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

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Definition[edit | edit source]

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[edit | edit source]

Step 1: Solve Associated Homogeneous Equation[edit | edit source]

Separate Variables[edit | edit source]

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[edit | edit source]

Solve SLPs[edit | edit source]

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[edit | edit source]

Step 2: Satisfy Initial Condition[edit | edit source]

Let's define the solution as an infinite sum:

With the initial condition:


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[edit | edit source]

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: