Solution of the inverse boundary value problem of heat transfer for an inhomogeneous ball
V.P. Tanana1,2, B.A. Markov3, A.I. Sidikova1
1South Ural State University, Chelyabinsk, Russia
2Chelyabinsk State University, Chelyabinsk, Russia
3Chelyabinsk Higher Military Aviation Red Banner School of Navigators, Chelyabinsk, Russia
Keywords: error estimation, modulus of continuity, Fourier transform, ill-posed problem
Abstract
This paper studies the problem of determining the boundary condition in the heat conduction equation for composite materials. Mathematically this problem is reduced to the equation of heat conduction in spherical coordinates for an inhomogeneous ball. The temperature inside the ball is assumed to be unknown for an infinite time interval. To find it, the temperature of the heat flow in the media interface is measured at the point r = r 0. An analytical study of the direct problem is carried out, which makes it possible to give a rigorous formulation of the inverse problem and to determine the functional spaces in which it is natural to solve the inverse problem. The main difficulty to be solved, is to obtain an error estimate of the approximate solution. The projection regularization method is used to estimate the modulus of conditional correctness. This allows one to obtain the order-accurate estimates.
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