Efficient spectral–Galerkin methods for fractional partial differential equations with variable coefficients

Mao, Zhiping; Shen, Jie

doi:10.1016/j.jcp.2015.11.047

Cited by 94 publications

(76 citation statements)

References 31 publications

(37 reference statements)

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“…In [19], with the introduction of an auxiliary variable, a mixed method approximation scheme for problem (1.1) and (1.2) was studied and error estimates derived. In [25], a spectral Galerkin method for the two-sided steady-state FDE with variable coefficient was analyzed, in which the outside and inside fractional derivatives are chosen carefully so that the corresponding Galerkin weak formulation are self-adjoint and coercive. Optimal error estimates were also derived under suitable smoothness assumption on the solution.…”

Section: Introductionmentioning

confidence: 99%

Spectral approximation of a variable coefficient fractional diffusion equation in one space dimension

Zheng

Ervin

Wang

2019

Applied Mathematics and Computation

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In this article we consider the approximation of a variable coefficient (two-sided) fractional diffusion equation (FDE), having unknown u. By introducing an intermediate unknown, q, the variable coefficient FDE is rewritten as a lower order, constant coefficient FDE. A spectral approximation scheme, using Jacobi polynomials, is presented for the approximation of q, q N . The approximate solution to u, u N , is obtained by post processing q N . An a priori error analysis is given for (q − q N ) and (u − u N ). Two numerical experiments are presented whose results demonstrate the sharpness of the derived error estimates.

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Section: Introductionmentioning

confidence: 99%

Spectral approximation of a variable coefficient fractional diffusion equation in one space dimension

Zheng

Ervin

Wang

2019

Applied Mathematics and Computation

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“…At the same time, the authors kept the properties of their numerical solutions with the weak formulas. Mao and Shen [32,33,34] utilized efficient spectral methods to solve some fractional problems, where there are pretty popular with researchers.…”

Section: Introductionmentioning

confidence: 99%

A generalized-Jacobi-function spectral method for space-time fractional reaction-diffusion equations with viscosity terms

Sun

et al. 2020

Applied Numerical Mathematics

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In this work, we study a new spectral Petrov-Galerkin approximation of space-time fractional reactiondiffusion equations with viscosity terms built by Riemann-Liouville fractional-order derivatives. The proposed method is reliant on generalized Jacobi functions (GJFs) for our problems. The contributions are threefold: First, thanks to the theoretical framework of variational problems, the well-posedness of the problem is proved. Second, new GJF-basis functions are established to fit weak solutions, which take full advantages of the global properties of fractional derivatives. Moreover, the basis functions conclude singular terms, in order to solve our problems with given smooth source term. Finally, we get a numerical analysis of error estimates to depend on GJF-basis functions. Numerical experiments confirm the expected convergence. In addition, they are given to show the effect of the viscosity terms in anomalous diffusion.

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“…For the one-dimensional two-sided FDEs, there are several available numerical methods, for example, the finite difference method [26,37], the finite element method [11,38], and the spectral method [19,43,24,22,10,23] and references therein. In the early works, the emphasis was on obtaining high accuracy by ignoring the issue of low regularity of the solution of FDEs, i.e., assuming that the solution is smooth, for example, [11,19,37,24]. However, solutions of fractional boundary value problems have endpoints singularities that limit the convergence rate of numerical discretizations significantly.…”

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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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Efficient spectral–Galerkin methods for fractional partial differential equations with variable coefficients

Cited by 94 publications

References 31 publications

Spectral approximation of a variable coefficient fractional diffusion equation in one space dimension

Spectral approximation of a variable coefficient fractional diffusion equation in one space dimension

A generalized-Jacobi-function spectral method for space-time fractional reaction-diffusion equations with viscosity terms

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

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