An analysis of the Grünwald–Letnikov scheme for initial-value problems with weakly singular solutions

Chen, Hu; Holland, Finbarr; Stynes, Martin

doi:10.1016/j.apnum.2019.01.004

Cited by 35 publications

(16 citation statements)

References 16 publications

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“…In this section, we present two different examples of type (1). The first one is a homogeneous with constant coefficients, and the second one is a nonhomogeneous with variable coefficients, whose exact solutions are unknown.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Riemann-Liouville fractional derivatives and Caputo fractional derivatives are the most important fractional derivatives that have been used very often in the literature. [1][2][3] In recent years, fractional calculus becomes the more powerful mathematical tool which has been employed to formulate several physical models in the fields of science and engineering such as spring-pot model (a linear viscoelastic element whose behavior is intermediate between that of an elastic element and a viscous element), the fractional Voigt model (a spring and a spring-pot in parallel), the fractional order Maxwell model, and so forth. Also, to describe an electromagnetic field arising in the motion of a train on an air-pillow, such models are very useful.…”

Section: Introductionmentioning

confidence: 99%

“…Notation: Throughout the paper, we use C as a generic constant which can take different values in different places but is independent of T and any mesh used for solving (1). For any continuous function Z ∶ Ω → R with Ω ⊂ R 2 and any mesh function Z n m for m = 0, 1, … , M and n = 0, 1, … , N, we define…”

Section: Introductionmentioning

confidence: 99%

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Analysis of the L1 scheme for a time fractional parabolic–elliptic problem involving weak singularity

Santra

Mohapatra

2020

Math Methods in App Sciences

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A time fractional initial boundary value problem of mixed parabolic-elliptic type is considered. The domain of such problem is divided into two subdomains. A reaction-diffusion parabolic problem is considered on the first domain, and on the second, a convection-diffusion elliptic type problem is considered. Such problem has a mild singularity at the initial time t = 0. The classical L1 scheme is introduced to approximate the temporal derivative, and a second order standard finite difference scheme is used to approximate the spatial derivatives. The domain is discretized with uniform mesh for both directions. It is shown that the order of convergence is more higher away from t = 0 than the order of convergence on the whole domain. To show the efficiency of the scheme, numerical results are provided.

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Section: Numerical Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Analysis of the L1 scheme for a time fractional parabolic–elliptic problem involving weak singularity

Santra

Mohapatra

2020

Math Methods in App Sciences

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“…With the help of above notations, we now prove some results which are useful in the derivation of fractional Gronwall type inequality and error analysis. For proving these results, we basically borrow the idea given in [15,23].…”

Section: A Priori Bound and Error Analysismentioning

confidence: 99%

“…In fact for the case u ∈ C [0, T], the Crank-Nicolson method based scheme (13) has same convergence order as Grünwald-Letnikov and L1 schemes which are of order O(Δt t −1 n ) (cf. [17,23]). We also confirm this fact in the section of numerical experiments.…”

Section: Theorem 43 Let U Be the Solution Of Problemmentioning

confidence: 99%

Fractional Crank–Nicolson–Galerkin finite element scheme for the time‐fractional nonlinear diffusion equation

Kumar

Chaudhary

Kumar

2019

Numerical Methods Partial

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This article presents a finite element scheme with Newton's method for solving the time-fractional nonlinear diffusion equation. For time discretization, we use the fractional Crank-Nicolson scheme based on backward Euler convolution quadrature. We discuss the existence-uniqueness results for the fully discrete problem. A new discrete fractional Gronwall type inequality for the backward Euler convolution quadrature is established. A priori error estimate for the fully discrete problem in L 2 (Ω) norm is derived. Numerical results based on finite element scheme are provided to validate theoretical estimates on time-fractional nonlinear Fisher equation and Huxley equation. KEYWORDS discrete fractional Gronwall type inequality, error estimates, fractional Crank-Nicolson method, time-fractional diffusion equation 1 Present address Sudhakar Chaudhary,

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Error estimate of GL‐ADI scheme for 2D multiterm nonlinear time‐fractional subdiffusion equation

Jiang,

Chen

2024

Math Methods in App Sciences

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In this paper, a 2D multiterm nonlinear problem of the form is considered, where each Caputo fractional derivative is of order . We use the Grünwald–Letnikov(GL) scheme on uniform mesh to discretize the multiterm Caputo fractional derivative and finite difference scheme is used for spatial discretization, and then we construct a fully discrete GL‐ADI scheme. A discrete Gronwall inequality is introduced for getting the sharp pointwise‐in‐time error estimate on uniform mesh. Numerical examples are provided to verify the sharpness of our error estimate.

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An analysis of the Grünwald–Letnikov scheme for initial-value problems with weakly singular solutions

Cited by 35 publications

References 16 publications

Analysis of the L1 scheme for a time fractional parabolic–elliptic problem involving weak singularity

Analysis of the L1 scheme for a time fractional parabolic–elliptic problem involving weak singularity

Fractional Crank–Nicolson–Galerkin finite element scheme for the time‐fractional nonlinear diffusion equation

Error estimate of GL‐ADI scheme for 2D multiterm nonlinear time‐fractional subdiffusion equation

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