The regular component U(x) is a sufficiently smooth solution of an inhomogeneous differential equation
; its first-order derivative is [epsilon]-uniformly bounded.
We consider the transformation f(r) = 1/[lambda](r) and obtain a first-order linear inhomogeneous differential equation
4: Solve the inhomogeneous differential equation (2.14) for [[phi].sub.1] with the setting [[psi].sub.1] = [v.sub.1,k], ..., [[psi].sub.k] = [v.sub.k,k] and p = k.
Note that (4.4) is just a second order inhomogeneous differential equation with one Dirichlet and one Robin boundary condition.
Equation (2) is an inhomogeneous differential equation
, where the function V(R) is called the perturbed scattering potential of the patterned structure relative to the unpatterned stratified structure with the dielectric constant of [[epsilon].sub.f] (R).
The corresponding axial displacement field is shown to be governed by a fourth-order inhomogeneous differential equation
. Boundary conditions are naturally inferred by performing a standard localization procedure of a variational problem formulated by making recourse to thermodynamic restrictions see, for example, [24-26], according to the geometric approach illustrated in [27-30].