Second order difference schemes for time-fractional KdV–Burgers’ equation with initial singularity

Cen, Dakang; Wang, Zhen; Mo, Yan

doi:10.1016/j.aml.2020.106829

Cited by 41 publications

(13 citation statements)

References 8 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%

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%

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

Roul

Kumari

Rohil

2022

Math Methods in App Sciences

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“…Such a regularity assumption with respect to time t is often used, see for instance [15,19,[30][31][32][33][34][35][36][37][38][39].…”

Section: Optimal Error Estimatementioning

confidence: 99%

“…where C is a positive constant dependent on u L ∞ ((0,T];H k+2 (Ω)) . Now we estimate the nonlinear term in (31). It is obvious to see that…”

Section: Optimal Error Estimatementioning

confidence: 99%

Local Discontinuous Galerkin Method Coupled with Nonuniform Time Discretizations for Solving the Time-Fractional Allen-Cahn Equation

2022

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This paper aims to numerically study the time-fractional Allen-Cahn equation, where the time-fractional derivative is in the sense of Caputo with order α∈(0,1). Considering the weak singularity of the solution u(x,t) at the starting time, i.e., its first and/or second derivatives with respect to time blowing-up as t→0+ albeit the function itself being right continuous at t=0, two well-known difference formulas, including the nonuniform L1 formula and the nonuniform L2-1σ formula, which are used to approximate the Caputo time-fractional derivative, respectively, and the local discontinuous Galerkin (LDG) method is applied to discretize the spatial derivative. With the help of discrete fractional Gronwall-type inequalities, the stability and optimal error estimates of the fully discrete numerical schemes are demonstrated. Numerical experiments are presented to validate the theoretical results.

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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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Second order difference schemes for time-fractional KdV–Burgers’ equation with initial singularity

Cited by 41 publications

References 8 publications

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

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

Local Discontinuous Galerkin Method Coupled with Nonuniform Time Discretizations for Solving the Time-Fractional Allen-Cahn Equation

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

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