The solution to a boundary value problem for a third-order equation with variable coefficients

In the paper we consider the second boundary value problem in a rectangular domain for an inhomogeneous partial differential equation of the third
order with multiple characteristics with variable coefficients. The uniqueness
of the solution of the posed problem is proved by the method of energy integrals. The uniqueness theorem is proved. A counter-example is constructed
in the case of violation of the conditions of the uniqueness theorem. By the
method of separation of variables, the solution of the problem is sought as
a product of two functions X (x) and Y (y). In order to determine Y (y),
we generate an ordinary differential equation of the second order with two
boundary conditions at the boundaries of the segment [0, q]. For this problem, the eigenvalues and the corresponding eigenfunctions, when n = 0 and
n ∈ N, are found. For determining X (x), we generate an ordinary differential
equation of the third order with three boundary conditions at the boundaries
of the segment [0, p]. A new function, which makes the boundary conditions
homogeneous, is introduced. The solution of the given problem is constructed
by means of the Green function. For n = 0 and for n ∈ N, the Green function
is individually constructed. It is verified that the obtained Green’s functions
satisfy the boundary conditions and properties of the Green function. The
solution X (x) is written by virtue of the constructed Green functions. After
some transformations, the Fredholm integral equation of the second kind is
generated and its solution is written in terms of the resolvent. Estimates
for the resolvent and the Green function are obtained. The uniform convergence of both solution and its possible partial derivatives are shown under the conditions on the given functions. The convergence of the third order derivative of the solution with respect to the variable x is proved using the Cauchy-Bunyakovsky and Bessel inequalities. When justifying the uniform
convergence of the solution, the absence of a “small denominator” is proved.