Non-Fourier heat conduction equation in a sphere; comparison of variational method and inverse Laplace transformation with exact solution

Small scale thermal devices, such as micro heater, have led researchers to consider more accurate models of heat in thermal systems. Moreover, biological applications of heat transfer such as simulation of temperature field in laser surgery is another pathway which urges us to re-examine thermal systems with modern ones. Non-Fourier heat transfer overcomes some shortcomings of Fourier heat transfer, when small scale systems as considered or non-homogeneous materials are under study. In this paper, the hyperbolic heat conduction problem in a sphere is solved by three approaches.۱. Finding the exact solution by using the method of separation of variables۲. Finding two approximate solutions by using the Laplace transformation and thena. applying the variational method for finding the Laplace inverseb. finding the solution of the problem in Laplace domain and using an asymptotic series to evaluate the solution for small values of timesVarious orders for the variational method are considered and compared against analytical solution. Since the two latter methods can be used in nonlinear problems such as those include radiation heat loss, the approximate solutions can be useful addition in the field of thermal analysis of non-Fourier problems.

Keywords:

Non-Fourier heat conduction , variational formulation , Laplace transformation , separation of variables , spherical coordinate