Contemporary Theoretical Challenges and Practical Advances in Fractional and Generalized Calculus

Fractional calculus, whose origins are as ancient as those of classical calculus, has undergone significant development in recent decades, proving its usefulness across diverse fields such as engineering, physics, biology, and the social sciences. Although the first ideas on non-integer order derivatives and integrals date back to the works of Leibniz and Liouville, it wasn't until the 1960s that formalizations began using local differential operators. The field reached greater theoretical maturity in 2014 with the introduction of definitions that enabled a comprehensive approach to memory effects and complex dynamic processes. This project aims to assess the current state of knowledge in fractional and generalized calculus, identify the main limitations of traditional methods, and develop new theoretical and numerical approaches to effectively address problems in differential equations, dynamical systems, and integral inequalities. Key objectives include unifying various definitions—such as those of Riemann–Liouville, Caputo, Grunwald–Letnikov, and other newer conformable and non-conformable operators—and creating robust numerical algorithms for simulating complex processes. The relevance of this research lies in its dual impact: advancing mathematical theory while simultaneously providing practical tools for modeling anomalous phenomena, with potential implications for the modernization of higher education.

General Objective:
To construct a comprehensive theoretical and methodological framework for systematically addressing challenges inherent to Fractional and Generalized Calculus across various application domains.

Specific Objectives:

To develop new fractional and generalized operators aimed at solving problems related to differential equations, mathematical modeling, and integral inequalities.

To achieve theoretical advancements, generalizations, and refinements of key topics in Qualitative Theory based on existing literature.

To define novel notions of convexity directly linked to new forms of integral inequalities, within the scope of fractional or generalized operators.

Fractional calculus, Generalized calculus, Fractional derivatives and integrals, Differential equations, Conformable operators, Integral inequalities.