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

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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:


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

From the linear independence of :

The undetermined coefficient satisfies the initial condition: