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In the paper the second boundary value problem in a rectangular domain for an inhomogeneous third-order partial differential equation with multiple characteristics with constant coefficients was considered. The uniqueness of the solution to the problem posed is proven by the method of energy integrals. A counterexample is constructed in case when the uniqueness theorem’s conditions are violated. Using the method of separation of variables, the solution to the problem is sought in the form of a product of two functions X(x) and Y (y). To determine Y (y), we obtain a second-order ordinary differential equation with two boundary conditions at the boundaries of the segment [0,q]. For this problem, the eigenvalues and the corresponding eigenfunctions are found for n = 0 and n ∈ N. To determine X(x), we obtain a third-order ordinary differential equation with three boundary conditions at the boundaries of the segment [0,p]. Using the Green’s function method, we constructed solution of the specified problem. A separate Green’s function for n = 0 and a separate Green’s function for the case when n is natural were constructed. The solution for X(x) is written in terms of the constructed Green’s function. After some transformations, an integral Fredholm equation of the second kind is obtained, the solution of which is written through the resolvent. Estimates for the resolvent and Green’s function are obtained. The uniform convergence of the solution and the possibility of its term-by-term differentiation under certain conditions on given functions are proven. When justifying the uniform convergence of the solution, the absence of a “small denominator” is proven.

Due to the inability to obtain ellipsoidal height data in the past, gravity anomaly is chosen as the boundary condition in the traditional third geodetic boundary value problem. The development of GNSS technology brings opportunities for the development of the second boundary value problem. The relatively mature third boundary value theory provides undoubtedly a good reference for the second boundary value problem. Therefore, this paper deals with how to use the Molodensky theory for the third boundary value problem to calculate the quasi-geoid for the second boundary value problem. In this paper, the relationship between the Hotine operator and gradient operator is deduced. Then the method of solving the second boundary value problem based on the Molodensky theory is presented. Experiments show that the accuracy of the quasi-geoid by this method is equivalent to that by the traditional Molodensky method in the third boundary value problem. Thus it is feasible to solve the second boundary value problem based on the Molodensky theory. [ABSTRACT FROM AUTHOR]