An implicit difference scheme for time-fractional diffusion equations with a time-invariant type variable order

Gu, Xian-Ming; Sun, Hai‐Wei; Zhao, Yongliang; Zheng, Xiangcheng

doi:10.1016/j.aml.2021.107270

Cited by 44 publications

(24 citation statements)

References 19 publications

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“…Consequently, the tests that we ran as part of this paper are relevant in the context of the Lame equation [5]. Furthermore, fractional differential equations have also become popular in recent years [6,7].…”

Section: Introductionmentioning

confidence: 99%

Scalability of k-Tridiagonal Matrix Singular Value Decomposition

et al. 2021

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Singular value decomposition has recently seen a great theoretical improvement for k-tridiagonal matrices, obtaining a considerable speed up over all previous implementations, but at the cost of not ordering the singular values. We provide here a refinement of this method, proving that reordering singular values does not affect performance. We complement our refinement with a scalability study on a real physical cluster setup, offering surprising results. Thus, this method provides a major step up over standard industry implementations.

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Section: Introductionmentioning

confidence: 99%

Scalability of k-Tridiagonal Matrix Singular Value Decomposition

et al. 2021

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“…From our review of several applications of FDM to solve Caputo's time-fractional mathematical equations, ref. [5] utilized the implicit FDM to solve the time-fractional diffusion equation with a time-invariant type variable. Subsequently, ref.…”

Section: Introductionmentioning

confidence: 99%

Quarter-Sweep Preconditioned Relaxation Method, Algorithm and Efficiency Analysis for Fractional Mathematical Equation

Sunarto¹,

Agarwal

Sulaiman

et al. 2021

Fractal Fract

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Research into the recent developments for solving fractional mathematical equations requires accurate and efficient numerical methods. Although many numerical methods based on Caputo’s fractional derivative have been proposed to solve fractional mathematical equations, the efficiency of obtaining solutions using these methods when dealing with a large matrix requires further study. The matrix size influences the accuracy of the solution. Therefore, this paper proposes a quarter-sweep finite difference scheme with a preconditioned relaxation-based approximation to efficiently solve a large matrix, which is based on the establishment of a linear system for a fractional mathematical equation. The paper presents the formulation of the quarter-sweep finite difference scheme that is used to approximate the selected fractional mathematical equation. Then, the derivation of a preconditioned relaxation method based on a quarter-sweep scheme is discussed. The design of a C++ algorithm of the proposed quarter-sweep preconditioned relaxation method is shown and, finally, efficiency analysis comparing the proposed method with several tested methods is presented. The contributions of this paper are the presentation of a new preconditioned matrix to restructure the developed linear system, and the derivation of an efficient preconditioned relaxation iterative method for solving a fractional mathematical equation. By simulating the solutions of time-fractional diffusion problems with the proposed numerical method, the study found that computing solutions using the quarter-sweep preconditioned relaxation method is more efficient than using the tested methods. The proposed numerical method is able to solve the selected problems with fewer iterations and a faster execution time than the tested existing methods. The efficiency of the methods was evaluated using different matrix sizes. Thus, the combination of a quarter-sweep finite difference method, Caputo’s time-fractional derivative, and the preconditioned successive over-relaxation method showed good potential for solving different types of fractional mathematical equations, and provides a future direction for this field of research.

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“…Recently, in [29], Garrappa et al investigated the main properties of the emerging variable-order operators and discussed some practical applications of the variable-order Scarpi integral and derivative. Gu et al [30] proposed an unconditionally stable implicit difference scheme for solving generalized time-space fractional diffusion equations with variable coefficients with numerical scheme that utilizes the L1-type formula for the generalized Caputo-fractional derivative in time discretization and the second-order weighted and shifted Gr € unwald difference formula in spatial discretization. Also, Fang et al [31] developed a fast finite difference method for solving a class of variable-order time fractional diffusion equations.…”

Section: Introductionmentioning

confidence: 99%

Variable-Order Fractional Diffusion Model-Based Medical Image Denoising

Abirami¹,

Prakash

2021

Mathematical Problems in Engineering

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Fractional differential models are playing a vital role in many applications such as diffusion, probability potential theory, and scattering theory. In this study, the variable-order space and time fractional diffusion model is employed for denoising the medical images. The finite difference approach is implemented to find the numerical solution of the proposed model. Convergence and stability of the numerical method are presented. The experimental outcomes of the variable-order model are analyzed with those of the fractional and integer-order diffusion models. It was noticed that the peak signal-to-noise ratio (PSNR) value is increased considerably for the proposed model.

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An implicit difference scheme for time-fractional diffusion equations with a time-invariant type variable order

Cited by 44 publications

References 19 publications

Scalability of k-Tridiagonal Matrix Singular Value Decomposition

Scalability of k-Tridiagonal Matrix Singular Value Decomposition

Quarter-Sweep Preconditioned Relaxation Method, Algorithm and Efficiency Analysis for Fractional Mathematical Equation

Variable-Order Fractional Diffusion Model-Based Medical Image Denoising

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