# Analytical and numerical treatment of the heat conduction equation obtained via time-fractional distributed-order heat conduction law

## TL;DR

This paper develops analytical and numerical methods to solve a generalized heat conduction equation incorporating fractional and distributed-order laws, demonstrating their effectiveness through numerical examples.

## Contribution

It introduces a combined analytical and numerical framework for solving distributed-order fractional heat conduction equations, extending classical models.

## Key findings

- Analytical solutions obtained via Fourier and Laplace transforms.
- Numerical solutions using finite difference schemes show good agreement with analytical results.
- The methods effectively handle multi-term and power-type distributed-order laws.

## Abstract

Generalization of the heat conduction equation is obtained by considering the system of equations consisting of the energy balance equation and fractional-order constitutive heat conduction law, assumed in the form of the distributed-order Cattaneo type. The Cauchy problem for system of energy balance equation and constitutive heat conduction law is treated analytically through Fourier and Laplace integral transform methods, as well as numerically by the method of finite differences through Adams-Bashforth and Gr\"{u}nwald-Letnikov schemes for approximation derivatives in temporal domain and leap frog scheme for spatial derivatives. Numerical examples, showing time evolution of temperature and heat flux spatial profiles, demonstrate applicability and good agreement of both methods in cases of multi-term and power-type distributed-order heat conduction laws.