A fast direct method for block triangular Toeplitz-like with tri-diagonal block systems from time-fractional partial differential equations

Ke, Rihuan; Ng, Michael K.; Sun, Hai‐Wei

doi:10.1016/j.jcp.2015.09.042

Cited by 85 publications

(53 citation statements)

References 30 publications

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“…Finally, we remark that the AIM here cannot be extended directly to the time-dependent diffusion coefficient case, that is, k = k(x, t), because the resulting matrix A in this situation will not be in the format of Equation 8. For the time-dependent diffusion coefficient case, one could employ the method proposed by Ke et al, 16 whose complexity is of (mn log 2 n) operations and the storage requirement is of (mn).…”

Section: Discussionmentioning

confidence: 99%

“…Tables 3 and 4 present the numerical results for a variety of . Note that the method proposed by Ke et al, 16 which combines the BFS method with the divide-and-conquer strategy, can also be used to solve the BLTT linear system. The cost of this method is typically (mn log 2 n) flops.…”

Section: Example 2 (See the Work Of Zhao Et Al 1 )mentioning

confidence: 99%

“…Note that if the numerical solutions in all time steps are stacked in one vector, then the finite-difference discretization of Equation 1 can lead to the following block lower triangular Toeplitz (BLTT) system 16,17 :…”

Section: Introductionmentioning

confidence: 99%

“…Obviously, when m is large, the block divide-and-conquer method is not competitive with the BFS method from both the computational cost and the storage requirement. A fast direct method, 16 which combines the BFS method with the divide-and-conquer strategy, was recently proposed to solve Equation 2. The computational cost of this method is of (mnlog 2 n) and the storage requirement is of (mn).…”

Section: Introductionmentioning

confidence: 99%

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Approximate inversion method for time‐fractional subdiffusion equations

Pang

Sun

et al. 2017

Numerical Linear Algebra App

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The finite-difference method applied to the time-fractional subdiffusion equation usually leads to a large-scale linear system with a block lower triangular Toeplitz coefficient matrix. The approximate inversion method is employed to solve this system. A sufficient condition is proved to guarantee the high accuracy of the approximate inversion method for solving the block lower triangular Toeplitz systems, which are easy to verify in practice and have a wide range of applications. The applications of this sufficient condition to several existing finite-difference schemes are investigated. Numerical experiments are presented to verify the validity of theoretical results. KEYWORDS approximate inversion method, block lower triangular Toeplitz matrix, fast Fourier transforms, matrix polynomial, time-fractional subdiffusion equations Numer Linear Algebra Appl. 2018;25:e2132.wileyonlinelibrary.com/journal/nla

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Section: Discussionmentioning

confidence: 99%

Section: Example 2 (See the Work Of Zhao Et Al 1 )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Approximate inversion method for time‐fractional subdiffusion equations

Pang

Sun

et al. 2017

Numerical Linear Algebra App

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“…Zigen Ouyang [14] obtained some conditions for the existence of the solutions of a class of nonlinear fractional order partial differential equations with delay. Rihuan et al [15] proposed the fast direct method for solving the linear block lower triangular Toeplitz-like with tridiagonal blocks system which arises from the time-fractional partial differential equation. Komal Singla and R.K. Gupta [16] introduced an extension of the concept of nonlinear self-adjointness and Noether operators for calculating conserved vectors of the time fractional nonlinear systems of partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

Modified least squares homotopy perturbation method for solving fractional partial differential equations

Thabet¹,

Kendre²

2018

Malaya J. Mat.

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This paper introduces a new modification of least squares homotopy perturbation method (LSHPM) for solving linear and nonlinear fractional partial differential equations (FPDEs). The main advantage of the new modification is to approximate the solution for FPDEs in a full general set. Moreover, the convergence of the proposed modification is shown. Analytical and numerical solutions for the linear Navier-Stokes equation and the nonlinear gas dynamic equation are successfully obtained to confirm the accuracy and efficiency of the proposed modification.

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An efficient quadratic finite volume method for variable coefficient Riesz space‐fractional diffusion equations

2020

Math Methods in App Sciences

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A quadratic finite volume (FV) method for steady‐state Riesz space‐fractional diffusion equations (sFDEs) with variable diffusivity coefficient is developed using piecewise quadratic basis functions, and a resulting linear algebra system with two‐by‐two block‐type Toeplitz‐like coefficient matrix is formulated. It is proved that the method requires a minimum memory of order scriptOfalse(Nfalse), where N is the number of spatial partitions. Moreover, as two of the produced Toeplitz‐like submatrices are not square, a new fast nonsquare Toeplitz‐like matrix‐vector product is specially designed, which requires an almost linear computational complexity of order scriptOfalse(Nlog2Nfalse). Then, a fast version of biconjugate gradient stabilized (BiCGSTAB) solution algorithm, named FBiCGSTAB, is proposed for the FV scheme. The FV method combined with Crank‐Nicolson (CN) time discretization is applied to solve time‐dependent sFDEs, and an efficient CN‐FV scheme is developed and analyzed. Finally, numerical results are presented to show the utility of the fast FV and fast CN‐FV methods.

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A fast direct method for block triangular Toeplitz-like with tri-diagonal block systems from time-fractional partial differential equations

Cited by 85 publications

References 30 publications

Approximate inversion method for time‐fractional subdiffusion equations

Approximate inversion method for time‐fractional subdiffusion equations

Modified least squares homotopy perturbation method for solving fractional partial differential equations

An efficient quadratic finite volume method for variable coefficient Riesz space‐fractional diffusion equations

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