Solving Multi-order Nonlinear Caputo Fractional Differential Equations and Numerical Simulations

This paper addresses the challenging problem of establishing the existence and uniqueness of solutions for coupled multi-order fractional differential equations involving multiple fractional orders, a class of equations that arises in various scientific and engineering applications. By employing the powerful Banach contraction principle, we derive novel sufficient conditions that guarantee the existence and uniqueness of solutions. The theoretical findings are further substantiated through the presentation and analysis of a detailed illustrative example, highlighting the practical relevance of our results. To validate the theoretical results, we implement a modified Adams-Bashforth-Moulton predictor-corrector method for the numerical solution of the proposed multi-term Caputo fractional differential systems. In particular, three illustrative examples are presented to demonstrate the impact of fractional orders and memory effects on system dynamics. These simulations, covering nonlinear coupled oscillators, neural feedback processes, and epidemic models. The results confirm the robustness of the theoretical framework and the applicability of the method to real-world problems governed by fractional dynamics.