Existence of solutions of fractal fractional partial differential equations through different contractions

In the past, the existence and uniqueness of the solutions of fractional differential equations have been investigated by many researchers theoretically in various approaches in the literature. In this paper, there is no discussion of the existence of solutions for the nonlinear differential equations with fractal fractional operators. The objective of this work is to present novel contraction approaches, notably the $\propto$- $\psi$-contraction $\propto$-type of the $\widetilde{\mathtt{\text{F}}}$-contraction, within the context of $\hat{F}$-metric and orbital metric spaces. The aim of this study is to illustrate certain fixed point theorems that offer a new and direct approach to establish the existence and uniqueness of the solution to the general partial differential equations by employing the fractal fractional operators.