Computation of Green's matrices for boundary value problems associated with a pair of mixed linear regular ordinary differential operators (Q1902931)

A pair of mixed linear regular ordinary differential operators of the type \(\tau_1u_1 = \sum^n_{k = 0}P_k D^ku_1 = \lambda u_1\), defined on the interval \(J_1 = [a,b]\) and \(\tau_2u_2 = \sum^m_{k = 0} Q_kD^ku_2 = \lambda u_2\) defined on the adjacent interval \(J_2 = [b,c]\), where \(\lambda\) is an unknown constant (eigenvalue) and the functions \(u_1\) and \(u_2\) satisfy certain mixed conditions at the interface \(x = b\), is studied. Algorithms are presented for the computation of Green's matrices for the boundary value problems associated with \((\tau_1, \tau_2)\). The developed algorithms are used in two physical examples, i.e. in the computation of Green's matrices encountered in the studies of acoustic waveguides in oceans (ocean surface-bottom) and in the studies of transversal vibrations in nonhomogeneous strings.