Padé approximants and continualization for 1D mass chain

Various continuum models (CM) for 1D discrete media are under consideration. As example we deal with a difference-differential equation, describing the system of connected oscillators. A common opinion is that the corresponding CM (string). String-type approximation justified for low part of frequency spectra, but for forced oscillations the solution of wave and chain equations can be quite different (splash effect). So, more appropriate CM of chain should be found. Intermediate CM (ICM). The difference operator makes analysis difficult due to its non-local form. Approximate equations can be gained by replacing it with a local derivative operator. If we use derivative of more then second order, we have ICM. ICM give possibility to take splash effect into account, but we have a higher order of approximate differential equation. Quasi-continuum approximation. The quasi-continuum approximation technique makes use of higher-order derivatives to form more accurate approximations of the discrete difference operator via one-point Padé approximants (PA). Pseudo-continuum. Two-point PA give the most accurate CM of difference operator. Possibilities of generalizations and applications are discussed.