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Lately, many phenomena in both applied and abstract mathematics and relateddisciplines have been expressed in terms of high order and fractional PDEs.Recently, Allouba introduced the Brownian-time Brownian sheet (BTBS) andconnected it to a new system of fourth order interacting PDEs. The interactionin this multiparameter BTBS-PDEs connection is novel, leads to anintimately-connected linear system variant of the celebratedKuramoto-Sivashinsky PDE, and is not shared with its one-time-parametercounterpart. It also means that these PDEs systems are to be solved for afamily of functions, a feature exhibited in well known fluids dynamics models.On the other hand, the memory-preserving interaction between the PDE solutionand the initial data is common to both the single and the multi parameterBrownian-time PDEs. Here, we introduce a new---even in the one parametercase---proof that judiciously combines stochastic analysis with analysis andfractional calculus to simultaneously link BTBS to a new system of temporallyhalf-derivative interacting PDEs as well as to the fourth order system provedearlier and differently by Allouba. We then introduce a general class of randomfields we call inverse-stable-L\'evy-time Brownian sheets (ISLTBSs), and welink 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 alsoconnects an ISLTBS to a system of memory-preserving $\nu$-Laplacian interactingPDEs. Memory is expressed via a sum of temporally-scaled $k$-Laplacians of theinitial data, $k=1,..., \nu-1$. Using aFourier-Laplace-transform-fractional-calculus approach, we give a conditionalequivalence result that gives a necessary and sufficient condition for theequivalence between the fractional and the high order systems. In the oneparameter case this condition automatically holds.

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