AI 中文总结
本文提出一类非线性多阶分数阶微分算子，定义相容多阶导数及其逆算子相容多阶积分，严格证明其性质，并展示在初等多阶微分方程求解中的应用。
AI 中文摘要
本文介绍了一类新颖的非线性多阶分数阶微分算子。首先，我们通过微分分析和Omega-导数公式，为多阶微分算子的存在性提供了直观且形式化的论证。随后，我们将相容多阶导数定义为分数阶导数的函数商，并严格建立了其解析性质。此外，利用数学分析的基本原理，我们构造了其逆算子，称为相容多阶积分。文中给出了所有理论性质的详细逐步证明。同时，深入探讨了示例及其在求解初等多阶微分方程中的应用。最后，讨论了全面的结论和未来研究方向的战略性展望。
英文摘要
This paper introduces a novel class of non-linear multiorder fractional differential operators. First, we provide intuitive and formal justifications for the existence of multiorder differential operators through differential analysis and Omega-derivative formulations. We then define the conformable multiorder derivative as a functional quotient of fractional derivatives and establish its analytical properties rigorously. Furthermore, leveraging fundamental principles of mathematical analysis, we construct its inverse operator, termed the conformable multiorder integral. Detailed step-by-step proofs of all theoretical properties are presented. Illustrative examples and applications to solving elementary multiorder differential equations are thoroughly examined. Finally, comprehensive conclusions and strategic outlooks for prospective research directions are discussed.
