Analysis of a generalized proportional fractional stochastic differential equation incorporating Carathéodory's approximation and applications

Fractional stochastic differential equations (FSDEs) with fractional derivatives describe the anomalous diffusion processes by incorporating the memory effects and spatial heterogeneities of the porous medium. The stochastic component addresses the random nature of the fluid flow due to the variability in pore sizes and connectivity. The first objective of this research is to prove the well-posedness of a class of generalized proportional FSDEs, and we acquire the global existence and uniqueness of findings under certain settings that are coherent with the classic SDEs. The secondary purpose is to evaluate the continuity of findings in fractional-order formulations. The Carathéodory approximation is taken into account for a class of generalized proportional FSDEs, which is pivotal and provides well-known bounds on the norm of the solutions. Carathéodory’s approximation aids in approximating the FSDEs governing turbulent flows, ensuring the solutions are mathematically robust and physically meaningful. As is widely documented, the existence and uniqueness of solutions to certain types of differential equations can be formed under Lipschitz and linear growth conditions. Furthermore, a class of generalized proportional FSDEs with time delays is considered according to certain new requirements. With the aid of well-known inequalities and Itô isometry technique, the Ulam–Hyers stability of the analyzed framework is addressed utilizing Lipschitz and non-Lipschitz characteristics, respectively. Additionally, we provide two illustrative examples as applications to demonstrate the authenticity of our interpretations. The demonstrated outcomes will generalize some previously published findings. Finally, this deviation from fractional Brownian motion necessitates a model that can capture the subdiffusive or superdiffusive behavior.