Analysis of existence and structure of solutions for Caputo and Grünwald–Letnikov fractional differential systems

Peng E; Weihong Zhou; Tingting Xu; Jie Cao; Yuxia Liu; Shangxi Li; Xueliang Zhou; Wei Zhou; Peng E; Weihong Zhou; Tingting Xu; Jie Cao; Yuxia Liu; Shangxi Li; Xueliang Zhou; Wei Zhou

doi:10.3934/math.20251307

AIMS Mathematics

2025, Volume 10, Issue 12: 29732-29764. doi: 10.3934/math.20251307

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Analysis of existence and structure of solutions for Caputo and Grünwald–Letnikov fractional differential systems

1.
School of Mathematics and Computer Science, Yunnan Minzu University, Kunming, Yunnan, 650504, China
2.
Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, Kunming, Yunnan, 650011, China
3.
Center for Astrophysics and Great Bay Center of National Astronomical Data Center, Guangzhou University, Guangzhou, 510006, China

Received: 15 October 2025 Revised: 01 December 2025 Accepted: 04 December 2025 Published: 17 December 2025
MSC : 26A33, 34A08, 34A30, 34K37

In this paper, we investigate the existence and structure of power-type solutions for Caputo fractional differential equation systems (CFDESs) and Grünwald–Letnikov fractional differential equation systems (GLFDESs). Building on the definitions of the Caputo fractional derivative (CFD) and the Grünwald–Letnikov fractional derivative (GLFD), we derive explicit expansion formulas for the fractional differential operators, construct joint coefficient-solution matrices for the considered systems, and, from these, obtain necessary and sufficient rank conditions for the existence of $ m $-th order power solutions. On this basis, we further (1) establish equivalent criteria that guarantee the uniqueness of the degree of power solutions and (2) derive rank-based conditions for the existence of two, and more generally $ p $, linearly independent power particular solutions with distinct degrees. Taken together, these results provide a unified matrix-based theoretical framework for analyzing the existence, uniqueness, and multiplicity of power-type solutions and the associated system structure of the two types of fractional differential systems. Two numerical examples are also provided to demonstrate the validity of the proposed results.
- Caputo fractional derivative,
- Grünwald–Letnikov fractional derivative,
- linearly independent solution,
- rank of coefficient matrices
Citation: Peng E, Weihong Zhou, Tingting Xu, Jie Cao, Yuxia Liu, Shangxi Li, Xueliang Zhou, Wei Zhou. Analysis of existence and structure of solutions for Caputo and Grünwald–Letnikov fractional differential systems[J]. AIMS Mathematics, 2025, 10(12): 29732-29764. doi: 10.3934/math.20251307

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Abstract

In this paper, we investigate the existence and structure of power-type solutions for Caputo fractional differential equation systems (CFDESs) and Grünwald–Letnikov fractional differential equation systems (GLFDESs). Building on the definitions of the Caputo fractional derivative (CFD) and the Grünwald–Letnikov fractional derivative (GLFD), we derive explicit expansion formulas for the fractional differential operators, construct joint coefficient-solution matrices for the considered systems, and, from these, obtain necessary and sufficient rank conditions for the existence of $ m $-th order power solutions. On this basis, we further (1) establish equivalent criteria that guarantee the uniqueness of the degree of power solutions and (2) derive rank-based conditions for the existence of two, and more generally $ p $, linearly independent power particular solutions with distinct degrees. Taken together, these results provide a unified matrix-based theoretical framework for analyzing the existence, uniqueness, and multiplicity of power-type solutions and the associated system structure of the two types of fractional differential systems. Two numerical examples are also provided to demonstrate the validity of the proposed results.

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