Bessel Functions - Revision history

Admin: Fix math notation error: changed \partial v to \partial n in general boundary condition (v was used for both Sherwood number and normal coordinate). Also fixed Python import (matplotlib.pyplot, not matplotlib.pylab) and explicit iv import. (via update-page on MediaWiki MCP Server)

2026-06-14T22:59:26Z

Fix math notation error: changed \partial v to \partial n in general boundary condition (v was used for both Sherwood number and normal coordinate). Also fixed Python import (matplotlib.pyplot, not matplotlib.pylab) and explicit iv import. (via update-page on MediaWiki MCP Server)

← Older revision		Revision as of 22:59, 14 June 2026
Line 10:		Line 10:

	<math>		<math>
	v (1-u_s) = \dfrac{\partial u}{\partial v}		v (1-u_s) = \dfrac{\partial u}{\partial n}
	</math>		</math>

Line 51:		Line 51:

	<source lang="python">		<source lang="python">
	import matplotlib.~~pylab~~ as plt		import matplotlib.pyplot as plt
	from scipy.special import *		from scipy.special import iv

	# Radial dimension		# Radial dimension

Admin: /* Bessel Functions with Scipy */

2016-08-18T02:50:42Z

Bessel Functions with Scipy

← Older revision		Revision as of 02:50, 18 August 2016
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	[[Image:BesselFunctionSolutions.png]]		[[Image:BesselFunctionSolutions.png]]

	~~[[Category:Mathematics]]~~
	~~[[Category:Python]]~~		=Flags=

			{{PythonFlag}}

Admin: /* Bessel Functions with Scipy */

2014-02-25T19:55:13Z

Bessel Functions with Scipy

← Older revision		Revision as of 19:55, 25 February 2014
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	which results in a plot of solutions for various Thiele modulii, for a given Sherwood number, shown below:		which results in a plot of solutions for various Thiele modulii, for a given Sherwood number, shown below:

			[[Image:BesselFunctionSolutions.png]]

	[[Category:Mathematics]]		[[Category:Mathematics]]
	[[Category:Python]]		[[Category:Python]]

Admin: /* Bessel Functions with Scipy */

2014-02-25T19:54:55Z

Bessel Functions with Scipy

← Older revision		Revision as of 19:54, 25 February 2014
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	# Sherwood number (rate of transfer of material across boundary)		# Sherwood number (rate of transfer of material across boundary)
	v = 100.0		v = 100.0

	~~# Create solution vector~~
	~~y1 = [iv(0,phirho) / ( iv(0,phi) + (phi/v)iv(1,phi) ) for rho in r]~~
	~~plot(r,y1)~~

	# Create parametric solution vector		# Create parametric solution vector

Admin: /* Bessel Functions with Scipy */

2014-02-25T19:54:13Z

Bessel Functions with Scipy

← Older revision		Revision as of 19:54, 25 February 2014
Line 71:		Line 71:
	y = [iv(0,phirho) / ( iv(0,phi) + (phi/v)iv(1,phi) ) for rho in r]		y = [iv(0,phirho) / ( iv(0,phi) + (phi/v)iv(1,phi) ) for rho in r]
	plt.plot(r,y)		plt.plot(r,y)
	plt.~~set_xlabel~~('r')		plt.xlabel('r')
	plt.~~set_ylabel~~('u')		plt.ylabel('u')
	plt.hold(True)		plt.hold(True)
	</source>		</source>

Admin at 19:53, 25 February 2014

2014-02-25T19:53:24Z

@@ Line 1: / Line 1: @@
-Dealing with Bessel Functions
+=The Problem Setup=
 I had to deal with some Bessel Functions in dealing with an analytical solution of steady state reaction-diffusion partial differential equation. The diffusion-reaction equation is a simplified, non-dimensionalized equation for a single species and single reaction, and is given by:
@@ Line 32: / Line 32: @@
 at <math>\rho=1</math>.

Admin: Created page with "Dealing with Bessel Functions I had to deal with some Bessel Functions in dealing with an analytical solution of steady state reaction-diffusion partial differential equation. T..."

2014-02-25T19:45:18Z

Created page with "Dealing with Bessel Functions I had to deal with some Bessel Functions in dealing with an analytical solution of steady state reaction-diffusion partial differential equation. T..."

New page

Dealing with Bessel Functions

I had to deal with some Bessel Functions in dealing with an analytical solution of steady state reaction-diffusion partial differential equation. The diffusion-reaction equation is a simplified, non-dimensionalized equation for a single species and single reaction, and is given by:

<math>
\nabla^2 u = \phi^2 u
</math>

with the following boundary condition on the boundary of the domain:

<math>
v (1-u_s) = \dfrac{\partial u}{\partial v}
</math>

For an infinite cylinder (analytical solutions on cylinders typically end up using Bessel Functions), the solution can be written in terms of the non-dimensionalized radial dimension <math>\rho</math>:

<math>
\dfrac{1}{\rho} \dfrac{d}{d \rho} \left( \rho \dfrac{du}{d \rho} \right) = \phi^2 u
</math>

over the region <math>0 \leq \rho \leq 1</math>, and the boundary conditions:

<math>
\dfrac{du}{d \rho} = 0
</math>

at <math>\rho=0</math>, and also

<math>
v(1-u) = \dfrac{\partial u}{\partial \rho}
</math>

at <math>\rho=1</math>.