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

TL;DR

This paper establishes novel connections between Brownian-time and inverse-stable-Lévy-time Brownian sheets with systems of fractional and high-order PDEs, revealing new interactions, memory effects, and equivalence conditions through stochastic and fractional calculus methods.

Contribution

It introduces a new proof linking Brownian-time sheets to fractional and high-order PDE systems, and defines inverse-stable-Lévy-time sheets connected to memory-preserving PDEs.

Findings

Connected BTBS to half-derivative PDEs and fourth order systems.

Introduced inverse-stable-Lévy-time Brownian sheets and linked them to fractional PDEs.

Provided a necessary and sufficient condition for equivalence of fractional and high-order PDE systems.

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.