User:Tohline/MathProjects/EigenvalueProblemN1 - Revision history

Tohline: /* Find Analytic Solutions to an Eigenvalue Problem */Comment out the recommendation by KV

2015-09-18T19:27:09Z

Find Analytic Solutions to an Eigenvalue Problem: Comment out the recommendation by KV

← Older revision		Revision as of 19:27, 18 September 2015
Line 518:		Line 518:
	describes the ''relative amplitude'' of the oscillation as a function of position, <math>~x</math>, within the star, and <math>~\sigma</math> gives the frequency of the oscillation.		describes the ''relative amplitude'' of the oscillation as a function of position, <math>~x</math>, within the star, and <math>~\sigma</math> gives the frequency of the oscillation.

			<!-- COMMENT OUT (by mutual agreement) on 9/18/2015
	==Suggested Eigenfunction by KV==		==Suggested Eigenfunction by KV==

Line 541:		Line 542:
	<math>~\sigma^2 = \frac{1}{\pi^2} \biggl[ (1 - 8\alpha)^{1/2} + 2\alpha -1 \biggr]</math>		<math>~\sigma^2 = \frac{1}{\pi^2} \biggl[ (1 - 8\alpha)^{1/2} + 2\alpha -1 \biggr]</math>
	</div>		</div>
			END COMMENT -->

	=Related Discussions=		=Related Discussions=

Tohline: /* Suggested Eigenfunction by K. Vogiatzis */

2015-09-16T18:29:56Z

Suggested Eigenfunction by K. Vogiatzis

← Older revision		Revision as of 18:29, 16 September 2015
Line 518:		Line 518:
	describes the ''relative amplitude'' of the oscillation as a function of position, <math>~x</math>, within the star, and <math>~\sigma</math> gives the frequency of the oscillation.		describes the ''relative amplitude'' of the oscillation as a function of position, <math>~x</math>, within the star, and <math>~\sigma</math> gives the frequency of the oscillation.

	==Suggested Eigenfunction by ~~K. Vogiatzis~~==		==Suggested Eigenfunction by KV==

	On 9/15/2015, ~~Konstantinos Vogiatzis recommends~~ the following eigenfunction:		On 9/15/2015, KV recommended the following eigenfunction:
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Find Analytic Solutions to an Eigenvalue Problem */ Vogiatzis suggestion

2015-09-15T23:30:53Z

Find Analytic Solutions to an Eigenvalue Problem: Vogiatzis suggestion

← Older revision		Revision as of 23:30, 15 September 2015
Line 517:		Line 517:
	</div>		</div>
	describes the ''relative amplitude'' of the oscillation as a function of position, <math>~x</math>, within the star, and <math>~\sigma</math> gives the frequency of the oscillation.		describes the ''relative amplitude'' of the oscillation as a function of position, <math>~x</math>, within the star, and <math>~\sigma</math> gives the frequency of the oscillation.

			==Suggested Eigenfunction by K. Vogiatzis==

			On 9/15/2015, Konstantinos Vogiatzis recommends the following eigenfunction:
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\mathcal{G}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\exp\biggl[\tfrac{1}{4}\sigma^2 x^2 - \alpha \ln x\biggr]</math>
			</td>
			</tr>
			</table>
			</div>
			where,
			<div align="center">
			<math>~\sigma^2 = \frac{1}{\pi^2} \biggl[ (1 - 8\alpha)^{1/2} + 2\alpha -1 \biggr]</math>
			</div>

	=Related Discussions=		=Related Discussions=

Tohline: /* The Challenge */ Correct definition of outer boundary condition

2015-08-27T20:56:41Z

The Challenge: Correct definition of outer boundary condition

← Older revision		Revision as of 20:56, 27 August 2015
Line 21:		Line 21:
	</div>		</div>
	<br />		<br />
	where, <math>~\alpha</math> is a known constant. The desired functional solution is subject to the following two boundary conditions: <math>~\mathcal{G}_\sigma = 0</math> and <math>~d\mathcal{G}_\sigma/dx = 0</math> at <math>~x = 0</math>. Note that, in the context of astrophysical discussions, the interval of <math>~x</math> that is of particular interest is <math>0 \le x \le \pi</math>.		where, <math>~\alpha</math> is a known constant. The desired functional solution is subject to the following two boundary conditions: <math>~\mathcal{G}_\sigma = 0</math> at <math>~x = 0</math>; and <math>~d\ln\mathcal{G}_\sigma/d\ln x = (\pi^2 \sigma^2/2 - \alpha)</math> at <math>~x = \pi</math>. Note that, in the context of astrophysical discussions, the interval of <math>~x</math> that is of particular interest is <math>0 \le x \le \pi</math>.

	</td></tr></table>		</td></tr></table>
	</td></tr>		</td></tr>
	</table>		</table>


	==Context==		==Context==

Tohline: /* Context */ Fix typo in one table expression

2015-08-14T16:54:19Z

Context: Fix typo in one table expression

← Older revision		Revision as of 16:54, 14 August 2015
Line 126:		Line 126:
	<td align="center">[[User:Tohline/SSC/Structure/Other_Analytic_Models#Linear_Density_Distribution\|Linear]]</td>		<td align="center">[[User:Tohline/SSC/Structure/Other_Analytic_Models#Linear_Density_Distribution\|Linear]]</td>
	<td align="center"><math>~1-x</math>		<td align="center"><math>~1-x</math>
	<td align="center"><math>~(1-x)^2(1 + ~~2\chi_0~~ - \tfrac{9}{5}x^2)</math>		<td align="center"><math>~(1-x)^2(1 + 2x - \tfrac{9}{5}x^2)</math>
	<td align="center"><math>~-\tfrac{12}{5}x(1-x)(4-3x)</math>		<td align="center"><math>~-\tfrac{12}{5}x(1-x)(4-3x)</math>
	</tr>		</tr>

Tohline: /* Context */ Insert another form of the LAWE

2015-08-14T16:45:02Z

Context: Insert another form of the LAWE

← Older revision		Revision as of 16:45, 14 August 2015
Line 82:		Line 82:
	<td align="left">		<td align="left">
	<math>~ \frac{P}{x^4} \frac{d}{dx}\biggl( x^4 \mathcal{G}_\sigma^' \biggr)		<math>~ \frac{P}{x^4} \frac{d}{dx}\biggl( x^4 \mathcal{G}_\sigma^' \biggr)
	+ \frac{P^'}{x^\alpha}\frac{d}{dx}\biggl(x^\alpha \mathcal{G}_\sigma\biggr) \, .		+ \frac{P^'}{x^\alpha}\frac{d}{dx}\biggl(x^\alpha \mathcal{G}_\sigma\biggr) \, ,
	</math>		</math>
			</td>
			</tr>
			</table>
			or, equivalently,

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

			<tr>
			<td align="right">
			<math>~\frac{d}{dx}\biggl(x^4 P \mathcal{G}_\sigma^'\biggr) + \biggl[ \biggl( \sigma^2 + \frac{\alpha P^'}{x\rho} \biggr) x^4 \rho
			\biggr] \mathcal{G}_\sigma</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ 0 \, .</math>
	</td>		</td>
	</tr>		</tr>
Line 128:		Line 145:

	Drawing the expressions for <math>~\rho(x)</math>, <math>~P(x)</math>, and <math>~P^'(x)</math> from the last row of this table and plugging them into this generic form of the LAWE leads to the ''specific'' statement of the astrophysically motivated eigenfunction problem presented above — inside the blue-framed box. As is discussed in the [[User:Tohline/MathProjects/EigenvalueProblemN1#Analogous_Problem_with_Known_Analytic_Solutions\|subsection that follows]], an analogous eigenvalue problem whose analytic solution is ''known'' comes from plugging expressions presented in the first row of this table into the generic form of the LAWE.		Drawing the expressions for <math>~\rho(x)</math>, <math>~P(x)</math>, and <math>~P^'(x)</math> from the last row of this table and plugging them into this generic form of the LAWE leads to the ''specific'' statement of the astrophysically motivated eigenfunction problem presented above — inside the blue-framed box. As is discussed in the [[User:Tohline/MathProjects/EigenvalueProblemN1#Analogous_Problem_with_Known_Analytic_Solutions\|subsection that follows]], an analogous eigenvalue problem whose analytic solution is ''known'' comes from plugging expressions presented in the first row of this table into the generic form of the LAWE.


	==Analogous Problem with Known Analytic Solutions==		==Analogous Problem with Known Analytic Solutions==

Tohline: /* Find Analytic Solutions to an Eigenvalue Problem */ Explain briefly how table expressions relate to various eigenvalue problems

2015-08-13T18:39:21Z

Find Analytic Solutions to an Eigenvalue Problem: Explain briefly how table expressions relate to various eigenvalue problems

← Older revision		Revision as of 18:39, 13 August 2015
Line 126:		Line 126:
	</tr>		</tr>
	</table>		</table>

			Drawing the expressions for <math>~\rho(x)</math>, <math>~P(x)</math>, and <math>~P^'(x)</math> from the last row of this table and plugging them into this generic form of the LAWE leads to the ''specific'' statement of the astrophysically motivated eigenfunction problem presented above — inside the blue-framed box. As is discussed in the [[User:Tohline/MathProjects/EigenvalueProblemN1#Analogous_Problem_with_Known_Analytic_Solutions\|subsection that follows]], an analogous eigenvalue problem whose analytic solution is ''known'' comes from plugging expressions presented in the first row of this table into the generic form of the LAWE.


	==Analogous Problem with Known Analytic Solutions==		==Analogous Problem with Known Analytic Solutions==

Tohline: /* Context */

2015-08-13T16:59:45Z

Context

← Older revision		Revision as of 16:59, 13 August 2015
Line 120:		Line 120:
	<tr>		<tr>
	<td align="center">[[User:Tohline/SSC/Stability/Polytropes#n_.3D_1_Polytrope\|<math>~n=1</math> Polytrope]]</td>		<td align="center">[[User:Tohline/SSC/Stability/Polytropes#n_.3D_1_Polytrope\|<math>~n=1</math> Polytrope]]</td>
	<td align="center"><math>~\frac{\sin~~(\pi x)~~}{~~\pi~~ x}</math>		<td align="center"><math>~\frac{\sin }{ x}</math>
	<td align="center"><math>~\biggl[\frac{\sin~~(\pi~~ x)}{~~\pi~~ x}\biggr]^2</math>		<td align="center"><math>~\biggl[\frac{\sin x}{x}\biggr]^2</math>
	<td align="center"><math>~\frac{2}{x} \biggl[ \cos~~(\pi~~ x) - \frac{\sin~~(\pi~~ x)}{~~\pi~~ x} \biggr]		<td align="center"><math>~\frac{2}{x} \biggl[ \cos x - \frac{\sin x}{x} \biggr]
	\frac{\sin~~(\pi~~ x)}{~~\pi~~ x}</math>		\frac{\sin x}{x}</math>
	</tr>		</tr>
	</table>		</table>

Tohline: /* Context */ Correct LAWE expressions

2015-08-13T16:45:32Z

Context: Correct LAWE expressions

← Older revision		Revision as of 16:45, 13 August 2015
Line 38:		Line 38:
	<td align="right">		<td align="right">
	<math>~\biggl[P \biggr]\frac{d^2\mathcal{G}_\sigma}{dx^2} + \biggl[\frac{4P}{x}		<math>~\biggl[P \biggr]\frac{d^2\mathcal{G}_\sigma}{dx^2} + \biggl[\frac{4P}{x}
	+ P^' \biggr]\frac{d\mathcal{G}_\sigma}{dx} + \biggl[ \sigma^2 \rho - \frac{\alpha P^'}{x} \biggr]\mathcal{G}_\sigma</math>		+ P^' \biggr]\frac{d\mathcal{G}_\sigma}{dx} + \biggl[ \sigma^2 \rho + \frac{\alpha P^'}{x} \biggr]\mathcal{G}_\sigma</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 57:		Line 57:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~- \sigma^2 x </math>		<math>~- \sigma^2 \mathcal{G}_\sigma </math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 75:		Line 75:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~- \sigma^2 \rho x </math>		<math>~- \sigma^2 \rho \mathcal{G}_\sigma </math>
	</td>		</td>
	<td align="center">		<td align="center">

Tohline: /* Context */ Finished simplifying table

2015-08-11T20:07:12Z

Context: Finished simplifying table

← Older revision		Revision as of 20:07, 11 August 2015
Line 57:		Line 57:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~- \sigma^2 ~~\rho~~ x </math>		<math>~- \sigma^2 x </math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 63:		Line 63:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~P \biggl[ \frac{4\mathcal{G}_\sigma^'}{x}+ \mathcal{G}_\sigma^{' '} \biggr]		<math>~\frac{P}{\rho} \biggl[ \frac{4\mathcal{G}_\sigma^'}{x}+ \mathcal{G}_\sigma^{' '} \biggr]
	+ P^' \biggl[ \frac{\alpha \mathcal{G}_\sigma}{x} + \mathcal{G}_\sigma^'\biggr] \, ,		+ \frac{P^'}{\rho} \biggl[ \frac{\alpha \mathcal{G}_\sigma}{x} + \mathcal{G}_\sigma^'\biggr] \, ,
	</math>		</math>
	</td>		</td>
Line 103:		Line 103:
	<td align="center">[[User:Tohline/SSC/UniformDensity#The_Stability_of_Uniform-Density_Spheres\|Uniform-density]]</td>		<td align="center">[[User:Tohline/SSC/UniformDensity#The_Stability_of_Uniform-Density_Spheres\|Uniform-density]]</td>
	<td align="center"><math>~1</math>		<td align="center"><math>~1</math>
	<td align="center"><math>~1 - ~~\chi_0~~^2</math>		<td align="center"><math>~1 - x^2</math>
	<td align="center"><math>~-~~2\chi_0~~</math>		<td align="center"><math>~-2x</math>
	</tr>		</tr>
	<tr>		<tr>
	<td align="center">[[User:Tohline/SSC/Structure/Other_Analytic_Models#Linear_Density_Distribution\|Linear]]</td>		<td align="center">[[User:Tohline/SSC/Structure/Other_Analytic_Models#Linear_Density_Distribution\|Linear]]</td>
	<td align="center"><math>~1-~~\chi_0~~</math>		<td align="center"><math>~1-x</math>
	<td align="center"><math>~(1-~~\chi_0~~)^2(1 + 2\chi_0 - \tfrac{9}{5}~~\chi_0~~^2)</math>		<td align="center"><math>~(1-x)^2(1 + 2\chi_0 - \tfrac{9}{5}x^2)</math>
	<td align="center"><math>~\tfrac{1}{5}[- ~~48\chi_0 + 84\chi_0^2~~ - ~~36\chi_0^3]~~</math>		<td align="center"><math>~-\tfrac{12}{5}x(1-x)(4-3x)</math>
	</tr>		</tr>
	<tr>		<tr>
	<td align="center">[[User:Tohline/SSC/Structure/Other_Analytic_Models#Parabolic_Density_Distribution\|Parabolic]]</td>		<td align="center">[[User:Tohline/SSC/Structure/Other_Analytic_Models#Parabolic_Density_Distribution\|Parabolic]]</td>
	<td align="center"><math>~1-~~\chi_0~~^2</math>		<td align="center"><math>~1-x^2</math>
	<td align="center"><math>~(1-~~\chi_0~~^2)^2(1 - \tfrac{1}{2} ~~\chi_0~~^2)</math>		<td align="center"><math>~(1-x^2)^2(1 - \tfrac{1}{2} x^2)</math>
	<td align="center"><math>~- 5~~\chi_0 + 8\chi_0^3~~ - ~~3\chi_0~~^5</math>		<td align="center"><math>~-x(1-x^2)(5-3x^2)</math>
	</tr>		</tr>
	<tr>		<tr>
	<td align="center">[[User:Tohline/SSC/Stability/Polytropes#n_.3D_1_Polytrope\|<math>~n=1</math> Polytrope]]</td>		<td align="center">[[User:Tohline/SSC/Stability/Polytropes#n_.3D_1_Polytrope\|<math>~n=1</math> Polytrope]]</td>
	<td align="center"><math>~\frac{\sin(\pi~~\chi_0~~)}{\pi~~\chi_0~~}</math>		<td align="center"><math>~\frac{\sin(\pi x)}{\pi x}</math>
	<td align="center"><math>~\biggl[\frac{\sin(\pi~~\chi_0~~)}{\pi~~\chi_0~~}\biggr]^2</math>		<td align="center"><math>~\biggl[\frac{\sin(\pi x)}{\pi x}\biggr]^2</math>
	<td align="center"><math>~\frac{2\~~sin~~(\pi\~~chi_0)}~~{(\pi~~^2\chi_0^3~~)} ~~\biggl[~~ \pi\~~chi_0~~ \~~cos(\pi\chi_0) -~~ \sin(\pi~~\chi_0~~) \~~biggr]~~</math>		<td align="center"><math>~\frac{2}{x} \biggl[ \cos(\pi x) - \frac{\sin(\pi x)}{\pi x} \biggr]
			\frac{\sin(\pi x)}{\pi x}</math>
	</tr>		</tr>
	</table>		</table>