Abstract
In this article analytical solution of one-dimensional heat equation in relaxation mode of heat generation and conduction using Laplace transforms method is presented. The model adopted takes into account finite velocity of heat propagation, and relaxation of heat source capacity. The properties of heat source terms in four different cases are incorporated in the model and investigated. Temperature distributions and variations with conduction mode and relaxation time are analyzed. High relaxation time is observed to lowers the temperature profile, whereas enhanced temperature distribution changes at particular values of α, andfor source capacity proportional to temperature. How the steady state solution is achieved for some selected values of coefficients is also discussed. Key words: Relaxation time, conduction mode, pulsed heat source, Laplace transforms.
Highlights
Cattaneo was the first to build an explicit mathematical theory to correct unacceptable properties of Fourier theory of heat diffusion
The model adopted takes into account finite velocity of heat propagation, and relaxation of heat source capacity
This leads to suitable heat conduction models that permit the finite speed of heat flow (Ozisik and Tzou, 1994; Joseph and Preziosi, 1990)
Summary
Cattaneo was the first to build an explicit mathematical theory to correct unacceptable properties of Fourier theory of heat diffusion. For example when the heat is of constant strength this differences slowly decrease for long times Solutions of both hyperbolic and parabolic heat conduction equation for temperature dependent heat source is reported to be use in analyzing normal zones in superconductors. The boundary value problem of Equation (9) was solved after including four different source terms namely: (i) source with constant capacity, (ii) source capacity proportional to temperature, (iii) Dirac delta energy pulse, and (iv) source capacity proportional to time. Riemann-sum approximation (Basant and Clement, 2013) is used for the inversion of the sets of Equation
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