Second Order Compact Difference Scheme for Time Fractional Sub-diffusion Fourth-Order Neutral Delay Differential Equations

Nandal, Sarita; Pandey, Dwijendra N.

doi:10.1007/s12591-020-00527-7

Cited by 8 publications

(2 citation statements)

References 25 publications

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“…Nandal and Pandey [21] proposed a linearized second‐order numerical method for solving the fourth‐order distributed‐fractional subdiffusion differential equation with time delay. In [22], a compact difference scheme was derived and analyzed for the time‐fractional subdiffusion fourth‐order neutral delay differential equations. Zhang and Pu [38] discussed the compact difference scheme for the fourth‐order fractional subdiffusion equation and the extension to the two‐dimensional problem was also considered.…”

Section: Introductionmentioning

confidence: 99%

Finite difference schemes for the fourth‐order parabolic equations with different boundary value conditions

Gao

Sun

2022

Numerical Methods Partial

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In this paper, the fourth-order parabolic equations with different boundary value conditions are studied. Six kinds of boundary value conditions are proposed. Several numerical differential formulae for the fourth-order derivative are established by the quartic interpolation polynomials and their truncation errors are given with the aid of the Taylor expansion with the integral remainders. Effective difference schemes are presented for the third Dirichlet boundary value problem, the first Neumann boundary value problem and the third Neumann boundary value problem, respectively. Some new embedding inequalities on the discrete function spaces are presented and proved. With the method of energy analysis, the unique solvability, unconditional stability and unconditional convergence of the difference schemes are proved. The convergence orders of derived difference schemes are all O(𝜏 2 + h 2 ) in appropriate norms. Finally, some numerical examples are provided to confirm the theoretical results.

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Section: Introductionmentioning

confidence: 99%

Finite difference schemes for the fourth‐order parabolic equations with different boundary value conditions

Gao

Sun

2022

Numerical Methods Partial

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“…Another numerical way is to reduce the order of the original fourth-order problem in space by introducing the second derivative of the unknown function as an auxiliary function and transform it into an equivalent second-order differential system, where the values of the unknown function and the introduced auxiliary function are both given at the boundary; hence, no special consideration will be needed at the boundary for numerical approximation. The authors adopted this numerical way in [14][15][16] to construct effective difference schemes for solving the nonlinear fourth-order reaction-diffusion equation, the fourth-order time-fractional wave equation and time-fractional subdiffusion fourth-order neutral delay differential equations, respectively.…”

Section: Introductionmentioning

confidence: 99%

Pointwise error estimate of the compact difference methods for the fourth‐order parabolic equations with the third Neumann boundary conditions

Gao,

Huang,

Sun

2023

Math Methods in App Sciences

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Various boundary value conditions have been endowed for the fourth‐order differential equations. In the current work, a class of the fourth‐order parabolic equations with the third Neumann boundary conditions is concerned, where the values of the second and third spatial derivatives of the unknown function are given at the boundary. A novel average is defined to get the compact approximation near the boundary. Then a compact difference scheme is derived by using the weighted average and the method of order reduction. Due to the special and highly accurate discretization at the boundary, the related terms have to be handled skillfully during the analysis. By the energy method, the unique solvability, unconditional stability, and convergence of the derived compact difference scheme are strictly proved. The analytical difficulties caused by the boundary approximation are successfully overcome. As far as we know, this is the first time that the global pointwise fourth‐order convergence in space of the difference approach for this problem is achieved. In addition, the generalization to solve the case with a space‐dependent reaction coefficient is discussed. Finally, two numerical examples are computed to verify the accuracy of proposed numerical schemes.

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Numerical technique for fractional variable-order differential equation of fourth-order with delay

Nandal

Pandey

2021

Applied Numerical Mathematics

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Second Order Compact Difference Scheme for Time Fractional Sub-diffusion Fourth-Order Neutral Delay Differential Equations

Cited by 8 publications

References 25 publications

Finite difference schemes for the fourth‐order parabolic equations with different boundary value conditions

Finite difference schemes for the fourth‐order parabolic equations with different boundary value conditions

Pointwise error estimate of the compact difference methods for the fourth‐order parabolic equations with the third Neumann boundary conditions

Numerical technique for fractional variable-order differential equation of fourth-order with delay

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