An ill-posed boundary value problem for a mixed type second-order differential equation with two degenerate lines
Abstract
This work is devoted to the study of ill-posed boundary value problem for a second-order mixed type differential equation with two degenerate lines. Boundary value problems for mixed type equations are applicable in various fields of the natural sciences: in problems of laser physics, in plasma modelling, and in mathematical biology. In this paper, based on the idea of A. N. Tikhonov, the conditional correctness of the problem, namely, uniqueness and conditional stability theorems are proved, as well as approximate solutions that are stable on the set of correctness are constructed. In obtaining an a priori estimate for the solution to the equation, we used the logarithmic convexity method and results for the spectral problem considered by S. G. Pyatkov. The regularization parameter is determined by the minimum value estimate for the norm of the difference between exact and approximate solutions.