A Compact Difference Scheme for a Class of Variable Coefficient Quasilinear Parabolic Equations with Delay

A linearized compact difference scheme is provided for a class of variable coefficient parabolic systems with delay. The unique solvability, unconditional stability, and convergence of the difference scheme are proved, where the convergence order is four in space and two in time. A numerical test is presented to illustrate the theoretical results.

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A fast compact difference scheme for the fourth-order multi-term fractional sub-diffusion equation with non-smooth solution

Filomat ◽

10.2298/fil2105495c ◽

2021 ◽

Vol 35 (5) ◽

pp. 1495-1509

Author(s):

Dakang Cen ◽

Zhibo Wang ◽

Yan Mo

Keyword(s):

Diffusion Equation ◽

Boundary Conditions ◽

Difference Scheme ◽

Fourth Order ◽

Neumann Boundary ◽

Neumann Boundary Conditions ◽

Stability And Convergence ◽

Compact Difference Scheme ◽

Compact Difference ◽

Graded Meshes

In this paper, we develop a fast compact difference scheme for the fourth-order multi-term fractional sub-diffusion equation with Neumann boundary conditions. Combining L1 formula on graded meshes and the efficient sum-of-exponentials approximation to the kernels, the proposed scheme recovers the losing temporal convergence accuracy and spares the computational costs. Meanwhile, difficulty caused by the Neumann boundary conditions and fourth-order derivative is also carefully handled. The unique solvability, unconditional stability and convergence of the proposed scheme are analyzed by the energy method. At last, the theoretical results are verified by numerical experiments.

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A Compact Difference Scheme for Solving Fractional Neutral Parabolic Differential Equation with Proportional Delay

Journal of Function Spaces ◽

10.1155/2017/3679526 ◽

2017 ◽

Vol 2017 ◽

pp. 1-8 ◽

Cited By ~ 1

Author(s):

Wei Gu ◽

Yanli Zhou ◽

Xiangyu Ge

Keyword(s):

Differential Equation ◽

Finite Difference ◽

Difference Scheme ◽

Numerical Test ◽

Unconditional Stability ◽

Parabolic Differential Equation ◽

Proportional Delay ◽

Compact Difference Scheme ◽

Compact Finite Difference Scheme ◽

Compact Difference

A linearized compact finite difference scheme is constructed for solving the fractional neutral parabolic differential equation with proportional delay. By the energy method, the unconditional stability of the scheme is proved, and the convergence order of the scheme is proved to be O(τ2-α+h4). A numerical test is also conducted to validate the accuracy and efficiency of the numerical algorithm.

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A Crank-Nicolson Difference Scheme for Solving a Type of Variable Coefficient Delay Partial Differential Equations

Journal of Applied Mathematics ◽

10.1155/2014/560567 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6 ◽

Cited By ~ 5

Author(s):

Wei Gu ◽

Peng Wang

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Difference Scheme ◽

Variable Coefficient ◽

Numerical Test ◽

Partial Differential ◽

The Difference ◽

Delay Partial Differential Equations ◽

Theoretical Results ◽

Crank Nicolson

A linearized Crank-Nicolson difference scheme is constructed to solve a type of variable coefficient delay partial differential equations. The difference scheme is proved to be unconditionally stable and convergent, where the convergence order is two in both space and time. A numerical test is provided to illustrate the theoretical results.

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Convergence and performance of iterative methods for solving variable coefficient convection-diffusion equation with a fourth-order compact difference scheme

Computers & Mathematics with Applications ◽

10.1016/s0898-1221(02)00162-1 ◽

2002 ◽

Vol 44 (3-4) ◽

pp. 457-479 ◽

Cited By ~ 17

Author(s):

S. Karaa ◽

Jun Zhang

Keyword(s):

Diffusion Equation ◽

Difference Scheme ◽

Iterative Methods ◽

Fourth Order ◽

Variable Coefficient ◽

Compact Difference Scheme ◽

Convection Diffusion ◽

Convection Diffusion Equation ◽

Compact Difference ◽

And Performance

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A High-Order Difference Scheme for the Generalized Cattaneo Equation

East Asian Journal on Applied Mathematics ◽

10.4208/eajam.110312.240412a ◽

2012 ◽

Vol 2 (2) ◽

pp. 170-184 ◽

Cited By ~ 19

Author(s):

Seak-Weng Vong ◽

Hong-Kui Pang ◽

Xiao-Qing Jin

Keyword(s):

Difference Scheme ◽

Fractional Part ◽

High Order ◽

Stability And Convergence ◽

Compact Difference Scheme ◽

Order Difference ◽

Compact Difference ◽

Fractional Cattaneo Equation ◽

The Stability ◽

High Order Finite Difference

AbstractA high-order finite difference scheme for the fractional Cattaneo equation is investigated. The L1 approximation is invoked for the time fractional part, and a compact difference scheme is applied to approximate the second-order space derivative. The stability and convergence rate are discussed in the maximum norm by the energy method. Numerical examples are provided to verify the effectiveness and accuracy of the proposed difference scheme.

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