Finite Time Blowup of Integer- and Fractional-Order Time-Delayed Diffusion Equations

TL;DR

This paper derives exact solutions for integer- and fractional-order time-delayed diffusion equations, demonstrating conditions for finite time blowup and contrasting behaviors with standard diffusion solutions.

Contribution

It introduces a method to obtain exact solutions for delayed diffusion equations and establishes conditions for finite time blowup, expanding understanding of their dynamics.

Findings

Solutions do not exhibit infinite speed of propagation for smooth initial conditions.

Finite time blowup can be explicitly calculated under certain initial conditions.

Contrasts with standard diffusion dynamics are highlighted.

Abstract

In this work, exact solutions are derived for an integer- and fractional-order time-delayed diffusion equation with arbitrary initial conditions. The solutions are obtained using Fourier transform methods in conjunction with the known properties of delay functions. It is observed that the solutions do not exhibit infinite speed of propagation for smooth initial conditions that are bounded and positive. Sufficient conditions on the initial condition are also established such that the finite time blowup of the solutions can be explicitly calculated. Examples are provided that highlight the contrasting behaviours of these exact solutions with the known dynamics of solutions to the standard diffusion equation.