  Fourier series multiscale method, a concise and efficient analytical approach
for multiscale computation, will be developed out of this series of papers. In
the seventh paper, the usual structural analysis of beams on an elastic
foundation is extended to a thorough multiscale analysis for a fourth order
linear differential equation for transverse deflection of the beam, where
general boundary conditions and a wide spectrum of model parameters are
prescribed. For this purpose, the solution function is expressed as a linear
combination of the boundary function and the internal function, to ensure the
series expression obtained uniformly convergent and termwise differentiable up
to fourth order. Meanwhile, the internal function corresponds to the particular
solution, and the boundary function corresponds to the general solution which
satisfies the homogeneous form of the governing differential equation. Since
the general solution has appropriately interpreted the meaning of the
differential equation, the spatial characteristics of the solution of the
equation are expected to be better captured. With the boundary function and the
internal function selected specifically as combination of the linearly
independent homogeneous solutions of the differential equation, and
one-dimensional half-range Fourier sine series over the solution interval, the
Fourier series multiscale solution of the bending problem of a beam on the
Pasternak foundation is derived. And then the convergence characteristics of
the Fourier series multiscale solution are investigated with numerical
examples, and the multiscale characteristics of the bending problem of a beam
on the Pasternak foundation are demonstrated for a wide spectrum of model
parameters.
