Analytical solution of PDEs - Revision history

Admin: /* Example 3: Heat Equation */

2012-05-25T07:11:27Z

Example 3: Heat Equation

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	== Example 3: Heat Equation ==		== Example 2: Heat Equation ==

	Separation of variables can be used to solve the heat equation under certain circumstances		Separation of variables can be used to solve the heat equation under certain circumstances

2012-05-25T07:11:13Z

Example 2: Second Derivatives

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	Which, when plugged in, satisfies the original PDE.		Which, when plugged in, satisfies the original PDE.



	~~== Example 2: Second Derivatives ==~~

	~~{{Stub\|section}}~~

2012-05-12T12:31:34Z

Example 3: Heat Equation

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	What circumstances?		What circumstances?

	''linear homogeneous boundary conditions... linear homogeneous PDE'''		''linear homogeneous boundary conditions... linear homogeneous PDE''

	Consider the heat equation in a normalized domain:		Consider the heat equation in a normalized domain:

2010-11-06T01:27:13Z

Step 1: Solution to PDE

2010-10-24T11:28:34Z

References

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	http://www.exampleproblems.com/wiki/index.php/PDE:Integration_and_Separation_of_Variables		http://www.exampleproblems.com/wiki/index.php/PDE:Integration_and_Separation_of_Variables

			Tannehill

			Anderson

2010-10-24T11:28:00Z

2010-10-24T11:10:05Z

Step 3: Solution to PDE, BCs, ICs

2010-10-24T11:06:18Z

Step 3: Solution to PDE, BCs, ICs

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	</math>		</math>

	Convince yourself of this by studying the plots of several sine waves:		(Convince yourself of this by studying the plots of several sine waves:)

	[[Image:Sine waves.png\|An orthogonal sequence of sine waves.]]		[[Image:Sine waves.png\|An orthogonal sequence of sine waves.]]

2010-10-24T10:55:40Z

Step 3: Solution to PDE, BCs, ICs

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 [[Image:Initial_T.png|A hypothetical initial temperature profile in a 1D rod.]]
-This temperature profile is given by the expression:
+This temperature profile plotted above (for <math>\alpha = 0.01</math>) is an expression consisting of multiple sine waves, each with a unique frequency.  The full expression for the temperature profile is:
 = Combination of Variables =

2010-10-24T08:14:26Z

Example 3: Heat Equation: adding final solution and interpretation...

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 ''<math>\alpha(\theta)</math> makes the equation quasi-linear - i.e. NOT LINEAR - so can't use the separation of variables (see [[Introduction to partial differential equations]] lecture)''
-Now a solution of the form <math>X(x) T(t)</math> can be assumed, and plugged into the PDE:
+=== Step 1: Solution to PDE ===
 <math>
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 where <math>A,B,\lambda</math> are arbitrary.
 Let's try these solutions at <math>t=0</math> - our initial condition
-So let's try the function for <math>A=1,B=1,\lambda=1</math> at <math>x=0</math>:
+Try the function for <math>A=1,B=1,\lambda=1</math> at <math>x=0</math>:
 <math>
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 '''Trivial solution''' - <math>A=0</math> is an example of a trivial solution.  It means our initial condition <math>\phi(x) = 0</math>, so we started out with a uniform temperature of 0.
 = Combination of Variables =