Pathway to Fractional Integrals, Fractional Differential Equations and the Role of H-function

TL;DR

This paper explores the pathway model's connections to fractional calculus, statistical mechanics, and integral transforms, emphasizing the central role of H-function representations in unifying these concepts.

Contribution

It demonstrates how the pathway parameter relates to fractional order and unifies various mathematical and physical models through H-function representations.

Findings

Pathway parameter linked to fractional order.

H-function serves as a unifying representation.

Connections established with statistical mechanics and integral transforms.

Abstract

The pathway model for the real scalar variable case is re-explored and its connections to fractional integrals, solutions of fractional differential equations, Tsallis statistics and superstatistics in statistical mechanics, reaction-rate probability integral, Kraetzel transform, and pathway transform are explored. It is shown that the common thread in these connections is the H-function representations. The pathway parameter is shown to be connected to the fractional order in fractional integrals and fractional differential equations.