Analyzing the Existence and Uniqueness of Solutions in Coupled Fractional Differential Equations

• Published in: International journal of applied and computational mathematics Vol. 11; no. 2
• Main Authors: Ansari, Intesham, Dubey, Rishika, Devi, Amita, Kumar, Anoop
• Format: Journal Article
• Language: English
• Published: New Delhi Springer India 01.04.2025; Springer Nature B.V
• Subjects: Applications of Mathematics; Computational Science and Engineering; Differential equations; Fixed points (mathematics); Fractional calculus; Green's functions; Mathematical and Computational Physics; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Nuclear Energy; Operations Research/Decision Theory; Original Paper; Theorems; Theoretical; Uniqueness; Krasnoselskii’s fixed point theorem; Banach Contraction Mapping principle; Caputo fractional derivative; Coupled fractional differential equation
• ISSN: 2349-5103; 2199-5796
• DOI: 10.1007/s40819-025-01876-z

This paper investigates a mixed fractional differential equation (FDE) involving both left-sided and right-sided Caputo fractional derivatives. Our primary contributions are as follows: First, we introduce foundational lemmas and definitions pertinent to the problem. Second, we develop a solution composed of Green’s function and an additional constant term, thoroughly determining the Green’s function and its properties. Third, we establish the existence of solutions using Krasnoselskii’s fixed point theorem and prove their uniqueness through the Banach fixed point theorem. These contributions collectively provide a robust framework for solving mixed FDEs, highlighting the utility of these mathematical methods in addressing complex fractional differential equations.