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A variational principle for the stationary single-velocity homogeneous transport equation (cf. [[Transport equations, numerical methods|Transport equations, numerical methods]])
 
A variational principle for the stationary single-velocity homogeneous transport equation (cf. [[Transport equations, numerical methods|Transport equations, numerical methods]])
  
<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/v/v096/v096820/v0968201.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
( \overline \Omega \; , \nabla \psi ) +
 +
\Sigma ( x) \psi  = \
 +
\lambda \int\limits _ {| {\overline \Omega \; }  ^  \prime  | = 1 }
 +
\theta ( x, \mu _ {0} )
 +
\psi  ( \overline \Omega \; {}  ^  \prime  , x)
 +
d \overline \Omega \; {}  ^  \prime  ,
 +
$$
  
<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/v/v096/v096820/v0968202.png" /></td> </tr></table>
+
$$
 +
\mu _ {0}  = ( \overline \Omega \; {}  ^  \prime  , \overline \Omega \; ),
 +
$$
  
 
with boundary condition
 
with boundary condition
  
<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/v/v096/v096820/v0968203.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\psi \mid  _ {x \in \Gamma }  = 0,\ \
 +
( \overline \Omega \; , \overline{n}\; ) < 0,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096820/v0968204.png" /> is the boundary of a convex bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096820/v0968205.png" />. If the scatter indicatrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096820/v0968206.png" /> is an even function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096820/v0968207.png" />, transition to the new unknown function
+
where $  \Gamma $
 +
is the boundary of a convex bounded domain $  G $.  
 +
If the scatter indicatrix $  \theta ( x, \mu _ {0} ) $
 +
is an even function of $  \mu _ {0} $,  
 +
transition to the new unknown function
  
<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/v/v096/v096820/v0968208.png" /></td> </tr></table>
+
$$
 +
=
 +
\frac{[ \psi ( \overline \Omega \; , x) +
 +
\psi (- \overline \Omega \; , x)] }{2}
  
reduces the problem (1), (2) to self-adjoint form. In the problem thus obtained, Vladimirov's variational principle for the smallest eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096820/v0968209.png" /> states that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096820/v09682010.png" /> is the minimum of the functional
+
$$
  
<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/v/v096/v096820/v09682011.png" /></td> </tr></table>
+
reduces the problem (1), (2) to self-adjoint form. In the problem thus obtained, Vladimirov's variational principle for the smallest eigenvalue  $  \lambda _ {1} $
 +
states that  $  \lambda _ {1} $
 +
is the minimum of the functional
  
<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/v/v096/v096820/v09682012.png" /></td> </tr></table>
+
$$
 +
\int\limits _ {| \overline \Omega \; | = 1 }
 +
\int\limits _  \Gamma
 +
| ( \overline \Omega \; , \overline{n}\; ) |
 +
u  ^ {2} ( \overline \Omega \; , x)
 +
d \overline \Omega \; dS _ {x} +
 +
$$
  
<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/v/v096/v096820/v09682013.png" /></td> </tr></table>
+
$$
 +
+
 +
\int\limits _ {| \overline \Omega \; | = 1 } \int\limits _ { G }
 +
\frac{1}{\Sigma ( x) }
 +
( \overline \Omega \; , \nabla u)  ^ {2}  dx  d \overline \Omega \; +
 +
$$
  
on the set of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096820/v09682014.png" /> which satisfy the condition
+
$$
 +
+
 +
\int\limits _ {| \overline \Omega \; | = 1 } \int\limits _ { G } \Sigma ( x)
 +
u  ^ {2} ( \overline \Omega \; , x)  dx  d \overline \Omega \;
 +
$$
  
<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/v/v096/v096820/v09682015.png" /></td> </tr></table>
+
on the set of functions  $  u ( \overline \Omega \; , x) $
 +
which satisfy the condition
 +
 
 +
$$
 +
\int\limits _ {| \overline \Omega \; | = 1 }
 +
\int\limits _ {| \overline \Omega \; {}  ^  \prime  | = 1 }
 +
\int\limits \theta ( x, \mu _ {0} )
 +
u ( \overline \Omega \; {}  ^  \prime  , x)
 +
dx  d {\overline \Omega \; {}  ^  \prime  }  d \overline \Omega \; = 1.
 +
$$
  
 
The corresponding (non-negative) eigenfunction realizes the minimum of the functional [[#References|[3]]]. In this variational principle the respective boundary conditions are natural. Variational principles for higher eigenvalues and for the non-homogeneous problem are formulated in a similar manner.
 
The corresponding (non-negative) eigenfunction realizes the minimum of the functional [[#References|[3]]]. In this variational principle the respective boundary conditions are natural. Variational principles for higher eigenvalues and for the non-homogeneous problem are formulated in a similar manner.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.S. Vladimirov,  "Mathematical problems in the theory of single-velocity particle transfer" , Moscow  (1961)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.I. Marchuk,  "Methods of calculation of nuclear reactors" , Moscow  (1961)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B. Davison,  "Neutron transport theory" , Oxford Univ. Press  (1957)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.S. Vladimirov,  "Mathematical problems in the theory of single-velocity particle transfer" , Moscow  (1961)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.I. Marchuk,  "Methods of calculation of nuclear reactors" , Moscow  (1961)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B. Davison,  "Neutron transport theory" , Oxford Univ. Press  (1957)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
Cf. also [[Vladimirov method|Vladimirov method]].
 
Cf. also [[Vladimirov method|Vladimirov method]].

Latest revision as of 08:28, 6 June 2020


A variational principle for the stationary single-velocity homogeneous transport equation (cf. Transport equations, numerical methods)

$$ \tag{1 } ( \overline \Omega \; , \nabla \psi ) + \Sigma ( x) \psi = \ \lambda \int\limits _ {| {\overline \Omega \; } ^ \prime | = 1 } \theta ( x, \mu _ {0} ) \psi ( \overline \Omega \; {} ^ \prime , x) d \overline \Omega \; {} ^ \prime , $$

$$ \mu _ {0} = ( \overline \Omega \; {} ^ \prime , \overline \Omega \; ), $$

with boundary condition

$$ \tag{2 } \psi \mid _ {x \in \Gamma } = 0,\ \ ( \overline \Omega \; , \overline{n}\; ) < 0, $$

where $ \Gamma $ is the boundary of a convex bounded domain $ G $. If the scatter indicatrix $ \theta ( x, \mu _ {0} ) $ is an even function of $ \mu _ {0} $, transition to the new unknown function

$$ u = \frac{[ \psi ( \overline \Omega \; , x) + \psi (- \overline \Omega \; , x)] }{2} $$

reduces the problem (1), (2) to self-adjoint form. In the problem thus obtained, Vladimirov's variational principle for the smallest eigenvalue $ \lambda _ {1} $ states that $ \lambda _ {1} $ is the minimum of the functional

$$ \int\limits _ {| \overline \Omega \; | = 1 } \int\limits _ \Gamma | ( \overline \Omega \; , \overline{n}\; ) | u ^ {2} ( \overline \Omega \; , x) d \overline \Omega \; dS _ {x} + $$

$$ + \int\limits _ {| \overline \Omega \; | = 1 } \int\limits _ { G } \frac{1}{\Sigma ( x) } ( \overline \Omega \; , \nabla u) ^ {2} dx d \overline \Omega \; + $$

$$ + \int\limits _ {| \overline \Omega \; | = 1 } \int\limits _ { G } \Sigma ( x) u ^ {2} ( \overline \Omega \; , x) dx d \overline \Omega \; $$

on the set of functions $ u ( \overline \Omega \; , x) $ which satisfy the condition

$$ \int\limits _ {| \overline \Omega \; | = 1 } \int\limits _ {| \overline \Omega \; {} ^ \prime | = 1 } \int\limits \theta ( x, \mu _ {0} ) u ( \overline \Omega \; {} ^ \prime , x) dx d {\overline \Omega \; {} ^ \prime } d \overline \Omega \; = 1. $$

The corresponding (non-negative) eigenfunction realizes the minimum of the functional [3]. In this variational principle the respective boundary conditions are natural. Variational principles for higher eigenvalues and for the non-homogeneous problem are formulated in a similar manner.

The principle was first obtained by V.S. Vladimirov [1], and it yielded the optimum boundary conditions in the method of spherical harmonics (cf. Spherical harmonics, method of). Vladimirov's variational principle, in conjunction with finite difference methods, is extensively employed in numerical computations of neutron physics.

References

[1] V.S. Vladimirov, "Mathematical problems in the theory of single-velocity particle transfer" , Moscow (1961) (In Russian)
[2] G.I. Marchuk, "Methods of calculation of nuclear reactors" , Moscow (1961) (In Russian)
[3] B. Davison, "Neutron transport theory" , Oxford Univ. Press (1957)

Comments

Cf. also Vladimirov method.

How to Cite This Entry:
Vladimirov variational principle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vladimirov_variational_principle&oldid=49156
This article was adapted from an original article by Yu.N. Drozhzhinov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article