Fractional approximation of solutions of evolution equations

TL;DR

This paper presents a method to approximate solutions of first-order linear evolution equations, including their analytic continuations, by solutions of time-fractional equations as the fractional order approaches one.

Contribution

It introduces a novel approximation technique connecting classical evolution equations with fractional-order equations for better analytical handling.

Findings

Approximation improves as fractional order approaches one.

Method applies to solutions with analytic continuation.

Provides a new perspective on fractional calculus in evolution equations.

Abstract

We show how to approximate a solution of the first order linear evolution equation, together with its possible analytic continuation, using a solution of the time-fractional equation of order $\delta >1$, where $\delta \to 1+0$.