On the stable approximate solution of the ill-posed boundary value problem for the Laplace equation with homogeneous conditions of the second kind on the edges at inaccurate data on the approximated boundary

In this paper, we consider the ill-posed continuation problem for harmonic functions from an ill-defined 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 first 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 specified boundary and on the value of the original function on this boundary. A numerical experiment is carried out to demonstrate the effectiveness of the proposed approach for a special case, for a flat boundary and a specific initial heat source (a set of sharpened sources).