Publikacje

Abstract:

The homotopy perturbation method (HPM) combined with Trefftz method is employed to find the solution
of two kinds of nonlinear inverse problems for heat conduction. The first one is a coefficient inverse
problem. The thermal conductivity coefficient, with the prescribed boundary conditions on a part of the
boundary and with some measured or anticipated values of the solution in some inner points, is determined.
The thermal conductivity is assumed to have a form of a linear function of temperature with
unknown coefficients. The number of T-functions in HPM is chosen to obtain the best fitting of the
approximate solution to the input data. Minimization of the difference between the input data and the
approximate solution of the problem, leads to the values of coefficients describing in an approximate
way the unknown coefficients. The second problem is determination of an unknown source term for nonlinear
stationary heat conduction with prescribed boundary conditions and some measured or anticipated
values of the solution in some inner points. The source term is assumed to have a form of a
polynomial with unknown coefficients. Number of the coefficients determines the number of functions
in HPM resulting from expansion of Hðv; pÞ with respect to the parameter p in order to find the components
of the approximate solution of the problem. The components consist of Trefftz functions for the linear
parts of the resultant equations based on powers of p-terms. Minimization of the difference between
the values prescribed or measured inside the considered domain and the approximate solution of the
problem, leads to the values of coefficients describing in an approximate way the source term.

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Temperature dependent thermal conductivity determination and source identification for nonlinear heat conduction by means of the Trefftz and homotopy perturbation methods