View source for Milne problem

A problem in [[Radiative transfer theory|radiative transfer theory]] concerning the single-velocity kinetic transport equation of quanta or particles in a half-space. The integral equation of the Milne problem with a source at infinity under zero incident flux of the radiation was first introduced by E. Milne [[#References|[1]]] for the case of isotropic scattering of quanta, diffusing without absorption in a stellar atmosphere.

The Milne equation takes the form

<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/m/m063/m063790/m0637901.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063790/m0637902.png" /> is the radiation (or particle) density and

<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/m/m063/m063790/m0637903.png" /></td> </tr></table>

is the exponential integral function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063790/m0637904.png" />.

In neutron physics the Milne problem is used for the formulation of approximate boundary conditions for solutions of the equations of a diffusion approximation in a bounded domain; in this connection one takes into account neutron capture by the medium, anisotropic scattering and the curvature of the boundary.

Here the Milne problem is to solve the integro-differential equation

<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/m/m063/m063790/m0637905.png" /></td> </tr></table>

with boundary conditions on the boundary of the half-space occupied by the matter, with a vacuum

<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/m/m063/m063790/m0637906.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063790/m0637907.png" /> is the mean number of secondary neutrons, colliding once with a nucleus (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063790/m0637908.png" /> for scattering and absorbing neutrons by the medium), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063790/m0637909.png" /> is the indicatrix of the scattering (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063790/m06379010.png" /> for isotropic scattering). The spherical or cylindrical Milne problem on the distribution of neutrons in the space outside absorbing spheres or cylinders is stated analogously.

The solution of the Milne problem is conveniently given by applying the [[Laplace transform|Laplace transform]] to the integro-differential transfer equation (see [[#References|[3]]]) and using the [[Wiener–Hopf method|Wiener–Hopf method]] to solve the functional equations thus obtained.

In order to solve the Milne problem it was suggested that expansion relative to generalized eigen functions and methods for solving singular integral equations be used (see [[#References|[4]]]). The solution of the Milne problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063790/m06379011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063790/m06379012.png" /> is sought in the form

<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/m/m063/m063790/m06379013.png" /></td> </tr></table>

where

<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/m/m063/m063790/m06379014.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/m/m063/m063790/m06379015.png" /></td> </tr></table>

are the eigen functions of the continuous spectrum, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063790/m06379016.png" /> denotes the Cauchy principal value, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063790/m06379017.png" /> is the Dirac <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063790/m06379018.png" />-function, and

<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/m/m063/m063790/m06379019.png" /></td> </tr></table>

The discrete eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063790/m06379020.png" /> are the roots of the characteristic equation

<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/m/m063/m063790/m06379021.png" /></td> </tr></table>

The eigen functions of the discrete spectrum take the form

<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/m/m063/m063790/m06379022.png" /></td> </tr></table>

The system of eigen functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063790/m06379023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063790/m06379024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063790/m06379025.png" />, turns out to be complete in the space of generalized functions on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063790/m06379026.png" /> and they are orthogonal with respect to a weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063790/m06379027.png" /> which is the solution of a singular integral equation (see [[#References|[4]]]).

The boundary condition (2) of the Milne problem gives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063790/m06379028.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/m/m063/m063790/m06379029.png" /></td> </tr></table>

that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063790/m06379030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063790/m06379031.png" /> are defined as the coefficients in the expansion of the 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/m/m063/m063790/m06379032.png" /></td> </tr></table>

The asymptotic density of the neutrons,

<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/m/m063/m063790/m06379033.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/m/m063/m063790/m06379034.png" /></td> </tr></table>

vanishes for

<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/m/m063/m063790/m06379035.png" /></td> </tr></table>

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063790/m06379036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063790/m06379037.png" />, the Hopf constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063790/m06379038.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.A. Milne,   ''Mon. Notices Roy. Astron. Soc.'' , '''81'''  (1921)  pp. 361–375</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Hopf,   "Mathematical problems of radiative equilibrium" , Cambridge Univ. Press  (1934)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I. Sneddon,   "Fourier transforms" , McGraw-Hill  (1951)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  K.M. Case,   P.F. Zweifel,   "Linear transport theory" , Addison-Wesley  (1967)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Greenberg,   C. van der Mee,   V. Protopopescu,   "Boundary value problems in abstract kinetic theory" , Birkhäuser  (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Cercignani,   "The Boltzmann equation and its applications" , Springer  (1988)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B. Davison,   J.B. Sykes,   "Neutron transport theory" , Clarendon Press  (1957)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M.M.R. Williams,   "The slowing down and thermalization of neutrons" , North-Holland  (1966)</TD></TR></table>