Investigation of an Exact Solution of a Mixed Boundary Value Problem Using the Residue Method

In this study, a finite differences method is proposed for solving a mixed problem, which represents phenomena in hydrodynamics. To evaluate the approximate solution of the problem, its analytical solution is also constructed by the residue method developed by M.L. Rasulov, which is applied to find the solution of the partial differential equations containing time-dependent derivatives at boundary conditions. The formula for expansion of an arbitrary function in a series of residues of the solution of the corresponding spectral problem is used to show that the solution of the mixed boundary-value problem can be represented by the given residue formula. The use of the residual method gives an exact solution for the mixed boundary value problem, represented as a rapidly decreasing series. The derived formula makes it possible to formally establish both the existence and the uniqueness of the solution. Moreover, the derived formula provides a framework for evaluation, allowing a comparative analysis between the exact solution and the numerical approach.