OICC Press Mathematical Sciences 2251-7456 20 3 2026 09 30 Solving Multi-order Nonlinear Caputo Fractional Differential Equations and Numerical Simulations 10.57647/mathsci.2026.2004.24 EN Manal Elzain Mohamed Abdalla Department of Mathematics, College of Science, King Khalid University, Abha 62521, Saudi Arabia https://orcid.org/0009-0001-8951-8185 Hasanen A. Hammad Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia; Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt https://orcid.org/0000-0001-8724-9367 Tarek Aboelenen Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia; Department of Mathematics, Faculty of Science, Assuit University, Assuit 71515, Egypt Journal Article 2026 09 30 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.