Fast Compact Difference Scheme for the Fourth-Order Time Multi-Term Fractional Sub-Diffusion Equations With the First Dirichlet Boundary

Gao, Guang‐Hua; Telecommunications, Nanjing; Xu, Peng; Tang, Rui

doi:10.11948/20200405

Cited by 3 publications

(3 citation statements)

References 38 publications

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“…A fourth‐order convergent difference scheme was derived along with some analyses by the discrete energy method. Gao, Tang and Yang [7] considered another average operator of two points near the boundary to construct a fourth‐order convergent difference scheme for solving the multiterm time‐fractional subdiffusion problem. For the fourth‐order fractional subdiffusion equation with spatially variable coefficient, Pu, Ran and Luo [23] introduced a new average operator of two points near the boundary to match with the overall accuracy of the algorithms based on variable coefficient problems and a second‐order finite difference scheme was constructed.…”

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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“…By introducing the first derivative of the unknown function as an auxiliary function, the Stephenson scheme is employed to approximate the spatial derivatives in Arshad et al [16] and Cui [17], and the three-point stencil compact approximation was derived to construct the fourth-order accurate scheme in Mohanty and Kaur [18,19]. By introducing the second derivative of the unknown function as an auxiliary function, the fourth-order fractional sub-diffusion equations were considered in Ji et al [20] and Gao et al [21] with the spatially constant coefficient and in Pu et al [22] with the spatially variable coefficient.…”

Section: Introductionmentioning

confidence: 99%

The compact difference approach for the fourth‐order parabolic equations with the first Neumann boundary value conditions

Gao

Huang

Sun

2023

Math Methods in App Sciences

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In this paper, the fourth‐order parabolic equation with the first Neumann boundary value conditions is concerned, where the values of the first and second spatial derivatives of the unknown function at the boundary are given. A compact difference scheme is established for this kind of problem by using the weighted average and order reduction methods. The difficulty lies in the challenges of handling the boundary conditions with high accuracy. The unique solvability, convergence and stability of the proposed compact difference scheme are proved by the energy method. Some novel techniques are introduced for the analysis. Then, the extension to a more general case with the reaction term is briefly explored. Finally, two numerical examples are numerically calculated to show the efficiency of the proposed numerical schemes.

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“…The weighted average of the second spatial derivatives at four points was approximated adjacent to the boundary, and the weighed average at three inner points was compactly approximated, where the global fourth-order convergence of the difference scheme in space was shown. Gao et al applied the method of order reduction in [11] to convert the original problem into an equivalent second-order system, and then they introduced a simple average of two points near the boundary to ensure that the derived difference scheme achieved the global second-order convergence in time and fourth-order convergence in space.…”

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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Fast Compact Difference Scheme for the Fourth-Order Time Multi-Term Fractional Sub-Diffusion Equations With the First Dirichlet Boundary

Cited by 3 publications

References 38 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

The compact difference approach for the fourth‐order parabolic equations with the first Neumann boundary value conditions

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

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