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Approximate solution of the nonlinear heat conduction equation in a semi-infinite domain. (English) Zbl 1203.80026
This paper develops an approximation method for solving a nonlinear heat conduction problem in a semi-infinite domain. Based on equivalent integral formulation of the considered problem, the partial differential equation is reduced to a set of first-order ordinary differential equations in time by means of the expansion of an energy density function (defined as the internal energy per unit volume) as a Taylor polynomial in a spatial domain. A systematic approach to derive approximate solutions using Taylor polynomials of a different degree is described. For a special case, an analytical solution is derived and compared with the result of a self-similar analysis. The comparison with both self-similar and numerical integrated solutions confirms the accuracy of the approximate results and shows the suitability and efficiency of the approximation method for more general and realistic cases since it assumes no restriction on the form of boundary energy density and heat conductivity. The consistent improvement of the quadratic approximation over the linear one indicates that the method can be systematically extended to achieve higher order of accuracy. Propagation of nonlinear heat waves is studied for different boundary energy density and the conductivity functions. The idea in the approximation method is quite general, as it can be applied to any nonlinear physical processes in which the solution exhibits wave front behavior. This makes the approximation methodology potentially useful in many application fields, including fluid dynamics, combustion, and environmental and material sciences.
MSC:
80M25Other numerical methods (thermodynamics)
35K05Heat equation
80A20Heat and mass transfer, heat flow
35K55Nonlinear parabolic equations
34B15Nonlinear boundary value problems for ODE
35Q79PDEs in connection with classical thermodynamics and heat transfer
35K86Nonlinear parabolic unilateral problems; nonlinear parabolic variational inequalities