Analysis of delay differential equations with dual caputo-type fractional derivatives using laplace transform methods

Area of Science:

• Fractional Calculus
• Nonlinear Dynamics
• Mathematical Modeling

Background:

• Systems with memory and delayed feedback are common in various scientific fields.
• Nonlinear fractional differential equations offer a powerful framework for modeling such complex behaviors.
• The generalized Caputo-Katugampola fractional derivative provides flexibility in modeling memory effects.

Purpose of the Study:

• To investigate initial value problems for nonlinear fractional differential equations with finite delay.
• To analyze the role of the generalized Caputo-Katugampola fractional derivative with parameter [Formula: see text].
• To establish the existence, uniqueness, and Ulam-Hyers stability of solutions.

Main Methods:

• ρ-Laplace transform to derive an equivalent integral formulation.
• Banach contraction principle and Schauder's fixed point theorem for existence and uniqueness.
• Analysis of Ulam-Hyers stability under specific conditions.

Main Results:

• Existence and uniqueness of solutions are proven under complementary assumptions.
• The generalized Caputo-Katugampola derivative parameter [Formula: see text] enhances modeling flexibility.
• Ulam-Hyers stability is demonstrated, indicating model robustness.

Conclusions:

• The study provides a comprehensive analysis of a class of nonlinear fractional delay differential equations.
• The employed fixed-point theorems and stability analysis offer valuable insights into the behavior of systems with memory.
• Numerical simulations using the L1 scheme validate the theoretical findings and demonstrate the method's applicability.