Boundary conditions for fractional diffusion

Baeumer, Boris; Kovács, Mihály; Meerschaert, Mark M.; Sankaranarayanan, Harish

doi:10.1016/j.cam.2017.12.053

Cited by 64 publications

(90 citation statements)

References 67 publications

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“…First, we will show that the eigenvalues of the matrix C have negative real parts. According to the Gershgorin theorem [26], the eigenvalues of the matrix C lie in the disks centered at c i, i with radii ∑ N−1 j=1,j≠i |c i,j |. When 1 ≤ i ≤ N − 2, according to Lemma 2, we have…”

Section: Theoremmentioning

confidence: 99%

“…Liu and Hou [25] developed an implicit finite difference method for fractional advection-dispersion equations with fractional derivative BCs. In [26], an explicit Euler scheme was proposed and numerical results presented for space-fractional diffusion equations with absorbing or reflecting BCs. Stable, consistent explicit and implicit Euler methods were generalized to two-sided fractional diffusion, and stability was established in [27].…”

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confidence: 99%

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A second‐order finite difference method for fractional diffusion equation with Dirichlet and fractional boundary conditions

Xie

Fang

2019

Numerical Methods Partial

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KEYWORDSCrank-Nicholson method, fractional boundary conditions, fractional diffusion equation, Riemann-Liouville fractional derivative, stability and convergence INTRODUCTIONFractional differential equations are generalizations of integer-order differential equations, which have been used in many models in physics, hydrology, geology, finance, and so on [1-6]. One of its most important applications is to describe, using fractional diffusion equations, the transport dynamics in various complex systems, where Gaussian statistics are no longer followed and the Fick's second law fails to describe the related transport behaviors [7]. A considerable number of numerical methods have been developed for space fractional diffusion equations [8][9][10][11][12][13][14][15]. Tadjeran et al. [8] used an approach based on the classical Crank-Nicholson method combined with spatial extrapolation to obtain temporally and spatially second-order accurate numerical estimates. In [9], combining the alternating directions implicit (ADI) approach with a Numer Methods Partial Differential Eq. 2019;35:1383-1395. wileyonlinelibrary.com/journal/num

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Section: Theoremmentioning

confidence: 99%

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confidence: 99%

A second‐order finite difference method for fractional diffusion equation with Dirichlet and fractional boundary conditions

Xie

Fang

2019

Numerical Methods Partial

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“…A completely different approach for solving (1) for p ≡ 1 with zero Dirichlet boundary conditions is employed in [12], where the authors obtained the viscosity solutions. Further discussion was made in [3] and [13] where the authors compare the problems with diffusive flux modeled by the Caputo and the Riemann-Liouville derivative and carry a numerical analysis.In the present paper we will present the results concerning solvability of (1) by means of the semigroup theory. At first, we will focus our attention on the case where p ≡ 1.…”

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confidence: 99%

An analytic semigroup generated by a fractional differential operator

Ryszewska

2020

Journal of Mathematical Analysis and Applications

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We study a space-fractional diffusion problem, where the non-local diffusion flux involves the Caputo derivative of the diffusing quantity. We prove the unique existence of regular solutions to this problem by means of the semigroup theory. We show that the operator defined as divergence of product of a positive function with the Caputo derivative is a generator of analytic semigroup. the simplified problem (1). We would like to emphasise that from the modelling point of view, it is important that (1) is a balance low.Our goal is to study the basic solvability problem for equation (1), by the semigroup approach. At this moment we would like to present a broader context. It is worth to mention here, that in the paper [2] the authors studied an array of fractional diffusion equations, for a variety of boundary conditions, but they constructed only the C 0 -semigroup. Besides, we note that the problem (1) was not explicitly studied in [2]. Another paper that presents the probabilistic point of view on space-fractional problems is [4], where the authors consider equations with time-fractional Caputo derivative and non-local space operators. A completely different approach for solving (1) for p ≡ 1 with zero Dirichlet boundary conditions is employed in [12], where the authors obtained the viscosity solutions. Further discussion was made in [3] and [13] where the authors compare the problems with diffusive flux modeled by the Caputo and the Riemann-Liouville derivative and carry a numerical analysis.In the present paper we will present the results concerning solvability of (1) by means of the semigroup theory. At first, we will focus our attention on the case where p ≡ 1. We will describe the domain of ∂ ∂x D α in terms of Sobolev spaces and as a final result we will construct an analytic semigroup. Precisely, we will show that the operator ∂ ∂x D α , considered on the domaingenerates an analytic semigroup. Here, by H α (0, 1) we mean the fractional Sobolev space (see [10, definition 9.1]) and the subspace 0 H α (0, 1) will be introduced in Proposition 3. Subsequently, we will extend our result for the case of any strictly positive lipschitz continuous p. We would like to emphasise that, developing the theory of analytic semigroups for the systems of type (1) seems to be especially important if we notice that the operator ∂ ∂x D α is not self-adjoint, thus we can not expect that its eigenfunctions generate the basis of any natural Hilbert space.The paper is organized as follows. In chapter 2 we give preliminary results concerning fractional operators. Chapter 3 is devoted to the proof of the main result. Namely, we will show that ∂ ∂x D α is a generator of C 0 -semigroup of contractions which can be extended to the analytic semigroup (Theorem 2). In chapter 4, we extend the results of the previous chapter to the case of operator ∂ ∂x (p(x)D α ), where p is a positive lipschitz function (Theorem 3). In chapter 5, we will present the basic applications of this result to solvability of (1). At last, for the sake of comple...

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“…Due to the non-locality of the fractional operator, the local BCs may not be suitable depending on the type of fractional derivative, hence non-local/fractional BCs have been considered in some other works, for instance, see [45,20,47,40,39,21]. Moreover, by imposing the no-flux BCs, namely, homogeneous fractional Neumann boundary conditions, we can recover the mass conservation [2,3,18]. However, the numerical implementation of non-local BCs is not straightforward and requires special treatment in order to preserve the accuracy of the numerical method used, especially in high order methods such as spectral Galerkin methods.…”

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confidence: 99%

“…• Conservative R-L FDEs, i.e., D The definitions of the fractional operators I s p , D s p , C D s p , s > 0 can be found in (2.13) and (2.14). From physical point of view, the FNBCs (R-L or Caputo) is the reflecting (no-flux) BCs, and the homogeneous classical Dirichlet BCs is the absorbing BCs [2]. For the R-L problem with FDBCs, although the physical meaning may not be clear, it is mathematically interesting, see [7] for the one-sided FDEs or [22] for the Riesz FDEs.…”

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confidence: 99%

A Spectral Penalty Method for Two-Sided Fractional Differential Equations with General Boundary Conditions

Wang¹,

Mao²,

Huang³

et al. 2019

SIAM J. Sci. Comput.

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We consider spectral approximations to the conservative form of the two-sided Riemann-Liouville (R-L) and Caputo fractional differential equations (FDEs) with nonhomogeneous Dirichlet (fractional and classical, respectively) and Neumann (fractional) boundary conditions. In particular, we develop a spectral penalty method (SPM) by using the Jacobi poly-fractonomial approximation for the conservative R-L FDEs while using the polynomial approximation for the conservative Caputo FDEs. We establish the well-posedness of the corresponding weak problems and analyze sufficient conditions for the coercivity of the SPM for different types of fractional boundary value problems. This analysis allows us to estimate the proper values of the penalty parameters at boundary points. We present several numerical examples to verify the theory and demonstrate the high accuracy of SPM, both for stationary and time dependent FDEs. Moreover, we compare the results against a Petrov-Galerkin spectral tau method (PGS-τ , an extension of [23]) and demonstrate the superior accuracy of SPM for all cases considered.

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Boundary conditions for fractional diffusion

Cited by 64 publications

References 67 publications

A second‐order finite difference method for fractional diffusion equation with Dirichlet and fractional boundary conditions

A second‐order finite difference method for fractional diffusion equation with Dirichlet and fractional boundary conditions

An analytic semigroup generated by a fractional differential operator

A Spectral Penalty Method for Two-Sided Fractional Differential Equations with General Boundary Conditions

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