Exact Solutions of Time-Delay Integer- and Fractional-Order Advection Equations

TL;DR

This paper derives exact solutions for time-delay advection equations with integer and fractional derivatives, revealing unique oscillatory and translatory behaviors distinct from standard wave solutions.

Contribution

It introduces a method to obtain exact solutions for delay differential advection equations using delay functions and separation of variables.

Findings

Solutions exhibit oscillatory behavior

Solutions show translatory dynamics

Behavior differs from classical wave solutions

Abstract

Transport phenomena play a vital role in various fields of science and engineering. In this work, exact solutions are derived for advection equations with integer- and fractional-order time derivatives and a constant time-delay in the spatial derivative. Solutions are obtained, for arbitrary separable initial conditions, by incorporating recently introduced delay functions in a separation of variables approach. Examples are provided showing oscillatory and translatory behaviours that are fundamentally different to standard propagating wave solutions.