Integral decomposition for the solutions of the generalized Cattaneo equation

TL;DR

This paper develops an integral decomposition method for solutions of the generalized Cattaneo equation with memory kernels, connecting it to subordination and extending beyond Gaussian-based approaches, especially for power-law kernels.

Contribution

It introduces a novel integral decomposition framework for the generalized Cattaneo equation using analytic methods, including Efross theorem and Bernstein functions, to handle complex memory kernels.

Findings

Solutions are represented by integral decompositions involving the standard Cattaneo solution and kernel-dependent functions.

The methodology extends the subordination concept to non-Gaussian kernels, including power-law cases.

The approach clarifies the limitations of Brownian motion-based subordination for certain power-law kernels.

Abstract

We present the integral decomposition for the fundamental solution of the generalized Cattaneo equation with both time derivatives smeared through convoluting them with some memory kernels. For power-law kernels $t^{-\alpha}$, $\alpha\in(0,1]$ this equation becomes the time fractional one governed by the Caputo derivatives which highest order is 2. To invert the solutions from the Fourier-Laplace domain to the space-time domain we use analytic methods based on the Efross theorem and find out that solutions looked for are represented by integral decompositions which tangle the fundamental solution of the standard Cattaneo equation with non-negative and normalizable functions being uniquely dependent on the memory kernels. Furthermore, the use of methodology arising from the theory of complete Bernstein functions allows us to assign such constructed integral decompositions the interpretation of subordination. This fact is preserved in two limit cases built into the generalized Cattaneo equations, i.e., either the diffusion or the wave equations. We point out that applying the Efross theorem enables us to go beyond the standard approach which usually leads to the integral decompositions involving the Gaussian distribution describing the Brownian motion. Our approach clarifies puzzling situation which takes place for the power-law kernels $t^{-\alpha}$ for which the subordination based on the Brownian motion does not work if $\alpha\in(1/2,1]$.