Sharp error estimate of a compact L1-ADI scheme for the two-dimensional time-fractional integro-differential equation with singular kernels

Wang, Zhen; Cen, Dakang; Mo, Yan

doi:10.1016/j.apnum.2020.09.006

Cited by 47 publications

(7 citation statements)

References 32 publications

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“…In this subsection, we consider the semi-discrete equation (12) subject to IC (10) and BCs (11) and use a collocation technique based on QBS function to approximate these equations in the spatial domain. For a given positive integer , m = 1, 2, … , M. To facilitate the QBS functions, we append eight additional mesh points outside the partition Ω M , positioned as…”

Section: Space Discretizationmentioning

confidence: 99%

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A computational technique for solving the time‐fractional Fokker‐Planck equation

Roul

Kumari

Rohil

2022

Math Methods in App Sciences

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In this paper, a high‐order computational scheme is constructed for the time‐fractional Fokker‐Planck (TFFP) equation. The L2−1σ$$ L2-{1}_{\sigma } $$ formula is used to approximate the fractional derivative in the model problem. The space derivatives in the resulting semi‐discrete equation are approximated by a collocation technique based on quartic B‐spline (QBS) basis functions. The method is rigorously analyzed for its convergence. Two examples are provided to show the applicability and robustness of the present method. Our results are compared with the results obtained using finite difference method (FDM) and high‐order compact finite difference method (HCFDM).

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Section: Space Discretizationmentioning

confidence: 99%

“…Fractional calculus, which deals with derivatives and integrals of any arbitrary real or complex order, have been evolved rapidly in last few decades due to its application in various scientific fields, for instance, see the papers [2][3][4][5][6][7][8][9][10][11][12][13][14] and references therein.…”

Section: Introductionmentioning

confidence: 99%

A computational technique for solving the time‐fractional Fokker‐Planck equation

Roul

Kumari

Rohil

2022

Math Methods in App Sciences

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“…Hadamard fractional calculus was introduced by Hadamard in 1892, and its theory and applications can be found in Hadamard and Kalbas et al 1,2 For example, Hadamard-type fractional differential equations are useful mathematical tools in the practical problems of fracture analysis or both planar and three-dimensional elasticities. 3 Currently, most of work focus on general fractional calculus, [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] but only few researches are related to Hadamard fractional calculus. Several definitions and properties of Hadamard fractional calculus are introduced in the following.…”

Section: Introductionmentioning

confidence: 99%

Efficient numerical algorithms of time fractional telegraph‐type equations involving Hadamard derivatives

Cen

Wang

2022

Math Methods in App Sciences

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In this paper, we study the finite difference/iterative method for the fractional telegraph equation with Hadamard derivatives. The model is first transformed into an equivalent fractional integro‐differential equation, then the technic of exponential type meshes is adopted. The resulting discrete coefficients of Hadamard fractional integral own several graceful properties, which are crucial in numerical analysis. To conquer the difficulty caused by weak regularity, we also propose iterative schemes. The error estimate is equal to O()false(logtfalse)n+γ,0.1em1

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“…The fractional-order derivative has recently been applied in modeling different phenomena including viscoelasticity, financial modeling, nanotechnology, control theory of dynamical systems, random walk, anomalous transport, biological modeling, and anomalous diffusion as well. For further applications of the fractional order derivative in the fields of engineering, physical sciences, we may refer to [1][2][3][4][5][6][7][8][9][10][11] and [12][13][14][15][16] and the references cited therein.…”

Section: Introductionmentioning

confidence: 99%

A pseudo-spectral method based on reproducing kernel for solving the time-fractional diffusion-wave equation

2022

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In this paper, we focus on the development and study of the finite difference/pseudo-spectral method to obtain an approximate solution for the time-fractional diffusion-wave equation in a reproducing kernel Hilbert space. Moreover, we make use of the theory of reproducing kernels to establish certain reproducing kernel functions in the aforementioned reproducing kernel Hilbert space. Furthermore, we give an approximation to the time-fractional derivative term by applying the finite difference scheme by our proposed method. Over and above, we present an appropriate technique to derive the numerical solution of the given equation by utilizing a pseudo-spectral method based on the reproducing kernel. Then, we provide two numerical examples to support the accuracy and efficiency of our proposed method. Finally, we apply numerical experiments to calculate the quality of our approximation by employing discrete error norms.

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Sharp error estimate of a compact L1-ADI scheme for the two-dimensional time-fractional integro-differential equation with singular kernels

Cited by 47 publications

References 32 publications

A computational technique for solving the time‐fractional Fokker‐Planck equation

A computational technique for solving the time‐fractional Fokker‐Planck equation

Efficient numerical algorithms of time fractional telegraph‐type equations involving Hadamard derivatives

A pseudo-spectral method based on reproducing kernel for solving the time-fractional diffusion-wave equation

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