Uniqueness of a nonlinear integro-differential equation with nonlocal boundary condition and variable coefficients

Area of Science:

• Mathematics
• Fractional Calculus
• Differential Equations

Background:

• Fractional integro-differential equations are crucial in modeling complex phenomena.
• Nonlocal boundary conditions and variable coefficients present significant analytical challenges.
• Existing methods may not fully address the uniqueness of solutions for such intricate equations.

Purpose of the Study:

• To establish the uniqueness of solutions for a specific two-term nonlinear fractional integro-differential equation.
• To incorporate nonlocal boundary conditions and variable coefficients within the analysis.
• To demonstrate the practical applicability of the derived theoretical results.

Main Methods:

• Application of the Mittag-Leffler function for fractional calculus.
• Utilizing Babenko's approach for problem analysis.
• Employing Banach's contractive principle to prove solution uniqueness.

Main Results:

• A rigorous mathematical theorem is presented, guaranteeing the existence and uniqueness of solutions.
• The methodology effectively handles the complexities introduced by variable coefficients and nonlocal conditions.
• An illustrative example validates the theoretical findings and showcases practical applications.

Conclusions:

• The study successfully proves the uniqueness of solutions for the investigated fractional integro-differential equation.
• The employed techniques provide a robust framework for analyzing similar complex differential equations.
• The findings contribute to the theoretical understanding and practical application of fractional calculus in science and engineering.