Study of impulsive problems under Mittag-Leffler power law

Area of Science:

• Mathematics
• Applied Mathematics
• Fractional Calculus

Background:

• Impulsive fractional differential equations (IFDEs) are crucial in modeling complex systems with sudden changes.
• The Atangana-Baleanu-Caputo (ABC) fractional derivative offers a nonsingular kernel, enhancing modeling capabilities.
• Understanding the existence and uniqueness of solutions for IFDEs is fundamental for their application.

Purpose of the Study:

• To derive the solution formula for two types of Cauchy problems involving IFDEs with the ABC fractional derivative.
• To establish the existence and uniqueness of solutions for these specific Cauchy problems.
• To illustrate the practical application and effectiveness of the derived methods through examples.

Main Methods:

• Utilizing principles of nonlinear functional analysis.
• Applying fixed-point theorems to guarantee solution properties.
• Developing analytical techniques for deriving solution formulas.

Main Results:

• A comprehensive solution formula for the considered Cauchy problems is derived.
• The existence and uniqueness of solutions are rigorously proven.
• Numerical examples are provided to validate the theoretical findings.

Conclusions:

• The study successfully provides a framework for analyzing IFDEs with ABC fractional derivatives.
• The established existence and uniqueness results are vital for the reliable application of these models.
• The presented methodology offers a robust approach for solving and understanding complex dynamical systems.