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We investigate a new type boundary value problem consisting of a differential-operator equation, eigendependent boundary conditions, and two supplementary conditions so-called interface conditions. We give a characterisation of some spectral properties of the considered problem. Particularly, it is established such properties as isomorphism and coerciveness, discreteness of the spectrum and found asymptotic formulas for eigenvalues.

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In this study, we developed a new modification of the well-known differential transformation method (DTM) that differs from the classical one by the algorithm for calculating the coefficients of an approximate solution given as a series. The proposed new algorithm we will call α-parameterized DTM (α-P DTM,for short). By using the proposed α-DTM we solve the boundary value transmission problem for the 3rd order differential equation. For the same problem, an approximate solution is found using also the classical DTM. The solutions obtained were compared graphically.