  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.
