On the stability of the difference analogue of the boundary value problem for a mixed type equation

This paper considers a difference problem for a mixed-type equation, to which a problem of integral geometry for a family of curves satisfying certain regularity conditions is reduced. These problems are related to numerous applications, including interpretation problem of seismic data, problem of interpretation of X-ray images, problems of computed tomography and technical diagnostics. The study of difference analogues of integral geometry problems has specific difficulties associated with the fact that for finite-difference analogues of partial derivatives, basic relations are performed with a certain shift in the discrete variable. In this regard, many relations obtained in a continuous formulation, when transitioned to a discrete analogue, have a more complex and cumbersome form, which requires additional studies of the resulting terms with a shift. Another important feature of the integral geometry problem is the absence of a theorem for existence of a solution in general case. Consequently, the paper uses the concept of correctness according to A.N. Tikhonov, particularly, it is assumed that there is a solution to the problem of integral geometry and its differential-difference analogue. The stability estimate of the difference analogue of the boundary value problem for a mixed-type equation obtained in this work is vital for understanding the effectiveness of numerical methods for solving problems of geotomography, medical tomography, flaw detection, etc. It also has a great practical significance in solving multidimensional inverse problems of acoustics, seismic exploration.