A mixed finite element method for a time-fractional fourth-order partial differential equation

Liu, Yang; Fang, Zhichao; Li, Hong; He, Shuping

doi:10.1016/j.amc.2014.06.023

Cited by 93 publications

(51 citation statements)

References 24 publications

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“…Moreover, in Fig. 4a, b, we plot the logarithmic graphs of MAEs (log 10 Error) at γ 1 (x, t) = 15+sin 5 (xt) 16 , γ 2 (x, t) = 9−x 2 +t 3 120 and γ 1 (x, t) = 16+(xt) 5 17 , γ 2 (x, t) = 10+(xt) 4 −(xt) 5…”

Section: Numerical Resultsmentioning

confidence: 99%

“…The subject is nowadays very active due to its many applications, not only in mathematics, but also in physics and engineering, and has proven to better describe certain complex phenomena in nature (see for instance [2][3][4][5][6][7][8][9]). Recently, several numerical techniques are investigated for achieving high accurate solutions for classical fractional differential equations (see for instance [10][11][12][13][14][15][16][17][18]). In 1993, Samko et al [19] proposed an interesting extension of the classical fractional calculus.…”

Section: Introductionmentioning

confidence: 99%

“…In addition, Fig. 5 plots the absolute error functions |u(x, y, t) − u 4,11,11 (x, y, t) for different values of t with γ 1 (x, y, t) = 9−x 5 +t 3 120 , γ 2 (x, y, t) = 15+sin 5 (yt) 16 and θ = ϑ = 0.…”

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confidence: 99%

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Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation

Bhrawy

Zaky²

2014

Nonlinear Dyn

202

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The cable equation plays a central role in many areas of electrophysiology and in modeling neuronal dynamics. This paper reports an accurate spectral collocation method for solving one-and twodimensional variable-order fractional nonlinear cable equations. The proposed method is based on shifted Jacobi collocation procedure in conjunction with the shifted Jacobi operational matrix for variable-order fractional derivatives, described in the sense of Caputo. The main advantage of the proposed method is to investigate a global approximation for spatial and temporal discretizations. In addition, the method reduces the variable-order fractional nonlinear cable equation to a simpler problem that consists of solving a system of algebraic equations. The validity and effectiveness of the method are demonstrated by solving three numerical examples. The convergence of the method is graphically analyzed. The results demonstrate that the proposed method is a powerful algorithm with high accuracy for solving the variable-order nonlinear partial differential equations.Keywords One-dimensional cable equation · Two-dimensional variable-order nonlinear cable equation · Collocation method · Jacobi polynomials · Operational matrix of fractional derivative · Variable-order derivative

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation

Bhrawy

Zaky²

2014

Nonlinear Dyn

202

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“…Most fractional differential equations (FDEs) do not have exact analytic solutions, therefore approximation and numerical techniques must be presented and developed. Finite element methods have been introduced in [11][12][13] to obtain numerical solutions of FDEs; also the numerical treatments based on finite difference methods have been developed in [14][15][16]. Moreover, several spectral techniques were designed for such equations (see for instance, [17][18][19][20][21][22][23][24][25][26][27]).…”

Section: Introductionmentioning

confidence: 99%

Efficient Spectral Collocation Algorithm for a Two-Sided Space Fractional Boussinesq Equation with Non-local Conditions

2015

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A spectral shifted Legendre Gauss-Lobatto collocation method is developed and analyzed to solve numerically one-dimensional two-sided space fractional Boussinesq (SFB) equation with non-classical boundary conditions. The method depends basically on the fact that an expansion in a series of shifted Legendre polynomials PL,n(x), x ∈ [0, L] is assumed, for the function and its space-fractional derivatives occurring in the two-sided SFB equation. The Legendre-Gauss-Lobatto quadrature rule is established to treat the non-local conservation conditions, and then the problem with its non-local conservation conditions is reduced to a system of ordinary differential equations (ODEs) in time. Thereby, the expansion coefficients are then determined by reducing the two-sided SFB with its boundary and initial conditions to a system of ODEs for these coefficients. This system may be solved numerically in a step-by-step manner by using implicit Runge-Kutta method of order four. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented.Mathematics Subject Classification. 34A08, 65M70, 33C45.

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“…In the series of works [22][23][24], Ervin and Roop presented a first rigorous analysis for the stationary fractional advection dispersion equation based on a variational formulation. Then the discontinuous Galerkin method [25], mixed finite element method [26][27][28][29][30], Petrov Galerkin method [31] and the least-squared mixed method are proposed [32] for stationary fractional diffusion equations, consecutively.…”

Section: Introductionmentioning

confidence: 99%

An expanded mixed finite element simulation for two-sided time-dependent fractional diffusion problem

Yuan

Chen

2018

Adv Differ Equ

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In this paper, we consider a time-dependent diffusion problem with two-sided Riemann-Liouville fractional derivatives. By introducing a fractional-order flux as auxiliary variable, we establish the saddle-point variational formulation, based on which we employ a locally conservative mixed finite element method to approximate the unknown function, its derivative and the fractional flux in space and use the backward Euler scheme to discrete the time derivative, and thus propose a fully discrete expanded mixed finite element procedure. We prove the well-posedness and the optimal order error estimates of the proposed procedure for a sufficiently smooth solution. Numerical experiments are presented to confirm our theoretical findings.

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A mixed finite element method for a time-fractional fourth-order partial differential equation

Cited by 93 publications

References 24 publications

Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation

Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation

Efficient Spectral Collocation Algorithm for a Two-Sided Space Fractional Boussinesq Equation with Non-local Conditions

An expanded mixed finite element simulation for two-sided time-dependent fractional diffusion problem

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