scholarly journals A fast method for variable-order Caputo fractional derivative with applications to time-fractional diffusion equations

2020 ◽  
Vol 80 (5) ◽  
pp. 1443-1458 ◽  
Author(s):  
Zhi-Wei Fang ◽  
Hai-Wei Sun ◽  
Hong Wang
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Variable Order ◽  
Fast Method ◽  
Fractional Diffusion Equations

Fast Evaluation of the Caputo Fractional Derivative and its Applications to Fractional Diffusion Equations: A Second-Order Scheme

2017 ◽  
Vol 22 (4) ◽  
pp. 1028-1048 ◽  
Author(s):  
Yonggui Yan ◽  
Zhi-Zhong Sun ◽  
Jiwei Zhang
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Computational Cost ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Fast Evaluation ◽  
Fractional Diffusion Equations ◽  
Long Time ◽  
Order Scheme ◽  
Speed Up

AbstractThe fractional derivatives include nonlocal information and thus their calculation requires huge storage and computational cost for long time simulations. We present an efficient and high-order accurate numerical formula to speed up the evaluation of the Caputo fractional derivative based on theL2-1σformula proposed in [A. Alikhanov,J. Comput. Phys., 280 (2015), pp. 424-438], and employing the sum-of-exponentials approximation to the kernel function appeared in the Caputo fractional derivative. Both theoretically and numerically, we prove that while applied to solving time fractional diffusion equations, our scheme not only has unconditional stability and high accuracy but also reduces the storage and computational cost.

scholarly journals A fast method for variable-order space-fractional diffusion equations

2020 ◽  
Vol 85 (4) ◽  
pp. 1519-1540 ◽  
Author(s):  
Jinhong Jia ◽  
Xiangcheng Zheng ◽  
Hongfei Fu ◽  
Pingfei Dai ◽  
Hong Wang
Keyword(s):  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Variable Order ◽  
Fast Method ◽  
Fractional Diffusion Equations

An Efficient Operational Matrix Technique for Multidimensional Variable-Order Time Fractional Diffusion Equations

2016 ◽  
Vol 11 (6) ◽  
Author(s):  
M. A. Zaky ◽  
S. S. Ezz-Eldien ◽  
E. H. Doha ◽  
J. A. Tenreiro Machado ◽  
A. H. Bhrawy
Keyword(s):  
Present Method ◽  
Operational Matrix ◽  
Solution Process ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Variable Order ◽  
Algebraic Equations ◽  
Global Approximation ◽  
Fractional Diffusion Equations ◽  
Accurate Numerical Algorithm

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.

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

2018 ◽  
Vol 19 (12) ◽  
pp. 3855-3881 ◽  
Author(s):  
Yavar Kian ◽  
Eric Soccorsi ◽  
Masahiro Yamamoto
Keyword(s):  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Variable Order ◽  
Fractional Diffusion Equations

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

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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