ON THE CONDITIONAL STABILITY OF A FINITE DIFFERENCE ANALOGUE OF A BOUNDARY PROBLEM FOR A HYPERBOLIC-PARABOLIC EQUATION - Bakanov Г.Б.; S.K. Meldebekova С.К. (Bulletin of Abai KazNPU. Series of Physical and Mathematical sciences, 2023)
In this paper we consider a finite-difference analog of the boundary value problem for an equation of hyperbolic-parabolic type. The problem of integral geometry for a family of curves is reduced to this problem. Assuming that the solution exists, an estimation of the stability of the discrete analogue of this problem on the space of sufficiently smooth functions is proved. These problems of integral geometry are associated with numerous applications, including problems of interpretation of seismic data, 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, the basic relations are performed with a certain shift in the discrete variable. In this regard, many relations obtained in a continuous formulation, when transitioning to a discrete analogue, have a more complex and cumbersome form, which requires additional studies of the resulting terms with a shift. Since there is no theorem of the existence of a solution in the general case, the concept of conditional correctness is used in the work. The estimate of the conditional stability of a finite-difference analog of a boundary value problem for a hyperbolic-parabolic equation obtained in this work is important for understanding the effectiveness of numerical methods for solving problems of geotomography, medical tomography, flaw detection, etc