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

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.