On solving the second boundary value problem for the Viscous Transonic Equation

In a rectangular domain, the second boundary value problem for the Viscous Transonic Equation is considered. The uniqueness of the solution to the problem is proved using the energy integral method. The existence of a solution is proved by the method of separation of variables, i.e. it is sought in the form of a product of two functions X (x) and Y (y). For definition Y (y), an ordinary differential equation of the second order with two boundary conditions on the boundaries of segment [0,q] is obtained. For this problem, the eigenvalues and the corresponding eigenfunctions are found at n ∈ N. For definition X (x), an ordinary differential equation of the third order with three boundary conditions on the boundaries of segment [0,q] is obtained. The solution to this problem is found in the form of an infinite series, uniform convergence, and the possibility of term-by-term differentiation under certain conditions on the given functions is proven. The convergence of the second-order derivative of the solution with respect to variable y is proved using the Cauchy-Bunyakovsky and Bessel inequalities. When substantiating the uniform convergence of the solution, the absence of a “small denominator” is proved.