In this paper, we extend a recently developed approximation method to nonlinear systems given by functional differential equations.
An element u ∈ K is called an extended best ordered approximation of f on K, if it satisfies d X ( u, f ( u ) ) ∈ Min { d X ( s, f ( u ) ) : s ∈ K }, where Min { d X ( s, f ( u ) ) : s ∈ K } is the set of minimal elements of the set { d X ( s, f ( u ) ) : s ∈ K } with respect to the ordering ≽ X on X.
For each interval, the recombination rate is estimated by using an extended composite likelihood approximation of the coalescent likelihood [12], [30].
From the extended LDF approximation, intraparticle concentration profiles during a cyclic process can be calculated and an insight can be gained on the dynamic changes during the process.
In their work [5], Bernardi and Maday extended the approximation to the spectral method.
Kang and Kurtz [ 9] extended this multiscale approximation method and gave a systematic way to obtain limiting models in the time scales of interest.
For example, Bell and Mohammed [14] extended the Carathéodory approximation scheme to the case of SDEs and showed the convergence of the Carathéodory approximate solution.
For this case, we also extended our impedance approximation to demonstrate how the SQUID saturates both by pump flux and by junction phase (or voltage).
In order to model the scattering of a group of objects embedded in a lossy medium under a rough boundary, we applied the method of Green's functions and extended the Kirchhoff approximation for the case of composed systems.
These results are extended to approximations of smooth functions by wavelets (see e.g. [2 4]), and to approximations of linear operators in Banach spaces by operators with finite ranks [5], and other similar approximations.
