The conjugation problems for some analogues of the equation of longitudinal waves with a discontinuous coefficient

It is studied the solvability of conjugation problems for a composite type equation with a discontinuous coefficient. In the first case it is considered the elliptic problem and in the second — an initial-boundary problem. By the method of continuation with respect to a parameter, existence and uniqueness theorems for regular solutions are proved. Also, various examples of the nonuniqueness of the existence of solutions are given, depending on how the discontinuous function behaves in one or another given region.