An efficient Chebyshev-tau method for solving the space fractional diffusion equations

Ren, Rong-Fen; Li, Hong; Jiang, Wei; Song, Ming-Yan

doi:10.1016/j.amc.2013.08.073

Cited by 24 publications

(13 citation statements)

References 11 publications

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“…A numerical approach was provided for the FDE based on a spectral tau method [12]. An efficient method based on the shifted Chebyshev-tau idea was presented for solving the space fractional diffusion equations [13]. Tau method is very effective for constant coefficient nonlinear problems, but the method is not generally adopted for nonlinear FDE.…”

Section: Introductionmentioning

confidence: 99%

An Operational Matrix of Fractional Differentiation of the Second Kind of Chebyshev Polynomial for Solving Multiterm Variable Order Fractional Differential Equation

Jian-ping

2016

Mathematical Problems in Engineering

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The multiterm fractional differential equation has a wide application in engineering problems. Therefore, we propose a method to solve multiterm variable order fractional differential equation based on the second kind of Chebyshev Polynomial. The main idea of this method is that we derive a kind of operational matrix of variable order fractional derivative for the second kind of Chebyshev Polynomial. With the operational matrices, the equation is transformed into the products of several dependent matrices, which can also be viewed as an algebraic system by making use of the collocation points. By solving the algebraic system, the numerical solution of original equation is acquired. Numerical examples show that only a small number of the second kinds of Chebyshev Polynomials are needed to obtain a satisfactory result, which demonstrates the validity of this method.

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Section: Introductionmentioning

confidence: 99%

An Operational Matrix of Fractional Differentiation of the Second Kind of Chebyshev Polynomial for Solving Multiterm Variable Order Fractional Differential Equation

Jian-ping

2016

Mathematical Problems in Engineering

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“…However, these studies verified the unconditional convergence of second‐order finite difference schemes often restrict diffusion coefficients positively bounded and relied on the spatial variable x . Besides, other numerical treatments including the Chebyshev‐tau, finite volume and finite element methods are proposed to solve the SFDEs with variable coefficients, refer, for example, to [6, 9, 19, 33–35, 42, 45] for details.…”

Section: Introductionmentioning

confidence: 99%

A fast implicit difference scheme for solving the generalized time–space fractional diffusion equations with variable coefficients

Huang

Zhao

et al. 2020

Numerical Methods Partial

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In this article, we first propose an unconditionally stable implicit difference scheme for solving generalized time–space fractional diffusion equations (GTSFDEs) with variable coefficients. The numerical scheme utilizes the L1‐type formula for the generalized Caputo fractional derivative in time discretization and the second‐order weighted and shifted Grünwald difference (WSGD) formula in spatial discretization, respectively. Theoretical results and numerical tests are conducted to verify the (2 − γ)‐order and 2‐order of temporal and spatial convergence with γ ∈ (0, 1) the order of Caputo fractional derivative, respectively. The fast sum‐of‐exponential approximation of the generalized Caputo fractional derivative and Toeplitz‐like coefficient matrices are also developed to accelerate the proposed implicit difference scheme. Numerical experiments show the effectiveness of the proposed numerical scheme and its good potential for large‐scale simulation of GTSFDEs.

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“…Another method for solving linear PDEs spectrally with Chebyshev polynomials is the Chebyshev-tau method, which involves solving linear PDEs in spectral space. In the Chebyshev-tau method, the boundary conditions of PDEs are directly enforced to the equation of the system [5][6][7]. This enforcement of boundary conditions produces tau lines, making numerical calculation slow because tau lines increase interactions between Chebyshev modes [8,9].…”

Section: Introductionmentioning

confidence: 99%

An Efficient Spectral Method to Solve Multi-Dimensional Linear Partial Different Equations Using Chebyshev Polynomials

2019

Mathematics

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We present a new method to efficiently solve a multi-dimensional linear Partial Differential Equation (PDE) called the quasi-inverse matrix diagonalization method. In the proposed method, the Chebyshev-Galerkin method is used to solve multi-dimensional PDEs spectrally. Efficient calculations are conducted by converting dense equations of systems sparse using the quasi-inverse technique and by separating coupled spectral modes using the matrix diagonalization method. When we applied the proposed method to 2-D and 3-D Poisson equations and coupled Helmholtz equations in 2-D and a Stokes problem in 3-D, the proposed method showed higher efficiency in all cases than other current methods such as the quasi-inverse method and the matrix diagonalization method in solving the multi-dimensional PDEs. Due to this efficiency of the proposed method, we believe it can be applied in various fields where multi-dimensional PDEs must be solved.

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An efficient Chebyshev-tau method for solving the space fractional diffusion equations

Cited by 24 publications

References 11 publications

An Operational Matrix of Fractional Differentiation of the Second Kind of Chebyshev Polynomial for Solving Multiterm Variable Order Fractional Differential Equation

An Operational Matrix of Fractional Differentiation of the Second Kind of Chebyshev Polynomial for Solving Multiterm Variable Order Fractional Differential Equation

A fast implicit difference scheme for solving the generalized time–space fractional diffusion equations with variable coefficients

An Efficient Spectral Method to Solve Multi-Dimensional Linear Partial Different Equations Using Chebyshev Polynomials

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