A finite element method for time fractional partial differential equations

Ford, Neville J.; Xiao, Jingyu; Yan, Yubin

doi:10.2478/s13540-011-0028-2

Cited by 186 publications

(102 citation statements)

References 32 publications

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“…The L1 scheme first appeared in the book [41] for the approximation of the Caputo fractional derivative. The L1 scheme may be obtained by direct approximation of the derivative in the definition of the Caputo fractional derivative, e.g., [28], [26], [18], [29], [47], [19], or by the approximation of the Hadamard finite-part integral, e.g., [9], [11], [15], [16], [17], [24], [51].…”

Section: Correction Of the Lubich Fractional Multistep Methodsmentioning

confidence: 99%

An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data

Ford

Yan

2017

FCAA

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In this paper, we shall review an approach by which we can seek higher order time discretisation schemes for solving time fractional partial differential equations with nonsmooth data. The low regularity of the solutions of time fractional partial differential equations implies standard time discretisation schemes only yield first order accuracy. To obtain higher order time discretisation schemes when the solutions of time fractional partial differential equations have low regularities, one may correct the starting steps of the standard time discretisation schemes to capture the singularities of the solutions. We will consider these corrections of some higher order time discretisation schemes obtained by using Lubich's fractional multistep methods, L1 scheme and its modification, discontinuous Galerkin methods, etc. Numerical examples are given to show that the theoretical results are consistent with the numerical results.

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Section: Correction Of the Lubich Fractional Multistep Methodsmentioning

confidence: 99%

An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data

Ford

Yan

2017

FCAA

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“…In short, the separation of variables can be portrayed as a tool to reduce a multidimensional problem to series of one-dimensional ones. It behaves similar to most of global numerical techniques for solving complex models arising in real-world systems [28][29][30][31]. This urges us to exploit the powerful properties of the separation-variables method for solving FPDEs as well as reducing the difficulty of working with fractional derivatives.…”

Section: Introductionmentioning

confidence: 99%

An Accurate Approximate-Analytical Technique for Solving Time-Fractional Partial Differential Equations

et al. 2017

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The demand of many scientific areas for the usage of fractional partial differential equations (FPDEs) to explain their real-world systems has been broadly identified. The solutions may portray dynamical behaviors of various particles such as chemicals and cells. The desire of obtaining approximate solutions to treat these equations aims to overcome the mathematical complexity of modeling the relevant phenomena in nature. This research proposes a promising approximate-analytical scheme that is an accurate technique for solving a variety of noninteger partial differential equations (PDEs). The proposed strategy is based on approximating the derivative of fractional-order and reducing the problem to the corresponding partial differential equation (PDE). Afterwards, the approximating PDE is solved by using a separation-variables technique. The method can be simply applied to nonhomogeneous problems and is proficient to diminish the span of computational cost as well as achieving an approximate-analytical solution that is in excellent concurrence with the exact solution of the original problem. In addition and to demonstrate the efficiency of the method, it compares with two finite difference methods including a nonstandard finite difference (NSFD) method and standard finite difference (SFD) technique, which are popular in the literature for solving engineering problems.

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“…We note that Definitions (4) and (5) are not equivalent [17]. For the deterministic space-fractional partial differential equations where the space-fractional derivative is defined by (5), or the Riemann-Liouville space-fractional derivative, or the Caputo space-fractional derivative, many numerical methods are available, for example, finite difference methods [18][19][20][21][22][23][24][25][26][27][28][29][30], finite element methods [14,[31][32][33][34][35][36][37][38][39][40] and spectral methods [41,42]. For the deterministic space-fractional partial differential equations where the space-fractional derivative is defined by (4), some numerical methods are also available, for example the matrix transfer method (MTT) [21,22,43] and the Fourier spectral method [44].…”

Section: Introductionmentioning

confidence: 99%

Fourier Spectral Methods for Some Linear Stochastic Space-Fractional Partial Differential Equations

2016

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Fourier spectral methods for solving some linear stochastic space-fractional partial differential equations perturbed by space-time white noises in the one-dimensional case are introduced and analysed. The space-fractional derivative is defined by using the eigenvalues and eigenfunctions of the Laplacian subject to some boundary conditions. We approximate the space-time white noise by using piecewise constant functions and obtain the approximated stochastic space-fractional partial differential equations. The approximated stochastic space-fractional partial differential equations are then solved by using Fourier spectral methods. Error estimates in the L 2 -norm are obtained, and numerical examples are given.

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A finite element method for time fractional partial differential equations

Cited by 186 publications

References 32 publications

An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data

An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data

An Accurate Approximate-Analytical Technique for Solving Time-Fractional Partial Differential Equations

Fourier Spectral Methods for Some Linear Stochastic Space-Fractional Partial Differential Equations

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