Finite Difference/Finite Element Methods for Distributed-Order Time Fractional Diffusion Equations

Bu, Weiping; Xiao, Aiguo; Zeng, Wei

doi:10.1007/s10915-017-0360-8

Cited by 81 publications

(32 citation statements)

References 33 publications

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“…We choose δ T = h t in (5.1). We solve equations (5.1) and (5.8) by an L1 scheme [17] in temporal direction and the finite element method in spatial direction [3]. In (5.8), the Dirac delta function δ(δ T − τ) at discrete points τ = kh t , k = 0, 1, .…”

Section: The Conjugate Gradient Algorithm For Solving the Minimizatiomentioning

confidence: 99%

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Inverse source problem for a distributed-order time fractional diffusion equation

Cheng

Yuan

Liang

2019

Journal of Inverse and Ill-Posed Problems

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This paper studies an inverse source problem for a time fractional diffusion equation with the distributed order Caputo derivative. The space-dependent source term is recovered from a noisy final data. The uniqueness, ill-posedness and a conditional stability for this inverse source problem are obtained. The inverse problem is formulated into a minimization functional with Tikhonov regularization method. Further, based on the series representation of the regularized solution, we give convergence rates of the regularized solution under an a-priori and an a-posteriori regularization parameter choice rule. With an adjoint technique for computing the gradient of the regularization functional, the conjugate gradient method is applied to reconstruct the space-dependent source term. Two numerical examples illustrate the effectiveness of the proposed method.

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Section: The Conjugate Gradient Algorithm For Solving the Minimizatiomentioning

confidence: 99%

“…The existence and uniqueness of the weak solution were studied in [16,18]. In addition, there were some studies on the numerical computation of DTFDEs, for example, finite difference methods [8][9][10][11] and finite element methods [3].…”

Section: Introductionmentioning

confidence: 99%

Inverse source problem for a distributed-order time fractional diffusion equation

Cheng

Yuan

Liang

2019

Journal of Inverse and Ill-Posed Problems

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“…There has been some pioneer work on the numerical treatment of distributed order fractional differential equations . In Diethelm and Ford and Katsikadelis, one‐dimensional linear and nonlinear distributed order initial value problems have been investigated.…”

Section: Introductionmentioning

confidence: 99%

“…With composite trapezoid/Simpson formula for the integral and WSGL method for the fractional terms, coupled with finite difference/compact finite difference method in the spatial dimension, their best convergence order is

O false(τ^{2} false| \ln τ false| + h_{1}^{4} + h_{2}^{4} + η^{4} false)

. Thereafter, Bu et al have applied finite element method in the spatial dimension to further improve the global convergence order to O ( τ 2 + h K +1 + η 4 ), where K indicates the degree of the piecewise polynomial used in each element. A large K ensures high accuracy but it will also incur high computation cost.…”

Section: Introductionmentioning

confidence: 99%

“…There has been some pioneer work on the numerical treatment of distributed order fractional differential equations. 25,[32][33][34][35][36][37] In Diethelm and Ford 33 and Katsikadelis, 35 one-dimensional linear and nonlinear distributed order initial value problems have been investigated. The integral term is discretized to give a multiterm equation by the quadrature formula in Diethelm and Ford,33 and by the trapezoidal rule in Katsikadelis.…”

Section: Introductionmentioning

confidence: 99%

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An efficient nonpolynomial spline method for distributed order fractional subdiffusion equations

Wong

2018

Math Methods in App Sciences

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In this paper, we propose an efficient numerical method for a distributed order fractional subdiffusion problem using nonpolynomial spline approach. The solvability, stability, and convergence of the scheme are established rigorously, and it is shown that the spatial convergence order improves some previous work done. Simulation is then conducted to verify the accuracy of the proposed scheme as well as to compare with earlier work.

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An efficient numerical method for the distributed‐order time‐fractional diffusion equation with the error analysis and stability properties

Irandoust‐Pakchin,

Hossein Derakhshan,

Rezapour

et al. 2024

Math Methods in App Sciences

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In this manuscript, we study and examine the time‐fractional modified anomalous sub‐diffusion model of distributed‐order. Two numerical approaches are used to study the approximate solutions of the presented model. For the first approach, we use a second‐order difference method based on the L1 formula for the temporal variable. In this case, stability and convergence analysis for time discretization using the L1 formula is displayed. For the second approach, we display and demonstrate the numerical method for the full‐discrete based on the Galerkin weak form based on various kernels and shape functions of reproducing kernel particle method which do not have the ‐Kronecker property. Moreover, in this manuscript, in order to be able to use the necessary boundary conditions, the two straight strategies are used: one is the Lagrange multiplier method, and the other one is the penalty method. By using penalty method, we can the main boundary value model is changed to a new BVP with Robin boundary condition. At the end of the article, some numerical examples are presented and analyzed for the efficiency of the proposed numerical method. Also, the presented numerical method is compared with other numerical approaches, and the results of this comparison are reported in the form of a table.

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Finite Difference/Finite Element Methods for Distributed-Order Time Fractional Diffusion Equations

Cited by 81 publications

References 33 publications

Inverse source problem for a distributed-order time fractional diffusion equation

Inverse source problem for a distributed-order time fractional diffusion equation

An efficient nonpolynomial spline method for distributed order fractional subdiffusion equations

An efficient numerical method for the distributed‐order time‐fractional diffusion equation with the error analysis and stability properties

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