A novel approach to fractional calculus and its applications to well-known problems

Abstract

This paper introduces a novel definition of fractional derivatives and fractional integrals using the conformable derivative approach. This new framework not only aligns more closely with the classical concept of derivatives but also provides a more practical and intuitive structure for fractional calculus. The proposed definition is applicable in two key ranges: 0 ≤ α < 1 and n−1 ≤ α < n, where n is a positive integer. We also demonstrate that when α =1, our definition seamlessly corresponds to the classical first-order derivative. The advantages of this approach include improved compatibility with classical calculus and enhanced computational convenience, making it a valuable tool for both theoretical investigations and practical applications. By bridging fractional calculus with traditional derivative concepts, our definition facilitates easier analysis and interpretation of fractional differential equations and their solutions. We further explore the implications of this definition in various contexts, including its impact on stability and convergence properties in numerical methods, and provide examples to illustrate its effectiveness and applicability.