Numerical Solutions for Chemotaxis Systems Using Stochastic Fractional Calculus Models

This paper addresses the numerical solutions of fractional differential equations (FDEs) using the Generalized Kudryashov Method (GKM) in the context of the conformable fractional derivative. Fractional calculus, particularly the conformable derivative, provides a versatile framework for modeling systems exhibiting memory and hereditary properties commonly found in complex physical phenomena. Traditional integer-order derivatives lack the capability to accurately represent such dynamics, which fractional derivatives effectively handle. The conformable derivative, a recent addition to fractional calculus, retains many advantageous properties of integer-order differentiation, such as the chain rule, while extending to non-integer orders. The Generalized Kudryashov Method, initially developed for solving nonlinear ordinary differential equations, is adapted here to address nonlinear FDEs involving conformable derivatives. By employing a traveling wave transformation, the study converts fractional partial differential equations into ordinary differential equations, facilitating the application of GKM. Through this approach, the study derives numerical solutions, demonstrating the method’s ability to capture complex dynamics in nonlinear fractional systems. The results indicate that GKM, in conjunction with the conformable derivative, offers a robust tool for accurately approximating solutions of FDEs, with potential applications across fields such as fluid mechanics, quantum mechanics, and anomalous diffusion (Anomalous diffusion refers to a type of diffusion process that deviates from classical, or "normal," diffusion as described by Brownian motion and Fick's laws. In normal diffusion, particles spread out over time in a predictable, linear manner, where the mean squared displacement (MSD) of particles scales).