10.57647/mathsci.2026.2004.24

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

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  • Manal Elzain Mohamed Abdalla1,
  • Hasanen A. Hammad2,3,
  • Tarek Aboelenen2,4,
  1. Department of Mathematics, College of Science, King Khalid University, Abha 62521, Saudi Arabia
  2. Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
  3. Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt
  4. Department of Mathematics, Faculty of Science, Assuit University, Assuit 71515, Egypt

Received: 2025-12-23

Revised: 2026-02-09

Accepted: 2026-04-18

Published Online: 2026-05-31

Copyright (c) 2026 Manal Elzain Mohamed Abdalla, Hasanen A. Hammad, Tarek Aboelenen (Author)

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Abdalla, M. E. M., Hammad, H. A., & Aboelenen, T. (2026). Solving Multi-order Nonlinear Caputo Fractional Differential Equations and Numerical Simulations. Mathematical Sciences. https://doi.org/10.57647/mathsci.2026.2004.24
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Abstract

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.

Keywords

  • Fractional derivative,
  • Fixed point technique,
  • Multi fractional order,
  • Boundary condition,
  • Numerical simulation
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