scholarly journals Regional Controllability of Anomalous Diffusion Generated by the Time Fractional Diffusion Equations

2015 ◽  
Author(s):  
Fudong Ge ◽  
YangQuan Chen ◽  
Chunhai Kou
Keyword(s):  
Anomalous Diffusion ◽  
Minimum Energy ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Fractional Diffusion Equations ◽  
Regional Controllability

This paper is concerned with the investigation of the regional controllability of the time fractional diffusion equations. First, some preliminaries and definitions of regional controllability of the system under consideration are introduced, which promote the existence contributions on controllability analysis. Then we analyze the regional controllability with minimum energy of the time fractional diffusion equations on two cases: B ∈ L (Rm, L2 (Ω)) and B ∉ L (Lm, L2 (Ω)). In the end, two applications are given to illustrate our obtained results.

An efficient localized collocation solver for anomalous diffusion on surfaces

2021 ◽  
Vol 24 (3) ◽  
pp. 865-894 ◽  
Author(s):  
Zhuochao Tang ◽  
Zhuojia Fu ◽  
HongGuang Sun ◽  
Xiaoting Liu
Keyword(s):  
Anomalous Diffusion ◽  
Early Period ◽  
Fractional Diffusion ◽  
Least Square ◽  
Diffusion Equations ◽  
Moving Least Square ◽  
Fractional Diffusion Equations ◽  
Time Stepping ◽  
Temporal Discretization ◽  
Surface Time

Abstract This paper introduces an efficient collocation solver, the generalized finite difference method (GFDM) combined with the recent-developed scale-dependent time stepping method (SD-TSM), to predict the anomalous diffusion behavior on surfaces governed by surface time-fractional diffusion equations. In the proposed solver, the GFDM is used in spatial discretization and SD-TSM is used in temporal discretization. Based on the moving least square theorem and Taylor series, the GFDM introduces the stencil selection algorithms to choose the stencil support of a certain node from the whole discretization nodes on the surface. It inherits the similar properties from the standard FDM and avoids the mesh generation, which is available particularly for high-dimensional irregular discretization nodes. The SD-TSM is a non-uniform temporal discretization method involving the idea of metric, which links the fractional derivative order with the non-uniform discretization strategy. Compared with the traditional time stepping methods, GFDM combined with SD-TSM deals well with the low accuracy in the early period. Numerical investigations are presented to demonstrate the efficiency and accuracy of the proposed GFDM in conjunction with SD-TSM for solving either single or coupled fractional diffusion equations on surfaces.

ANOMALOUS DIFFUSION AND GENERALIZED DIFFUSION EQUATIONS

2005 ◽  
Vol 05 (02) ◽  
pp. L275-L282 ◽  
Author(s):  
I. M. SOKOLOV ◽  
A. V. CHECHKIN
Keyword(s):  
Anomalous Diffusion ◽  
Diffusion Processes ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Fractional Operator ◽  
Fractional Diffusion Equations ◽  
Whole Number ◽  
Simple Scaling ◽  
Special Case ◽  
Distributed Order

Fractional diffusion equations are widely used to describe anomalous diffusion processes where the characteristic displacement scales as a power of time. The forms of such equations might differ with respect to the position of the corresponding fractional operator in addition to or instead of the whole-number derivative in the Fick's equation. For processes lacking simple scaling the corresponding description may be given by distributed-order equations. In the present paper different forms of distributed-order diffusion equations are considered. The properties of their solutions are discussed for a simple special case.

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