Finite difference methods with non-uniform meshes for nonlinear fractional differential equations

Li, Changpin; Yi, Qian; Chen, An

doi:10.1016/j.jcp.2016.04.039

Cited by 151 publications

(55 citation statements)

References 28 publications

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“…[5][6][7][8]). In [10], Li et al presented the stability and convergence analysis under the nonuniform meshes for solving nonlinear fractional PDEs. Then, they proposed a linearized finite difference scheme to compute the reduced problem to avoid the iterative process for the classical implicit schemes.…”

Section: Introductionmentioning

confidence: 99%

“…Then, they proposed a linearized finite difference scheme to compute the reduced problem to avoid the iterative process for the classical implicit schemes. In [10], Li et al presented the stability and convergence analysis under the nonuniform meshes for solving nonlinear fractional PDEs. In [11], Li et al established a discrete L1-type Gronwall inequality and applied it to nonlinear time fractional PDEs.…”

Section: Introductionmentioning

confidence: 99%

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High accuracy error estimates of a Galerkin finite element method for nonlinear time fractional diffusion equation

Ren

Shi

Vong

2019

Numerical Methods Partial

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In this work, an effective and fast finite element numerical method with high‐order accuracy is discussed for solving a nonlinear time fractional diffusion equation. A two‐level linearized finite element scheme is constructed and a temporal–spatial error splitting argument is established to split the error into two parts, that is, the temporal error and the spatial error. Based on the regularity of the time discrete system, the temporal error estimate is derived. Using the property of the Ritz projection operator, the spatial error is deduced. Unconditional superclose result in H1‐norm is obtained, with no additional regularity assumption about the exact solution of the problem considered. Then the global superconvergence error estimate is obtained through the interpolated postprocessing technique. In order to reduce storage and computation time, a fast finite element method evaluation scheme for solving the nonlinear time fractional diffusion equation is developed. To confirm the theoretical error analysis, some numerical results are provided.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

High accuracy error estimates of a Galerkin finite element method for nonlinear time fractional diffusion equation

Ren

Shi

Vong

2019

Numerical Methods Partial

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“…This excellent approach in [21,22] allows to obtain a (relatively) high convergence order without the otherwise required very unnatural smoothness assumptions on the given solution. Other works for solving fractional differential equations with nonuniform meshes may be found in, for example, [12,19,24,25]. Motivated by the ideas in Diethelm [4] and Stynes et al [22], we will introduce a numerical method for solving (1.1) with the graded meshes and we prove that the optimal convergence order uniformly in t n for the proposed numerical method can be recovered when C 0 D α t y(t), α > 0 behaves as t σ , 0 < σ < 1.…”

Section: Introductionmentioning

confidence: 99%

Detailed error analysis for a fractional Adams method with graded meshes

2017

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We consider a fractional Adams method for solving the nonlinear fractional differential equation C 0 D α t y(t) = f (t, y(t)), α > 0, equipped with the initial conditions y (k) Here, α may be an arbitrary positive number and α denotes the smallest integer no less than α and the differential operator is the Caputo derivative. Under the assumption 2004) introduced a fractional Adams method with the uniform meshes t n = T (n/N), n = 0, 1, 2, . . . , N and proved that this method has the optimal convergence order uniformly in t n , that is, the optimal convergence order of this method cannot be obtained with the uniform meshes. However, it is well-known that for y ∈ C m [0, T ] for some m ∈ N and 0 < α < m, the Caputo fractional derivative C 0 D α t y(t) takes the form " C 0 D α t y(t) = ct α −α + smoother terms" (Diethelm et al. Numer. Algor. 36, 2004), which implies that C 0 D α t y behaves as t α −α which is not in C 2 [0, T ]. By using the graded meshes t n = T (n/N) r , n = 0, 1, 2, . . . , N with some suitable r > 1, we show that the optimal convergence order of this method can be recovered uniformly in t n even if C 0 D α t y behaves as

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“…As a result, the analytic solutions of most fractional differential equations cannot be obtained explicitly. Hence, developing numerical methods for these equations are becoming more and more necessary and important, one can refers to the papers [5][6][7][8][9][10][11][12][13] and reference therein. Generally speaking, the first step of solving fractional differential equations is to approximate the fractional derivatives.…”

Section: Introductionmentioning

confidence: 99%

High‐order algorithms for Riesz derivative and their applications (V)

Ding

2017

Numerical Methods Partial

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In this article, based on the idea of combing symmetrical fractional centred difference operator with compact technique, a series of even‐order numerical differential formulas (named the fractional‐compact formulas) are established for the Riesz derivatives with order α ∈ ( 1 , 2 ) . Properties of coefficients in the derived formulas are studied in details. Then applying the constructed fourth‐order formula, a difference scheme is proposed to solve the Riesz spatial telegraph equation. By the energy method, the constructed numerical algorithm is proved to be stable and convergent with order O ( τ 4 + h 4 ) , where τ and h are the temporal and spatial stepsizes, respectively. Finally, several numerical examples are presented to verify the theoretical results.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1754–1794, 2017

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Finite difference methods with non-uniform meshes for nonlinear fractional differential equations

Cited by 151 publications

References 28 publications

High accuracy error estimates of a Galerkin finite element method for nonlinear time fractional diffusion equation

High accuracy error estimates of a Galerkin finite element method for nonlinear time fractional diffusion equation

Detailed error analysis for a fractional Adams method with graded meshes

High‐order algorithms for Riesz derivative and their applications (V)

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