Abstract
We use an approximation method to study the solution to a nonlinear heat conduction equation in a semi‐infinite domain. By expanding an energy density function (defined as the internal energy per unit volume) as a Taylor polynomial in a spatial domain, we reduce the partial differential equation to a set of first‐order ordinary differential equations in time. We describe a systematic approach to derive approximate solutions using Taylor polynomials of a different degree. For a special case, we derive an analytical solution and compare it with the result of a self‐similar analysis. A comparison with the numerically integrated results demonstrates good accuracy of our approximate solutions. We also show that our approximation method can be applied to cases where boundary energy density and the corresponding effective conductivity are more general than those that are suitable for the self‐similar method. Propagation of nonlinear heat waves is studied for different boundary energy density and the conductivity functions.
Highlights
The nonlinear heat equation, as given in 1.1, has applications in various branches of science and engineering, including thermal processing of materials 1, liquid movement in porous media 2, and radiation heat wave 3
We focus on analysis of nonlinear heat waves governed by a nonlinear heat equation in a semi-infinite domain, with zero initial condition and a prescribed boundary condition at the origin
We have developed an approximation method for solving nonlinear heat conduction problem
Summary
The nonlinear heat equation, as given in 1.1 , has applications in various branches of science and engineering, including thermal processing of materials 1 , liquid movement in porous media 2 , and radiation heat wave 3. In order to obtain a self-similar solution to the parabolic differential equation, boundary temperature was assumed to be either an exponential 10 or a power function 3 of time and conductivity was assumed to be a power function of temperature. This limited the application of self-similar method to more general and realistic cases. Another approach, initiated by Parlange, is based on the observation that even though the conductivity may be a complicated function, its integral is far easier to handle By assuming such a chosen integral a quadratic polynomial in spatial coordinate, Parlange et al 2 obtained approximate analytical solution of the nonlinear diffusion equation for arbitrary boundary conditions.
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