation is introduced to speed-up the evaluation of the Caputo fractional derivative. The resulting algorithm is highly efficient for long time simulations. Finally, we end the paper by reporting some numerical simulations to validate the properties (accuracy and efficiency) of the derived scheme.

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Absorbing boundary conditions for the fractional wave equation

Cited by 8 publications

References 36 publications

Fast Evaluation of the Caputo Fractional Derivative and its Applications to Fractional Diffusion Equations

Fast Evaluation of the Caputo Fractional Derivative and its Applications to Fractional Diffusion Equations

On the stability of exact ABCs for the reaction-subdiffusion equation on unbounded domain

Efficient Numerical Computation of Time-Fractional Nonlinear Schrödinger Equations in Unbounded Domain

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