Application of the method of summation identities in solving a boundary-value problem for the Lame equations

A boundary-value problem on an interval for a one-dimensional system of the Lame equations corresponding to a physical problem of propagation of an elastic wave through the gradient layer is considered. In this case, the coefficients of the equations are complex-valued continuous functions. The boundary conditions of the most general type corresponding to an additional condition, which, from the physical point of view, means absence of any surface waves at the working frequency, are considered. The concept of the generalized solution in the Sobolev space is formulated. Equivalence of the generalized and classical solutions is proven. A finite-difference scheme is constructed by the method of summation identities. For the case in which the equation coefficients and the desired functions are sufficiently smooth, it is shown that the error of approximation is of the order $O(h^2)$.