On Estimated Accuracy of the Approximate Solution of Inverse Boundary Problem for Parabolic Equation

This paper is devoted to the development of projection regularization method, analysis of the efficiency with the help of accurate error estimates of this method obtained in order and its application for the solution of the inverse boundary value problems of heat transfer. In the article there is one-dimensional problem on the restoration of heat exchange conditions at one end of the homogeneous rod of a finite length by the results of temperature measurements with finite error at the point, which is located at some distance from the end. The inverse problem is ill-posed. The paper provides an analytical solution of this ill-posed problem in terms of the Fourier transform, regularizing operator is discharged, the method for the selection of the regularization parameter is given and its optimality in order is proved. It is established, that the accuracy of the approximations is of order $\ln^{-1}\delta$.
Now while using computational methods great attention is paid to the error estimate of the algorithms used, its accuracy and optimality. These questions are of a great importance at numerical calculation of ill-posed problems with the application of various regularizers. A new technology of error estimates to solve the inverse boundary value problems of heat transfer is developed in the paper. The results can be used for numerical calculation of the thermal characteristics of the inverse problems of heat exchange as well as for the development of new regularizing algorithms of similar problems.