Evolution, its Fractional Extension and Generalization

TL;DR

This paper explores the fractional extension of evolution equations, replacing the first time derivative with a fractional derivative, and establishes a relationship between their solutions, broadening the understanding of such equations.

Contribution

It introduces a fractional extension of evolution equations and derives a relationship between solutions of the classical and fractional forms, enhancing the theoretical framework.

Findings

Derived a relationship between classical and fractional evolution solutions

Extended evolution equations to fractional derivatives of order 0<α≤1

Provided theoretical insights into fractional evolution dynamics

Abstract

The evolution of a quantity, described by a function of space and time, relates the first derivative in time of this function to a spatial operator applied to the function. The initial value of the function at time $t=0$ is given. The fractional extension of this evolution consists of replacing the first derivative in time by a fractional derivative of order $\alpha$, $0 < \alpha \le 1$. We give a relationship between the solution of the equation of evolution and the solution of the equation belonging to its fractional extension.