Fast algorithms for high-dimensional variable-order space-time fractional diffusion equations

This paper derives a new operational matrix of the variable-order (VO) time fractional partial derivative involved in anomalous diffusion for shifted Chebyshev polynomials. We then develop an accurate numerical algorithm to solve the 1 + 1 and 2 + 1 VO and constant-order fractional diffusion equation with Dirichlet conditions. The contraction of the present method is based on shifted Chebyshev collocation procedure in combination with the derived shifted Chebyshev operational matrix. The main advantage of the proposed method is to investigate a global approximation for spatial and temporal discretizations, and it reduces such problems to those of solving a system of algebraic equations, which greatly simplifies the solution process. In addition, we analyze the convergence of the present method graphically. Finally, comparisons between the algorithm derived in this paper and the existing algorithms are given, which show that our numerical schemes exhibit better performances than the existing ones.

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On Time-Fractional Diffusion Equations with Space-Dependent Variable Order

Annales Henri Poincaré ◽

10.1007/s00023-018-0734-y ◽

2018 ◽

Vol 19 (12) ◽

pp. 3855-3881 ◽

Cited By ~ 15

Author(s):

Yavar Kian ◽

Eric Soccorsi ◽

Masahiro Yamamoto

Keyword(s):

Fractional Diffusion ◽

Diffusion Equations ◽

Variable Order ◽

Fractional Diffusion Equations

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The Maximum Principle for Variable-Order Fractional Diffusion Equations and the Estimates of Higher Variable-Order Fractional Derivatives

Frontiers in Physics ◽

10.3389/fphy.2020.580554 ◽

2020 ◽

Vol 8 ◽

Author(s):

Guangming Xue ◽

Funing Lin ◽

Guangwang Su

Keyword(s):

Maximum Principle ◽

Fractional Derivatives ◽

Arbitrary Point ◽

Initial Boundary ◽

Fractional Diffusion ◽

Diffusion Equations ◽

Maximum Principles ◽

Variable Order ◽

The Maximum Principle ◽

Fractional Diffusion Equations

In this paper, the maximum principle of variable-order fractional diffusion equations and the estimates of fractional derivatives with higher variable order are investigated. Firstly, we deduce the fractional derivative of a function of higher variable order at an arbitrary point. We also give an estimate of the error. Some important inequalities for fractional derivatives of variable order at arbitrary points and extreme points are presented. Then, the maximum principles of Riesz-Caputo fractional differential equations in terms of the multi-term space-time variable order are proved. Finally, under the initial-boundary value conditions, it is verified via the proposed principle that the solutions are unique, and their continuous dependance holds.

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Crank-Nicolson-weighted-shifted-Grünwald-difference schemes for space Riesz variable-order fractional diffusion equations

Numerical Algorithms ◽

10.1007/s11075-020-00980-z ◽

2020 ◽

Author(s):

Fu-Rong Lin ◽

Qiu-Ya Wang ◽

Xiao-Qing Jin

Keyword(s):

Fractional Diffusion ◽

Difference Schemes ◽

Diffusion Equations ◽

Variable Order ◽

Fractional Diffusion Equations ◽

Crank Nicolson

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Fast algorithms for high-order numerical methods for space-fractional diffusion equations

International Journal of Computer Mathematics ◽

10.1080/00207160.2016.1149579 ◽

2016 ◽

Vol 94 (5) ◽

pp. 1062-1078 ◽

Cited By ~ 18

Author(s):

Siu-Long Lei ◽

Yun-Chi Huang

Keyword(s):

Numerical Methods ◽

Fast Algorithms ◽

Fractional Diffusion ◽

High Order ◽

Diffusion Equations ◽

Fractional Diffusion Equations

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Multilevel Circulant Preconditioner for High-Dimensional Fractional Diffusion Equations

East Asian Journal on Applied Mathematics ◽

10.4208/eajam.060815.180116a ◽

2016 ◽

Vol 6 (2) ◽

pp. 109-130 ◽

Cited By ~ 20

Author(s):

Siu-Long Lei ◽

Xu Chen ◽

Xinhe Zhang

Keyword(s):

Krylov Subspace ◽

Fractional Diffusion ◽

Diffusion Equations ◽

Low Rank ◽

High Dimensional ◽

Fractional Diffusion Equations ◽

Implicit Finite Difference Scheme ◽

Small Norm ◽

Variable Diffusion ◽

Variable Diffusion Coefficients

AbstractHigh-dimensional two-sided space fractional diffusion equations with variable diffusion coefficients are discussed. The problems can be solved by an implicit finite difference scheme that is proven to be uniquely solvable, unconditionally stable and first-order convergent in the infinity norm. A nonsingular multilevel circulant pre-conditoner is proposed to accelerate the convergence rate of the Krylov subspace linear system solver efficiently. The preconditoned matrix for fast convergence is a sum of the identity matrix, a matrix with small norm, and a matrix with low rank under certain conditions. Moreover, the preconditioner is practical, with an O(NlogN) operation cost and O(N) memory requirement. Illustrative numerical examples are also presented.

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