Invariant imbedding

It turns out that one cannot propagate Y using standard numerical methods because Y blows up whenever g is zero. To circumvent this one must propagate Y by invariant imbedding . The basic idea here is to construct a propagator Y which satisfies... 

Another method of solution of the above nonlinear, two-point boundary-value problem would be based on the invariant imbedding concept. 

Lee, E. S. Quasilinearization and Invariant Imbedding, Academic Press, New York (1968). 

Bi L, Yang P Modeling of light scattering by biconcave and deformed red blood ceUs with the invariant imbedding T-matrix method. Journal of biomedical optics 18(5) 055001-1-055001-13, 2013. 

Invariant imbedding and neutron transport theory. II. Functional egua-... 

In effect, we are emphasizing the power of the semi-grouj) concept not only in time, the classical parameter, but also in other physically meaningful variables, such as space and energy, as well. A formal statement of the principle of invariant imbedding as well as of its history and relationships to other general principles of mathematical physics can be found in [1]. Let us, however, emphasize the fundamental pioneering work of Ambarzumian [2], and Chandrasekhar [3], in the use of invariance principles in the field of radiative transfer. 

We shall first determine the internal fluxes along classical lines and then treat the process from the invariant imbedding viewpoint. 

It is interesting to note that the partial differential Equation (6.7) for the generating function can be derived directly by invariant imbedding methods as is shown in 5 of [8]. This is an illustration of the general notion that whenever an equation for the expected value can be derived, an equation for the more meaningful generating function can also be obtained. 

Discussion. For a discussion from the invariant imbedding viewpoint of neutron transport in media with more general geometries than discussed here, the reader may consult [8]. Although a uniform method is used, the details can become onerous. 

R. Bellman and R. Kalaba, On the principle of invariant imbedding and diffuse reflection from cylindrical regions, Proc. Nat. Acad. Sci. USA vol. 43 (1957) pp. 514-517. 

Invariant imbedding and wave propagation in stochastic media. Electro-... 

Functional equations, wave propagation and invariant imbedding, J. Math. 

Random walk, scattering and invariant imbedding. I One-dimensional... 

Invariant imbedding, random walk and scattering. II Discrete versions,... 

Quasilinearization and Invariant Imbedding, E. S. Lee, Academic Press, New York, 1968. Digital Computation for Chemical Engineers, L. Lapidus, McGraw-Hill, New York, 1962. 

An example from the field of Chemical Engineering will now be given. This example is from the book. Quasilinearization and Invariant Imbedding, by E. S Lee (Academic Press, 1968). A set of equations is used to model the concentration of a particular reactant, in an adiabatic tubular chemical reactor wifli axial mixing ... 

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