Abstract
Hyperbolic heat conduction equation with temperature dependent thermal properties is considered. The thermal conductivity, specific heat and density are assumed to be functions of temperature. The equation is cast into a non-dimensional form suitable for perturbation analysis. By employing a newly developed approximate symmetry theory, the approximate symmetries of the equation are calculated for the case of small variations in thermal properties. Various similarity solutions corresponding to the symmetries of first order equations are presented. For second order equations, the method of constructing approximate symmetries and similarity solutions are discussed. A linear functional variation is assumed for the thermal properties and a similarity solution is constructed using one of the first order solutions as an example.
Highlights
The hyperbolic heat conduction equation with temperature dependent thermal properties is given as follows [1] τ∂ ∂t ρ(T )C p (T) ∂T ∂t + ∂T ∂t = ∂ ∂x k(T) ∂T ∂x (1)where T is the temperature, x and t are the spatial and time variables and τ=α0/c02=k0/ρ0Cp0c02 is the relaxation time. α0 is the reference thermal diffusivity, c0, k0, ρ0 and Cp0 are the reference propagation speed, thermal conductivity, density and specific heat respectively
Corresponding to different first order solutions, various h functions can be calculated which leads to different symmetries and different solutions corresponding to the symmetries
An approximate symmetry theory newly developed is applied to the resulting equations
Summary
The hyperbolic heat conduction equation with temperature dependent thermal properties is given as follows [1]. A group classification is needed for the equation since the thermal properties are arbitrary functions of the temperature This analysis might be involved since there are three arbitrary functional dependences on the dependent variable. In the first method due to Baikov, Gazizov and Ibragimov[4], the dependent variable is not expanded in a perturbation series as should be done in an ordinary perturbation problem, rather, the infinitesimal generator is expanded in a perturbation series In this way, an approximate generator is found from which approximate solutions can be retrieved. The nonhomogenous term is a known function but different at each order of approximation Requiring this term to be an arbitrary function, the approximate symmetry of the original equation is defined as the exact symmetry of the non-homogenous linear equation [9, 10] in the third method. For the second order equations, the general method to retrieve similarity solutions is discussed and an example solution is presented
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