Wellposedness and regularity of the variable-order time-fractional diffusion equations
Wellposedness and regularity of the variable-order time-fractional diffusion equations
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变阶时间分数扩散方程的适定性和正则性
DOI：
10.1016/j.jmaa.2019.03.052
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发表时间：
2019-07
影响因子：
1.3
通讯作者：
Xiangcheng Zheng
中科院分区：
文献类型：
--
作者：
Hong Wang;Xiangcheng Zheng
证明了时间分数阶偏微分方程的解在初始时刻t=0时缺乏正则性，这往往与所建模的物理问题不一致，尽管时间分数阶偏微分方程很好地捕捉了远离t=0的物理问题的本质.我们认为，这一现象发生的根本原因在于时间分数阶偏微分方程的非局部性与经典初始条件在t=0时的局部性之间的不相容性.我们证明了变阶时间分数阶偏微分方程，其中阶数随着时间t变为0而变化，以适应时间t=0时局部初始条件的影响，原来是一个自然的候选人，以消除非物理奇异的解决方案，（constant-order）时间-分数阶偏微分方程，并开辟了机会，从非本地到本地动态建模多物理现象，反之亦然。证明了多维变阶线性时间分数阶扩散偏微分方程的适定性。我们还证明了它的解决方案的正则性依赖于行为的变量阶（及其衍生物）在时间t=0，除了通常的光滑性假设。更准确地说，我们证明了如果变量阶在t=0时具有整数限制，则其解与其整数阶类似物一样具有完全正则性，或者如果变量阶在t = 0时具有非整数值，则其解在t = 0时表现出奇异行为，就像常阶时间分数偏微分方程的情况一样。
It was proved that the solution to a time-fractional partial differential equation lacks regularity at the initial time t=0, which is often inconsistent with the physical problem being modeled even though the nature of the physical problem away from t=0 is well captured by a time-fractional partial differential equation. We argue that the fundamental reason why this phenomenon occurs lies in the incompatibility between the nonlocality of the time-fractional partial differential equations and the locality of the classical initial condition at t=0. We show that the variable-order time-fractional partial differential equations, in which the order varies in time as tgoes to 0 to accommodate the impact of the local initial condition at time t=0, turn out to be a natural candidate to eliminate the nonphysical singularity of the solutions to the (constant-order) time-fractional partial differential equations and open up opportunities for modeling multiphysics phenomena from nonlocal to local dynamics and vice versa. We prove the wellposedness of a variable-order linear time-fractional diffusion partial differential equation in multiple space dimensions. We also prove that the regularity of its solution depends on the behavior of the variable order (and its derivatives) at time t=0, in addition to the usual smoothness assumptions. More precisely, we prove that its solution has full regularity like its integer-order analogue if the variable order has an integer limit at t=0 or exhibits singular behaviors at t=0 like in the case of the constant-order time-fractional partial differential equations if the variable order has a non-integer value at time t=0.
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DOI：
10.1103/physreve.93.032151
发表时间：
2016-01
期刊：
Physical review. E
影响因子：
--
作者：
Xiaochao Wu;W. Deng;E. Barkai
通讯作者：
Xiaochao Wu;W. Deng;E. Barkai
使用多保真贝叶斯优化从现场数据发现平流弥散方程的可变分数阶
影响因子：
4.1
作者：
Guofei Pang;Paris Perdikaris;Wei Cai;George Em Karniadakis
通讯作者：
George Em Karniadakis
DOI：
10.1007/978-0-387-49319-0
发表时间：
1941
期刊：
--
影响因子：
--
作者：
J. Jost
通讯作者：
J. Jost
影响因子：
3.5
作者：
R. Courant;David Hilbert;T. Teichmann
通讯作者：
R. Courant;David Hilbert;T. Teichmann
影响因子：
0.8
作者：
K. Diethelm;N. Ford
通讯作者：
K. Diethelm;N. Ford