Heat equation/Solution to the 2-D Heat Equation

Definition

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The solution to the 2-dimensional heat equation (in rectangular coordinates) deals with two spatial and a time dimension,  . The heat equation, the variable limits, the Robin boundary conditions, and the initial condition are defined as:

 

Solution

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The solution is just an advanced version of the solution in 1 dimension. If you have questions about the steps shown here, review the 1-D solution.

Step 1: Partition Solution

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Just as in the 1-D solution, we partition the solution into a "steady-state" and a "variable" portion:

 

We substitute this equation into the initial boundary value problem (IBVP):

 

We want to set some conditions on s and v:

  1. Let   satisfy the Laplace equation:  
  2. Let   satisfy the non-homogeneous boundary conditions.
  3. Let v satisfy the non-homogeneous equation and homogeneous boundary conditions.

We end up with 2 separate IBVPs:

 

 

Step 2: Solve Steady-State Portion

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Solving for the steady-state portion is exactly like solving the Laplace equation with 4 non-homogeneous boundary conditions. Using that technique, a solution can be found for all types of boundary conditions.

Step 3: Solve Variable Portion

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Step 3.1: Solve Associated Homogeneous BVP

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The associated homogeneous BVP equation is:

 

The boundary conditions for v are the ones in the IBVP above.

Separate Variables
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By similar methods, you obtain the following ODEs:

 

Translate Boundary Conditions
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Solve SLPs
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We have obtained eigenfunctions that we can use to solve the nonhomogeneous IBVP.

Step 3.2: Solve Non-homogeneous IBVP

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Setup Problem
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Just like in the 1-D case, we define v(x,y,t) and q(x,y,t) as infinite sums:

 

 

Determine Coefficients
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We then substitute expansion into the PDE:

 

 

 

 

This implies that   forms an orthogonal basis. This means that we can write the following:

 

This is a first-order ODE which can be solved using the integration factor:

 

Solving for our coefficient we get:

 

Satisfy Initial Condition
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We apply the initial condition to our equation above:

 

The Fourier coefficients can be solved using the inner product definition:

 

We have all the necessary information about the variable portion of the function.

Step 4: Combine Solutions

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We now have solved for the "steady-state" and "variable" portions, so we just add them together to get the complete solution to the 2-D heat equation.