Heat equation/Solution to the 3-D Heat Equation in Cylindrical Coordinates

Definition

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We are adding to the equation found in the 2-D heat equation in cylindrical coordinates, starting with the following definition:

 

By changing the coordinate system, we arrive at the following nonhomogeneous PDE for the heat equation:

 

We choose for the example the Robin boundary conditions and initial conditions as follows:

 

Solution

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All of the boundary conditions are homogeneous, so we don't have to partition the solution into a "steady-state" portion and a "variable" portion. Otherwise, that would be the way to solve this problem.

Step 1: Solve Associated Homogeneous Equation

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Separate Variables

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There is a separation constant   that both sides of the equation are equivalent to. This yields:

 

 

The second equation yields the equations:

 

 

 

This yields the following equations:

 

 

Translate Boundary Conditions

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Just like in the 2-D heat equation, the boundary conditions yield:

 

Solve SLPs

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Solve Time Equation

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The solution to the equation is:

 

Step 2: Satisfy Initial Condition

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Define:

 

Applying the initial condition:

 

This is the orthogonal expansion of   in terms of   Hence,

 

Step 3: Solve the Non-homogeneous Equation

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Let:

 

 

Substitute the expansions for u and h into the non-homogeneous equation:

 

 

From the linear independence of  :

 

 

The undetermined coefficient satisfies the initial condition: