On an approximate solution to the ill-posed continuation problem for harmonic functions given with an error from an approximated boundary under boundary conditions of the second-type

In this paper we consider the ill-posed continuation problem for harmonic functions from an ill-de ned boundary in a cylindrical domain with homogeneous boundary conditions of the second type on the side faces. The value of the function and its normal derivative (Cauchy conditions) is known approximately on an approximated surface of arbitrary shape bounding the cylinder. In this case, the Cauchy problem for the Laplace equation has the property of instability with respect to the error in the Cauchy data, that is, it is ill-posed. On the basis of an idea about the source function of the original problem, the exact solution is represented as a sum of two functions, one of which depends explicitly on the Cauchy conditions, and the second one can be obtained as a solution of the Fredholm integral equation of the rst kind in the form of Fourier series on the eigenfunctions of the second boundary value problem for the Laplace equation. To obtain an approximate stable solution of the integral equation, the Tikhonov regularization method is applied when the solution is obtained as an extremal of the Tikhonov functional. For an approximated surface, we consider the calculation of the normal to this surface and its convergence to the exact value depending on the error with which the original surface is given. The convergence of the obtained approximate solution to the exact solution is proved when the regularization parameter is compared with the errors in the data both on the inexactly speci ed boundary and on the value of the original function on this boundary. A numerical experiment is carried out to demonstrate the e ectiveness of the proposed approach for a special case, for a at boundary and a speci c initial heat source (a set of sharpened sources).