Heat equation/Solution to the 2-D Heat Equation in Cylindrical Coordinates

Definition

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We start by changing the Laplacian operator in the 2-D heat equation from rectangular to cylindrical coordinates by 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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Step 1: Solve Associated Homogeneous Equation

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Separate Variables

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This means that a separation constant can be found that both sides will equal. Let's define it to be   This yields:

 

and multiplying the other side by   yields:

 

After defining another separation constant  , it yields:

 

Multiplying the other side by R yields:

 

We now have separate differential equations for each variable.

Translate Boundary Conditions

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Solve SLPs

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The SLP for   is a singular Bessel type, whose eigenvalues   depends on   and are non-negative solutions to the following equation:

 

and the eigenfunction is:

 

where   is the Bessel function of the first kind of order  .

Solve Time Equation

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Step 2: Satisfy Initial Condition

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Let's define the solution as an infinite sum:

 

With the initial condition:

 

where  

The weight function in the inner product   in integrals involving the Bessel functions. The Bessel functions   are orthogonal relative to the "weighted" scalar product  

Step 3: Solve Non-homogeneous Equation

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Solving the non-homogeneous equation involves defining the following functions:

 

 

Substitute the new definitions into the non-homogeneous equations:

 

We will use the following substitutions in our equation above:

 

We can eliminate the derivatives by substituting:

 

From the linear independence of  , it follows that:

 

This first-order ODE can be solved with the following integration factor:

 

Thus, the equation becomes:

 

 

We satisfy the initial condition: