Integral decomposition for the solutions of the generalized Cattaneo equation

Area of Science:

• Mathematical Physics
• Fractional Calculus
• Partial Differential Equations

Background:

• The Cattaneo equation describes heat transport, but standard models struggle with memory effects.
• Fractional calculus offers a way to incorporate non-local time dependencies via memory kernels.

Purpose of the Study:

• To present a novel integral decomposition for the fundamental solution of the generalized Cattaneo equation with memory kernels.
• To analyze time-fractional Cattaneo equations using Caputo derivatives and power-law kernels.
• To interpret these decompositions via subordination theory and explore limiting cases.

Main Methods:

• Integral decomposition of the fundamental solution.
• Analytic inversion from Fourier-Laplace domain using the Efross theorem.
• Application of complete Bernstein function theory for subordination interpretation.

Main Results:

• Solutions are integral decompositions linking standard Cattaneo solutions with memory-dependent functions.
• The Efross theorem allows inversion beyond Gaussian distributions.
• Subordination interpretation holds for diffusion and wave equation limits.

Conclusions:

• The developed method provides a new framework for solving generalized Cattaneo equations with memory effects.
• This approach resolves issues with Brownian motion subordination for specific power-law kernels (α∈(1/2,1]).
• The findings extend the understanding of anomalous transport phenomena governed by fractional differential equations.