Heat equation/Solution to the 2-D Heat Equation

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Definition[edit | edit source]

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[edit | edit source]

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[edit | edit source]

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 s satisfy the Laplace equation:
  2. Let s 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[edit | edit source]

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[edit | edit source]

Step 3.1: Solve Associated Homogeneous BVP[edit | edit source]

The associated homogeneous BVP equation is:

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

Separate Variables[edit | edit source]

By similar methods, you obtain the following ODEs:

Translate Boundary Conditions[edit | edit source]

Solve SLPs[edit | edit source]

We have obtained eigenfunctions that we can use to solve the nonhomogeneous IBVP.

Step 3.2: Solve Non-homogeneous IBVP[edit | edit source]

Setup Problem[edit | edit source]

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

Determine Coefficients[edit | edit source]

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[edit | edit source]

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[edit | edit source]

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.