Existence, uniqueness, stability results for tripled system which containing of mixed fractional derivatives with laplacian operator

Decades ago, fractional calculus arose to generalize ordinary derivation and integration, and then became a means of modeling and interpreting many phenomena in various fields such as engineering, physics, chemistry, biology and signal processing. The definition of the fractional derivative began with a derivative with a singular kernel, such as the Riemann-Liouville and Caputo derivative. The study of existence and uniqueness of solution of fractional partial differential equations are then proved by mani methods as: the well-known Lax–Milgram theorem, Energy estimate and fixed point theorem... Among them, In this work we investigate the existence and uniqueness of solutions for a new class of tripled system, which containing of mixed fractional derivatives between the Riemann-Liouville and the Caputo fractional derivatives of different orders with Laplacian operator, by using Leray-Schauder alternative fixed point theorem type in generalized Banach space and Banach contraction principle. The stability of the Ulam type of the proposed coupled system is also studied. Due to the importance of variable-order fractional calculus, both from a theoretical and an applied perspective, we think that the results obtained can be of interest to the research pursuits of many readers. At the end, to conclude this paper we provide an example illustrating our findings and approach.