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Galerkin fem for fractional order parabolic equations with initial data
 in H−s, 0 ≤ s ≤ 1. Proc. 5th Conf. Numer. Anal. Appl
, 2013
"... Abstract. We investigate semidiscrete numerical schemes based on the standard Galerkin and lumped mass Galerkin finite element methods for an initialboundary value problem for homogeneous fractional diffusion problems with nonsmooth initial data. We assume that Ω ⊂ Rd, d = 1, 2, 3 is a convex po ..."
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Cited by 4 (4 self)
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Abstract. We investigate semidiscrete numerical schemes based on the standard Galerkin and lumped mass Galerkin finite element methods for an initialboundary value problem for homogeneous fractional diffusion problems with nonsmooth initial data. We assume that Ω ⊂ Rd, d = 1, 2, 3 is a convex polygonal (polyhedral) domain. We theoretically justify optimal order error estimates in L2 and H1norms for initial data in H−s(Ω), 0 ≤ s ≤ 1. We confirm our theoretical findings with a number of numerical tests that include initial data v being a Dirac δfunction supported on a (d − 1)dimensional manifold. 1.
4 InitialBoundary Value Problems for MultiTerm TimeFractional Diffusion Equations with
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Generalized Jacobi functions and their applications to fractional differential equations
 Math. Comp
"... Abstract. In this paper, we consider spectral approximation of fractional differential equations (FDEs). A main ingredient of our approach is to define a new class of generalized Jacobi functions (GJFs), which is intrinsically related to fractional calculus, and can serve as natural basis functions ..."
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Cited by 3 (1 self)
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Abstract. In this paper, we consider spectral approximation of fractional differential equations (FDEs). A main ingredient of our approach is to define a new class of generalized Jacobi functions (GJFs), which is intrinsically related to fractional calculus, and can serve as natural basis functions for properly designed spectral methods for FDEs. We establish spectral approximation results for these GJFs in weighted Sobolev spaces involving fractional derivatives. We construct efficient GJFPetrovGalerkin methods for a class of prototypical fractional initial value problems (FIVPs) and fractional boundary value problems (FBVPs) of general order, and show that with an appropriate choice of the parameters in GJFs, the resulted linear systems are sparse and wellconditioned. Moreover, we derive error estimates with convergence rate only depending on the smoothness of data, so truly spectral accuracy can be attained if the data are smooth enough. The ideas and results presented in this paper will be useful to deal with more general FDEs involving RiemannLiouville or Caputo fractional derivatives. 1.
A discontinuous PetrovGalerkin method for timefractional diffusion equations
 SIAM J. Numer. Anal
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An analysis of the RayleighStokes problem for the generalized second grade fluid
, 2014
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ERROR ANALYSIS OF SEMIDISCRETE FINITE ELEMENT METHODS FOR INHOMOGENEOUS TIMEFRACTIONAL DIFFUSION
"... Abstract. We consider the initial boundary value problem for the inhomogeneous timefractional diffusion equation with a homogeneous Dirichlet boundary condition and a nonsmooth right hand side data in a bounded convex polyhedral domain. We analyze two semidiscrete schemes based on the standard Gal ..."
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Cited by 2 (2 self)
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Abstract. We consider the initial boundary value problem for the inhomogeneous timefractional diffusion equation with a homogeneous Dirichlet boundary condition and a nonsmooth right hand side data in a bounded convex polyhedral domain. We analyze two semidiscrete schemes based on the standard Galerkin and lumped mass finite element methods. Almost optimal error estimates are obtained for right hand side data f(x, t) ∈ L∞(0, T; Ḣq(Ω)), −1 < q ≤ 1, for both semidiscrete schemes. For lumped mass method, the optimal L2(Ω)norm error estimate requires symmetric meshes. Finally, numerical experiments for one and twodimensional examples are presented to verify our theoretical results. 1.
EFFICIENT SPECTRALGALERKIN METHODS FOR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS
"... ABSTRACT. Efficient SpectralGalerkin algorithms are developed to solve multidimensional fractional elliptic equations with variable coefficients in conserved form as well as nonconserved form. These algorithms are extensions of the spectralGalerkin algorithms for usual elliptic PDEs developed i ..."
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ABSTRACT. Efficient SpectralGalerkin algorithms are developed to solve multidimensional fractional elliptic equations with variable coefficients in conserved form as well as nonconserved form. These algorithms are extensions of the spectralGalerkin algorithms for usual elliptic PDEs developed in [24]. More precisely, for separable FPDEs, we construct a direct method by using a matrix diagonalization approach, while for nonseparable FPDEs, we employ an preconditioned BICGSTAB method with a suitable separable FPDE with constantcoefficients as preconditioner. The cost of these algorithms are of O(Nd+1) flops where d is the space dimension. We derive rigorous weighted error estimates which provide more precise convergence rate for problems with singularities at boundaries. We also present ample numerical results to validate the algorithms and error estimates. 1.
IMPROVED ERROR ESTIMATES OF A FINITE DIFFERENCE/SPECTRAL METHOD FOR TIMEFRACTIONAL DIFFUSION EQUATIONS
"... Abstract. In this paper, we first consider the numerical method that Lin and Xu proposed and analyzed in [Finite difference/spectral approximations for the timefractional diffusion equation, JCP 2007] for the timefractional diffusion equation. It is a method basing on the combination of a finite d ..."
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Abstract. In this paper, we first consider the numerical method that Lin and Xu proposed and analyzed in [Finite difference/spectral approximations for the timefractional diffusion equation, JCP 2007] for the timefractional diffusion equation. It is a method basing on the combination of a finite different scheme in time and spectral method in space. The numerical analysis carried out in that paper showed that the scheme is of (2 − α)order convergence in time and spectral accuracy in space for smooth solutions, where α is the timefractional derivative order. The main purpose of this paper consists in refining the analysis and providing a sharper estimate for both time and space errors. More precisely, we improve the error estimates by giving a more accurate coefficient in the time error term and removing the factor in the space error term, which grows with decreasing time step. Then the theoretical results are validated by a number of numerical tests. Key words. Error estimates, finite difference methods, spectral methods, time fractional diffusion equation. 1.
θFUNCTION METHOD FOR A TIMEFRACTIONAL REACTIONDIFFUSION EQUATION
"... Abstract. In this paper, the initialboundaryvalue problems with both the Dirichlet and the Neumann boundary conditions for a nonlinear timefractional reactiondiffusion equation are considered. The proposed solution method consists in employing a suitable generalization of the θfunction that i ..."
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Abstract. In this paper, the initialboundaryvalue problems with both the Dirichlet and the Neumann boundary conditions for a nonlinear timefractional reactiondiffusion equation are considered. The proposed solution method consists in employing a suitable generalization of the θfunction that is constructed based on the fundamental solution of the corresponding linear timefractional diffusion equation. For the solutions of the initialboundaryvalue problems for the timefractional reactiondiffusion equation, the integral equations of the Volterra type with the generalized θfunction in the kernel are obtained. These equations are useful, e.g., for the numerical solutions of the problems under consideration. 1.