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## Interacting time-fractional and $\Delta^{\nu}$ PDEs systems via Brownian-time and Inverse-stable-L\'evy-time Brownian sheets

### Abstract

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-L\'evy-time Brownian sheets (ISLTBSs), and we link them to $\beta$-fractional-time-derivative systems of interacting PDEs for $0<\beta<1$. When $\beta=1/\nu$, $\nu\in\lbr2,3,...\rbr$, our proof also connects an ISLTBS to a system of memory-preserving $\nu$-Laplacian interacting PDEs. Memory is expressed via a sum of temporally-scaled $k$-Laplacians of the initial data, $k=1,..., \nu-1$. 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.Comment: 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 Dynamic

Topics: Mathematics - Probability, Mathematical Physics, Mathematics - Analysis of PDEs, 35C15, 35G31, 35G46, 60H30, 60G60, 60J45, 60J35 (Primary) 60J60, 60J65 (Secondary)
Year: 2012
OAI identifier: oai:arXiv.org:1105.0652