User:Tohline/SSC/IsothermalSimilaritySolution - Revision history

Tohline: /* Key References */

2017-07-21T20:56:21Z

Key References

← Older revision		Revision as of 20:56, 21 July 2017
Line 2,110:		Line 2,110:
	* [http://adsabs.harvard.edu/abs/1977ApJ...214..488S F. H. Shu (1977, ApJ, 214, 488-497)]:   ''Self-Similar Collapse of Isothermal Spheres and Star Formation''		* [http://adsabs.harvard.edu/abs/1977ApJ...214..488S F. H. Shu (1977, ApJ, 214, 488-497)]:   ''Self-Similar Collapse of Isothermal Spheres and Star Formation''
	* [http://adsabs.harvard.edu/abs/1977ApJ...218..834H C. Hunter (1977, ApJ, 218, 834-845)]:   ''The Collapse of Unstable Isothermal Spheres''		* [http://adsabs.harvard.edu/abs/1977ApJ...218..834H C. Hunter (1977, ApJ, 218, 834-845)]:   ''The Collapse of Unstable Isothermal Spheres''
			* [http://adsabs.harvard.edu/abs/1985MNRAS.214….1W A. Whitworth & D. Summers (1985, MNRAS, 214, 1 - 25)]:   ''Self-Similar Condensation of Spherically Symmetric Self-Gravitating Isothermal Gas Clouds''


	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Numerical Integration of Coupled ODEs */

2017-07-14T23:16:12Z

Numerical Integration of Coupled ODEs

← Older revision		Revision as of 23:16, 14 July 2017
Line 1,551:		Line 1,551:
	=Numerical Integration of Coupled ODEs=		=Numerical Integration of Coupled ODEs=

	Let's develop a finite-difference expression that ~~allow~~ us to straightforwardly integrate the [[#CoupledODEs\|above pair of coupled ODEs]], namely,		Let's develop a finite-difference expression that allows us to straightforwardly integrate the [[#CoupledODEs\|above pair of coupled ODEs]], namely,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Limiting Behavior */

2017-07-14T23:15:38Z

Limiting Behavior

← Older revision		Revision as of 23:15, 14 July 2017
Line 1,118:		Line 1,118:
	\|<math>~\Rho \approx \Rho_0 - \zeta^2\biggl[\frac{1}{2} \Rho_0(\Rho_0 -2)\biggr] + \frac{1}{3} \zeta^3 U_0 \Rho_0(\Rho_0-4) \, ,</math><p> </p>		\|<math>~\Rho \approx \Rho_0 - \zeta^2\biggl[\frac{1}{2} \Rho_0(\Rho_0 -2)\biggr] + \frac{1}{3} \zeta^3 U_0 \Rho_0(\Rho_0-4) \, ,</math><p> </p>
	\|-		\|-
	\|where, <math>~U_0</math> and <math>~\Rho_0</math> are positive ~~constans~~.		\|where, <math>~U_0</math> and <math>~\Rho_0</math> are positive constants.
	\|}		\|}
	<p> </p>		<p> </p>

Tohline: /* Numerical Integration of Coupled ODEs */

2017-07-14T23:14:17Z

Numerical Integration of Coupled ODEs

← Older revision		Revision as of 23:14, 14 July 2017
Line 1,611:		Line 1,611:
	</div>		</div>

			==First ODE==
	We'll begin by using the first ODE to provide one expression for <math>~\Rho_2</math> in terms of <math>~U</math> and <math>~\Rho_1</math>. We have,		We'll begin by using the first ODE to provide one expression for <math>~\Rho_2</math> in terms of <math>~U</math> and <math>~\Rho_1</math>. We have,
	<div align="center">		<div align="center">
Line 1,664:		Line 1,665:
	where,		where,
	<div align="center">		<div align="center">
	<math>~\ell \equiv (\zeta \bar{U} + 1)\, .</math>		<math>~\ell \equiv (\zeta \bar{U} + 1)</math>       <math>~\Rightarrow</math>       <math>~\bar{U} =\biggl[\frac{\ell - 1}{\zeta}\biggr] \, .</math>
	</div>		</div>

			==Second ODE==
	The second ODE can presumably provide a second, independent expression for <math>~\Rho_2</math> in terms of <math>~U</math> and <math>~\Rho_1</math>. In this case we have,		The second ODE can presumably provide a second, independent expression for <math>~\Rho_2</math> in terms of <math>~U</math> and <math>~\Rho_1</math>. In this case we have,
	<div align="center">		<div align="center">
Line 1,881:		Line 1,882:
	</div>		</div>

			==Combined==
	Now, before returning to the first ODE, let's write <math>~\Delta U</math> in terms of <math>~\ell</math> and, hereafter, use <math>~\ell</math> as the unknown instead of <math>~U_2</math>.		Now, before returning to the first ODE, let's write <math>~\Delta U</math> in terms of <math>~\ell</math> and, hereafter, use <math>~\ell</math> as the unknown instead of <math>~U_2</math>.
	<div align="center">		<div align="center">
Line 1,976:		Line 1,978:
	<td align="right">		<td align="right">
	<math>~\Rightarrow~~~		<math>~\Rightarrow~~~
	(\zeta \ell^2 \Delta\zeta )\Rho_2		(2 \zeta \ell^2 \Delta\zeta )\Rho_2
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			2(4\ell - \ell^2 P_1) \zeta\Delta\zeta + 8( \ell^2 - \zeta^2 ) (\ell - 1 - \zeta U_1 ) \, .
			</math>
			</td>
			</tr>
			</table>
			</div>

			Finally, using this to replace <math>~\Rho_2</math> in the second ODE expression gives,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~
			2(4\ell - \ell^2 P_1) \zeta\Delta\zeta + 8( \ell^2 - \zeta^2 ) (\ell - 1 - \zeta U_1 )
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\ell [ (4 - 2\Rho_1 \ell )\zeta \Delta\zeta - 4( \ell^2 - \zeta^2) ]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			\pm \ell \biggl\{
			[ (4 - 2\Rho_1 \ell )\zeta \Delta\zeta - 4( \ell^2 - \zeta^2) ]^2
			+ 4\ell \zeta \Delta\zeta[ ( 4\Rho_1 - \Rho_1^2 \ell )\zeta \Delta\zeta + 4( \ell^2 - \zeta^2) P_1 ]
			\biggr\}^{1 / 2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~
			\frac{8}{\ell} ( \ell^2 - \zeta^2 ) (\ell - 1 - \zeta U_1 )
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			-4 \zeta \Delta\zeta - 4( \ell^2 - \zeta^2)
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			\pm \biggl\{
			\biggl[ (4 - 2\Rho_1 \ell )\zeta \Delta\zeta - 4( \ell^2 - \zeta^2) \biggr]^2
			+ 4\ell \zeta \Delta\zeta \biggl[ ( 4\Rho_1 - \Rho_1^2 \ell )\zeta \Delta\zeta + 4( \ell^2 - \zeta^2) P_1 \biggr]
			\biggr\}^{1 / 2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~
			\frac{1}{\ell} ( \ell^2 - \zeta^2 )\biggl[ 2 (\ell - 1 - \zeta U_1 ) + \ell \biggr] + \zeta \Delta\zeta
	</math>		</math>
	</td>		</td>
Line 1,984:		Line 2,074:
	<td align="left">		<td align="left">
	<math>~		<math>~
	(4\ell - \ell^2 ~~P_1~~) \zeta\Delta\zeta + 4( \ell^2 - \zeta^2 ) (\ell - 1 - \zeta U_1 ) \, .		\pm \frac{1}{4} \biggl\{
			\biggl[ (4 - 2\Rho_1 \ell )\zeta \Delta\zeta - 4( \ell^2 - \zeta^2) \biggr]^2
			+ 4\ell \zeta \Delta\zeta \biggl[ ( 4\Rho_1 - \Rho_1^2 \ell )\zeta \Delta\zeta + 4( \ell^2 - \zeta^2) P_1 \biggr]
			\biggr\}^{1 / 2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~
			( \ell^2 - \zeta^2 )\biggl[ 3\ell - 2 - 2\zeta U_1 \biggr] + \ell \zeta \Delta\zeta
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\pm \frac{\ell}{4} \biggl\{
			\biggl[ (4 - 2\Rho_1 \ell )\zeta \Delta\zeta - 4( \ell^2 - \zeta^2) \biggr]^2
			+ 4\ell \zeta \Delta\zeta \biggl[ ( 4\Rho_1 - \Rho_1^2 \ell )\zeta \Delta\zeta + 4( \ell^2 - \zeta^2) P_1 \biggr]
			\biggr\}^{1 / 2} \, .
	</math>		</math>
	</td>		</td>
Line 1,990:		Line 2,102:
	</table>		</table>
	</div>		</div>
			This doesn't look particularly useful because, after squaring both sides, it is a sixth-order polynomial in <math>~\ell</math>, which generally has no analytic solution.

	=Key References=		=Key References=

Tohline: /* Numerical Integration of Coupled ODEs */

2017-07-14T22:23:12Z

Numerical Integration of Coupled ODEs

← Older revision		Revision as of 22:23, 14 July 2017
Line 1,930:		Line 1,930:
	<td align="right">		<td align="right">
	<math>~		<math>~
	\Rho_2		\ell^2 \Rho_2
	</math>		</math>
	</td>		</td>
Line 1,938:		Line 1,938:
	<td align="left">		<td align="left">
	<math>~		<math>~
	\~~frac~~{2}{\ell} \biggl\{ 2+\~~biggl[~~ \frac{ \ell^2 - \zeta^2 }{ \ell } \~~biggr]~~\frac{\~~Delta U~~}{\Delta\zeta} \~~biggr~~\} - P_1		2\biggl\{ 2\ell +\biggl[ \ell^2 - \zeta^2 \biggr]\frac{\Delta U}{\Delta\zeta} \biggr\} - \ell^2 P_1
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~
			\Delta\zeta \ell^2 \Rho_2
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			(4\ell - \ell^2 P_1)\Delta\zeta + 2( \ell^2 - \zeta^2 ) \Delta U
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			(4\ell - \ell^2 P_1)\Delta\zeta + 4( \ell^2 - \zeta^2 ) \biggl[\frac{(\ell - 1)}{\zeta} - U_1 \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~
			(\zeta \ell^2 \Delta\zeta )\Rho_2
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			(4\ell - \ell^2 P_1) \zeta\Delta\zeta + 4( \ell^2 - \zeta^2 ) (\ell - 1 - \zeta U_1 ) \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Numerical Integration of Coupled ODEs */

2017-07-14T22:12:34Z

Numerical Integration of Coupled ODEs

← Older revision		Revision as of 22:12, 14 July 2017
Line 1,643:		Line 1,643:
	<math>~		<math>~
	\frac{2}{(\zeta \bar{U} +1)} \biggl\{ 2+\biggl[ \frac{ (\zeta \bar{U} +1)^2 - \zeta^2 }{ (\zeta \bar{U} +1) } \biggr]\frac{\Delta U}{\Delta\zeta} \biggr\} - P_1 \, .		\frac{2}{(\zeta \bar{U} +1)} \biggl\{ 2+\biggl[ \frac{ (\zeta \bar{U} +1)^2 - \zeta^2 }{ (\zeta \bar{U} +1) } \biggr]\frac{\Delta U}{\Delta\zeta} \biggr\} - P_1 \, .
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{2}{\ell} \biggl\{ 2+\biggl[ \frac{ \ell^2 - \zeta^2 }{ \ell } \biggr]\frac{\Delta U}{\Delta\zeta} \biggr\} - P_1 \, ,
	</math>		</math>
	</td>		</td>
Line 1,648:		Line 1,662:
	</table>		</table>
	</div>		</div>
			where,
			<div align="center">
			<math>~\ell \equiv (\zeta \bar{U} + 1)\, .</math>
			</div>


	The second ODE can presumably provide a second, independent expression for <math>~\Rho_2</math> in terms of <math>~U</math> and <math>~\Rho_1</math>. In this case we have,		The second ODE can presumably provide a second, independent expression for <math>~\Rho_2</math> in terms of <math>~U</math> and <math>~\Rho_1</math>. In this case we have,
	<div align="center">		<div align="center">
Line 1,806:		Line 1,826:
	=		=
	- [ ( 4\Rho_1 - \Rho_1^2 \ell )\zeta \Delta\zeta + 4( \ell^2 - \zeta^2) P_1 ]		- [ ( 4\Rho_1 - \Rho_1^2 \ell )\zeta \Delta\zeta + 4( \ell^2 - \zeta^2) P_1 ]
	\, ,		\, .
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
			</table>
			</div>

			The pair of roots of this equation are, then,

			<div align="center">
			<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\ell</math>		<math>~2 \ell \zeta \Delta\zeta P_2</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~~~\equiv~~</math>		<math>~=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>~
	(\zeta \~~bar~~{U} +1)		[ (4 - 2\Rho_1 \ell )\zeta \Delta\zeta - 4( \ell^2 - \zeta^2) ]
	\, .		\pm \biggl\{
			[ (4 - 2\Rho_1 \ell )\zeta \Delta\zeta - 4( \ell^2 - \zeta^2) ]^2
			+ 4\ell \zeta \Delta\zeta[ ( 4\Rho_1 - \Rho_1^2 \ell )\zeta \Delta\zeta + 4( \ell^2 - \zeta^2) P_1 ]
			\biggr\}^{1 / 2} \, .
	</math>		</math>
	</td>		</td>
Line 1,827:		Line 1,857:
	</table>		</table>
	</div>		</div>
	~~The pair of roots of this equation are, then~~,		Or,

	<div align="center">		<div align="center">
Line 1,834:		Line 1,864:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~2 \ell \zeta \Delta\zeta P_2</math>		<math>~\biggl\{ \frac{2 \ell \zeta \Delta\zeta}{ [ (4 - 2\Rho_1 \ell )\zeta \Delta\zeta - 4( \ell^2 - \zeta^2) ] } \biggr\}P_2</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 1,841:		Line 1,871:
	<td align="left">		<td align="left">
	<math>~		<math>~
	~~[ (4 - 2~~\~~Rho_1~~ \~~ell )~~\~~zeta~~ \~~Delta\zeta -~~ 4( \ell~~^2 -~~ \zeta~~^2) ]~~		1
	\pm \~~biggl\{~~		\pm \biggl\{ 1
	[ (4 - 2\Rho_1 \ell )\zeta \Delta\zeta - 4( \ell^2 - \zeta^2) ]^2		+ \frac{4\ell \zeta \Delta\zeta[ ( 4\Rho_1 - \Rho_1^2 \ell )\zeta \Delta\zeta + 4( \ell^2 - \zeta^2) P_1 ]}{[ (4 - 2\Rho_1 \ell )\zeta \Delta\zeta - 4( \ell^2 - \zeta^2) ]^2 }
	~~+ 4\ell \zeta \Delta\zeta~~[ ( 4~~\Rho_1~~ - \Rho_1^2 \ell )\zeta \Delta\zeta + 4( \ell^2 - \zeta^2) ~~P_1~~ ]
	\biggr\}^{1 / 2} \, .		\biggr\}^{1 / 2} \, .
			</math>
			</td>
			</tr>
			</table>
			</div>

			Now, before returning to the first ODE, let's write <math>~\Delta U</math> in terms of <math>~\ell</math> and, hereafter, use <math>~\ell</math> as the unknown instead of <math>~U_2</math>.
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\Delta U \equiv U_2 - U_1</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~(2\bar{U} - U_1) - U_1</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~2\biggl[\bar{U} - U_1 \biggr]</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~2\biggl[\frac{(\ell - 1)}{\zeta} - U_1 \biggr]</math>
			</td>
			</tr>
			</table>
			</div>

			Hence, the first ODE gives,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~
			\Rho_2
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{2}{\ell} \biggl\{ 2+\biggl[ \frac{ \ell^2 - \zeta^2 }{ \ell } \biggr]\frac{\Delta U}{\Delta\zeta} \biggr\} - P_1
	</math>		</math>
	</td>		</td>

Tohline: /* Numerical Integration of Coupled ODEs */

2017-07-14T04:14:03Z

Numerical Integration of Coupled ODEs

← Older revision		Revision as of 04:14, 14 July 2017
Line 1,769:		Line 1,769:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~ (\zeta \bar{U} +1)\zeta \Delta\zeta \, ,</math>		<math>~
			(\zeta \bar{U} +1)\zeta \Delta\zeta
			= \ell \zeta \Delta\zeta
			\, ,</math>
	</td>		</td>
	</tr>		</tr>
Line 1,784:		Line 1,787:
	- \{ [4 - 2\Rho_1 (\zeta \bar{U} +1) ]\zeta \Delta\zeta		- \{ [4 - 2\Rho_1 (\zeta \bar{U} +1) ]\zeta \Delta\zeta
	- 4[ (\zeta \bar{U} +1)^2 - \zeta^2] \}		- 4[ (\zeta \bar{U} +1)^2 - \zeta^2] \}
			=
			- [ (4 - 2\Rho_1 \ell )\zeta \Delta\zeta - 4( \ell^2 - \zeta^2) ]
	\, ,</math>		\, ,</math>
	</td>		</td>
Line 1,798:		Line 1,803:
	<math>~		<math>~
	- \{ [ 4\Rho_1 - \Rho_1^2 (\zeta \bar{U} +1) ]\zeta \Delta\zeta		- \{ [ 4\Rho_1 - \Rho_1^2 (\zeta \bar{U} +1) ]\zeta \Delta\zeta
	+ 4[ (\zeta \bar{U} +1)^2 - \zeta^2] P_1 \} \, .		+ 4[ (\zeta \bar{U} +1)^2 - \zeta^2] P_1 \}
			=
			- [ ( 4\Rho_1 - \Rho_1^2 \ell )\zeta \Delta\zeta + 4( \ell^2 - \zeta^2) P_1 ]
			\, ,
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\ell</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~
			(\zeta \bar{U} +1)
			\, .
			</math>
			</td>
			</tr>
			</table>
			</div>
			The pair of roots of this equation are, then,

			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~2 \ell \zeta \Delta\zeta P_2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			[ (4 - 2\Rho_1 \ell )\zeta \Delta\zeta - 4( \ell^2 - \zeta^2) ]
			\pm \biggl\{
			[ (4 - 2\Rho_1 \ell )\zeta \Delta\zeta - 4( \ell^2 - \zeta^2) ]^2
			+ 4\ell \zeta \Delta\zeta[ ( 4\Rho_1 - \Rho_1^2 \ell )\zeta \Delta\zeta + 4( \ell^2 - \zeta^2) P_1 ]
			\biggr\}^{1 / 2} \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Numerical Integration of Coupled ODEs */

2017-07-14T03:58:15Z

Numerical Integration of Coupled ODEs

Show changes

Tohline: /* Numerical Integration of Coupled ODEs */

2017-07-13T22:30:46Z

Numerical Integration of Coupled ODEs

← Older revision		Revision as of 22:30, 13 July 2017
Line 1,577:		Line 1,577:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{\zeta \Rho [2-\Rho (\zeta U +1)]}{[ (\zeta U +1)^2 - \zeta^2]} \, ,</math>		<math>~\frac{\zeta \Rho [2-\Rho (\zeta U +1)]}{[ (\zeta U +1)^2 - \zeta^2]} \, .</math>
	</td>		</td>
	</tr>		</tr>
Line 1,583:		Line 1,583:
	</div>		</div>

			Assume that the starting values of <math>~U</math> and <math>~\Rho</math> have been provided by, for example, one of the above detailed series expansions. Let the subscript "1" denote these known values at coordinate-location, <math>~\zeta_1</math>, and let the subscript "2" denote the unknown values of <math>~U</math> and <math>~\Rho</math> at <math>~\zeta_2 = \zeta_1 + \Delta\zeta</math>. We should be able to construct a 2<sup>nd</sup>-order accurate integration scheme by treating <math>~U</math> and <math>~\Rho</math> as ''average values'' everywhere they occur on the right-hand sides of the pair of ODEs. That is,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~U</math>
			</td>
			<td align="center">
			<math>~\rightarrow</math>
			</td>
			<td align="left">
			<math>~\frac{1}{2}\biggl(U_2 + U_1 \biggr) \, ,</math>
			</td>
			<td align="center">      and
			<td align="right">
			<math>~\Rho</math>
			</td>
			<td align="center">
			<math>~\rightarrow</math>
			</td>
			<td align="left">
			<math>~\frac{1}{2}\biggl(\Rho_2 + \Rho_1 \biggr) \, .</math>
			</td>
			</tr>
			</table>
			</div>

	=Key References=		=Key References=

Tohline at 22:09, 13 July 2017

2017-07-13T22:09:19Z

← Older revision		Revision as of 22:09, 13 July 2017
Line 1,548:		Line 1,548:
	</table>		</table>
	</div>		</div>

			=Numerical Integration of Coupled ODEs=

			Let's develop a finite-difference expression that allow us to straightforwardly integrate the [[#CoupledODEs\|above pair of coupled ODEs]], namely,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{dU}{d\zeta}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{(\zeta U +1) [\Rho (\zeta U +1) -2)]}{[ (\zeta U +1)^2 - \zeta^2]} \, ,
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\frac{dP}{d\zeta}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{\zeta \Rho [2-\Rho (\zeta U +1)]}{[ (\zeta U +1)^2 - \zeta^2]} \, ,</math>
			</td>
			</tr>
			</table>
			</div>



	=Key References=		=Key References=