Stability analysis for linear heat conduction with memory kernels described by Gamma functions

TL;DR

This paper investigates the stability of heat conduction with memory kernels modeled by Gamma functions, transforming the non-local problem into a local PDE system and deriving stability criteria and thresholds.

Contribution

It introduces a novel approach to analyze heat equations with Gamma function kernels by converting them into hyperbolic PDE systems and deriving stability conditions.

Findings

Derived stability conditions for heat equations with Gamma kernels.

Established sharp instability thresholds for specific kernel combinations.

Reformulated non-local equations as local hyperbolic PDE systems.

Abstract

This paper analyzes heat equation with memory in the case of kernels that are linear combinations of Gamma distributions. In this case, it is possible to rewrite the non-local equation as a local system of partial differential equations of hyperbolic type. Stability is studied in details by analyzing the corresponding dispersion relation, providing sufficient stability condition for the general case and sharp instability thresholds in the case of linear combination of the first three Gamma functions.