Time fractional diffusion equations: solution concepts, regularity, and long-time behavior

2019 ◽  
pp. 159-180 ◽  
Author(s):  
Rico Zacher
Keyword(s):  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Solution Concepts ◽  
Fractional Diffusion Equations ◽  
Long Time

Long-time behavior of solutions and chaos in reaction-diffusion equations

2017 ◽  
Vol 99 ◽  
pp. 91-100 ◽  
Author(s):  
Kamal N. Soltanov ◽  
Anatolij K. Prykarpatski ◽  
Denis Blackmore
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time

LONG-TIME BEHAVIOR OF SOLUTIONS TO NONLINEAR REACTION DIFFUSION EQUATIONS INVOLVING L1 DATA

2012 ◽  
Vol 14 (01) ◽  
pp. 1250007 ◽  
Author(s):  
CHENGKUI ZHONG ◽  
WEISHENG NIU
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Smooth Bounded Domain ◽  
Bootstrap Argument ◽  
Nonlinear Reaction ◽  
L1 Data ◽  
Long Time

In this paper we consider the long-time behavior of solutions to nonlinear reaction diffusion equations involving L1 data, [Formula: see text] where Ω is a smooth bounded domain and u0, g ∈ L1(Ω). Using a decomposition technique combined with a bootstrap argument we establish some uniform regularity results on the solutions, by which we prove that the solution semigroup generated by the problem above possesses a global attractor [Formula: see text] in L1(Ω). Moreover, we obtain that the attractor is actually invariant, compact in [Formula: see text], q < max {N/(N-1), (2p-2)/p}, and attracts every bounded subset of L1(Ω) in the norm of [Formula: see text], 1 ≤ r < ∞.

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.

A Novel Finite Element Method for a Class of Time Fractional Diffusion Equations

2011 ◽  
Author(s):  
H. G. Sun ◽  
W. Chen ◽  
K. Y. Sze
Keyword(s):  
Fractional Diffusion ◽  
Transport Processes ◽  
Anomalous Transport ◽  
Diffusion Equations ◽  
Time Range ◽  
Main Task ◽  
Fractional Diffusion Equations ◽  
Long Time ◽  
Range Computation ◽  
Good Agreement

Anomalous transport of contaminants in groundwater or porous soil is a research focus in hydrology and soil science for decades. Because fractional diffusion equations can well characterize early breakthrough and heavy tail decay features of contaminant transport process, they have been considered as promising tools to simulate anomalous transport processes in complex media. However, the efficient and accurate computation of fractional diffusion equations is a main task in their applications. In this paper, we introduce a novel numerical method which captures the critical Mittag-Leffler decay feature of subdiffusion in time direction, to solve a class of time fractional diffusion equations. A key advantage of the new method is that it overcomes the critical problem in the application of time fractional differential equations: long-time range computation. To illustrate its efficiency and simplicity, three typical academic examples are presented. Numerical results show a good agreement with the exact solutions.

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