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基于布朗时间和逆稳定Lévy时间布朗单的时间分数阶和$Δ^ν$偏微分方程系统

Interacting time-fractional and $Δ^ν$ PDEs systems via Brownian-time and Inverse-stable-Lévy-time Brownian sheets

Hassan Allouba, Erkan Nane

备注：
29 pages. Added examples of initial functions (Section 0.6) and added a stronger version of our main results in Section 3 that weakens the boundedness conditions on f and its derivatives. Other minor improvements throughout. To appear in Stochastics and Dynamics
发表信息：
Stochastics and Dynamics. 13 (2013), no. 1, 1250012, 31 pp
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摘要：

最近，应用和抽象数学及相关学科中的许多现象都用高阶和分数偏微分方程来表达。最近，Allouba引入了布朗时间布朗单（BTBS），并将其连接到一个新的四阶相互作用偏微分方程系统。这种多参数BTBS-PDE连接中的相互作用是新颖的，导致了著名的Kuramoto-Sivashinsky PDE的紧密连接的线性系统变体，并且不与其一次性参数对应物共享。这也意味着，这些偏微分方程系统是解决一个家庭的功能，在众所周知的流体动力学模型中表现出的功能。另一方面，单参数和多参数布朗时间偏微分方程的解和初始数据之间的记忆保持相互作用是共同的。在这里，我们引入了一个新的-即使在一个参数的情况下-证明，明智地结合了随机分析与分析和分数阶微积分，同时链接BTBS到一个新的系统的时间半导数相互作用偏微分方程，以及四阶系统证明较早和不同的Allouba。然后，我们介绍了一般类的随机场，我们称之为逆稳定-列维-时间布朗单（ISLTBS），我们把它们连接到$β$ -分数时间导数系统的相互作用偏微分方程$0<β<1$。当$β=1/ν$，$ν\in\lbr2,3,...\rbr$，我们的证明也连接一个ISLTBS的系统的内存保存$ν$ -拉普拉斯互动PDE。存储器通过初始数据的时间缩放的$k$ -拉普拉斯算子的总和$k=1,..., ν-1$来表示。利用Fourier-Laplace变换-分数阶微积分方法，给出了分数阶系统与高阶系统等价的条件等价性结果，给出了分数阶系统与高阶系统等价的充要条件.在一个参数的情况下，这个条件自动成立。

英文摘要：

Lately, many phenomena in both applied and abstract mathematics and related disciplines have been expressed in terms of high order and fractional PDEs. Recently, Allouba introduced the Brownian-time Brownian sheet (BTBS) and connected it to a new system of fourth order interacting PDEs. The interaction in this multiparameter BTBS-PDEs connection is novel, leads to an intimately-connected linear system variant of the celebrated Kuramoto-Sivashinsky PDE, and is not shared with its one-time-parameter counterpart. It also means that these PDEs systems are to be solved for a family of functions, a feature exhibited in well known fluids dynamics models. On the other hand, the memory-preserving interaction between the PDE solution and the initial data is common to both the single and the multi parameter Brownian-time PDEs. Here, we introduce a new---even in the one parameter case---proof that judiciously combines stochastic analysis with analysis and fractional calculus to simultaneously link BTBS to a new system of temporally half-derivative interacting PDEs as well as to the fourth order system proved earlier and differently by Allouba. We then introduce a general class of random fields we call inverse-stable-Lévy-time Brownian sheets (ISLTBSs), and we link them to $β$-fractional-time-derivative systems of interacting PDEs for $0<β<1$. When $β=1/ν$, $ν\in\lbr2,3,...\rbr$, our proof also connects an ISLTBS to a system of memory-preserving $ν$-Laplacian interacting PDEs. Memory is expressed via a sum of temporally-scaled $k$-Laplacians of the initial data, $k=1,..., ν-1$. Using a Fourier-Laplace-transform-fractional-calculus approach, we give a conditional equivalence result that gives a necessary and sufficient condition for the equivalence between the fractional and the high order systems. In the one parameter case this condition automatically holds.

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