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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

$$ \tag{a1 }

\frac{du}{dz}
  =  A ( z) u + B ( z) v ,
$$

$$ 
- 
\frac{dv}{dz}
  =  C ( z) u + D ( z) v ,
$$

$$ 
x  \leq   z  \leq   y,
$$

$$ 
u ( x)  =  0,\  v ( y)  =  1.
$$

(When  $  B $
and  $  C $
are non-negative, (a1) may be considered as describing a flow of right-moving particles,  $  u $, 
and left-moving particles,  $  v $, 
with unit input at  $  y $
and no input at  $  x $. 
This interpretation is often helpful but is by no means necessary.) One asks for  $  R _ {r} ( x, y) \equiv u ( y) $. 
(In the flow model,  $  R _ {r} $
is a reflection coefficient.)

Write

$$ \tag{a2 }
u ( z)  =  R _ {r} ( x, z) v ( z).
$$

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

$$ \tag{a3 }

\frac{dR _ {r} }{dz}
  = \ 
B + ( A + D) R _ {r} + CR _ {r}  ^ {2} ,\ \ 
R _ {r} ( x, x)  =  0.
$$

This [[Riccati equation|Riccati equation]] may, under certain conditions, be integrated from  $  x $
to  $  y $
to yield  $  u ( y) $. 
If one defines  $  T _ {r} ( x, y) \equiv v ( x) $, 
then

$$ \tag{a4 }

\frac{d}{dz}

T _ {r} ( x, z)  = \ 
[ D + CR _ {r} ( x, z)]
T _ {r} ( x, z),\ \ 
T _ {r} ( y, y)  =  1.
$$

If the boundary conditions in (a1) are simply switched, analogous functions,  $  R _ {l} $
and  $  T _ {l} $, 
can be defined. Like  $  T _ {r} $, 
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

$$ \tag{a5 }
u ( x)  =  s _ {l} ,\ \ 
v ( y)  =  s _ {r} .
$$

Then, for any  $  z  ^ {*} $, 
$  x \leq  z  ^ {*} \leq  y $,

$$ \tag{a6 }
u ( z  ^ {*} ) =
$$

$$ 
= \ 
\rho ( x, y, z  ^ {*} ) [ s _ {r} T _ {r} ( z  ^ {*} , y)
R _ {r} ( x, z  ^ {*} ) + s _ {l} T _ {l} ( x, z  ^ {*} )],
$$

$$ 
v ( z  ^ {*} ) =
$$

$$ 
= \ 
\rho ( x, y, z  ^ {*} ) [ s _ {l} T _ {l} ( x, z  ^ {*} )
R _ {l} ( z  ^ {*} , y) + s _ {r} T _ {r} ( z  ^ {*} , y)],
$$

$$ 
\rho ( x, y, z  ^ {*} )  =  [ 1 - R _ {r} ( x,\ 
z  ^ {*} ) R _ {l} ( z  ^ {*} , y)]  ^ {-} 1 .
$$

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

a) Equation (a1) may be replaced by a system with  $  u $
an  $  n $-
vector and  $  v $
a  $  k $-
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  $  z $, 
$  x \leq  z \leq  y $, 
various devices may be employed to cross the singularity.

In all cases the function  $  R _ {r} $
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  $  R _ {l} $, 
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>