Nonlinear heat conduction equations with memory: physical meaning and analytical results

TL;DR

This paper explores nonlinear heat conduction equations with memory effects using fractional calculus, deriving models with finite thermal propagation velocity and providing analytical solutions that relate to classical results.

Contribution

It introduces new nonlinear fractional models for heat conduction with memory, employing Mittag-Leffler and power law functions, and offers explicit analytical solutions.

Findings

Derived nonlinear time-fractional telegraph and wave equations

Provided explicit analytical solutions using generalized separating variable method

Connected fractional models to classical heat conduction results

Abstract

We study nonlinear heat conduction equations with memory effects within the framework of the fractional calculus approach to the generalized Maxwell-Cattaneo law. Our main aim is to derive the governing equations of heat propagation, considering both the empirical temperature-dependence of the thermal conductivity coefficient (which introduces nonlinearity) and memory effects, according to the general theory of Gurtin and Pipkin of finite velocity thermal propagation with memory. In this framework, we consider in detail two different approaches to the generalized Maxwell-Cattaneo law, based on the application of long-tail Mittag-Leffler memory function and power law relaxation functions, leading to nonlinear time-fractional telegraph and wave-type equations. We also discuss some explicit analytical results to the model equations based on the generalized separating variable method and discuss their meaning in relation to some well-known results of the ordinary case.