View source for Invariant imbedding

A method of converting certain two-point boundary value problems to initial value problems. It is an outgrowth of the method of invariance, introduced by V.A. Ambarzumyan (V.A. Ambartsumyan) (cf. [[#References|[a1]]]) and used so successfully by S. Chandrasekhar in radiative transfer problems (cf. [[#References|[a2]]]). Invariant imbedding is closely related to the sweep method (cf. also [[Double-sweep method|Double-sweep method]]; [[Shooting method|Shooting method]]).

Consider the system of linear ordinary scalar differential equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i0522301.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i0522302.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i0522303.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i0522304.png" /></td> </tr></table>

(When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i0522305.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i0522306.png" /> are non-negative, (a1) may be considered as describing a flow of right-moving particles, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i0522307.png" />, and left-moving particles, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i0522308.png" />, with unit input at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i0522309.png" /> and no input at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223010.png" />. This interpretation is often helpful but is by no means necessary.) One asks for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223011.png" />. (In the flow model, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223012.png" /> is a reflection coefficient.)

Write

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

Substitution of (a2) into (a1) yields, formally,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

This [[Riccati equation|Riccati equation]] may, under certain conditions, be integrated from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223015.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223016.png" /> to yield <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223017.png" />. If one defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223018.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>

If the boundary conditions in (a1) are simply switched, analogous functions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223021.png" />, can be defined. Like <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223022.png" />, they satisfy linear initial value equations. Only (a3) is non-linear (cf. [[#References|[a3]]]).

The approach becomes much more valuable when the boundary conditions in (a1) are replaced by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>

Then, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223025.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223027.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223028.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223029.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223030.png" /></td> </tr></table>

There are many extensions and generalizations of these ideas. A few are:

a) Equation (a1) may be replaced by a system with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223031.png" /> an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223032.png" />-vector and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223033.png" /> a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223034.png" />-vector.

b) Inhomogeneous terms may be appended to (a1).

c) Conditions (a5) may be replaced by mixed boundary conditions.

d) If (a3) fails to have a solution for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223036.png" />, various devices may be employed to cross the singularity.

In all cases the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223037.png" /> plays the crucial role and is the only function obtained by solving a non-linear equation. All other problems are linear. (Note: the theory may also be built around <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052230/i05223038.png" />, which then satisfies a Riccati equation.)

Invariant imbedding concepts have been employed to investigate problems in transport theory (cf. also [[Transport equations, numerical methods|Transport equations, numerical methods]]), wave propagation, difference equations, integral equations, quantum mechanics, and many other areas of mathematics and theoretical physics.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.A. Ambarzumyan,   "Diffuse reflection of light by a foggy medium"  ''C.R. Acad. Sci. SSSR'' , '''38'''  (1943)  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Chandrasekhar,   "Radiative transfer" , Dover, reprint  (1960)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Bellman,   G.M. Wing,   "An introduction to invariant imbedding" , Wiley  (1975)</TD></TR></table>