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:
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
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:
- Let s satisfy the Laplace equation:
- Let s satisfy the non-homogeneous boundary conditions.
- Let v satisfy the non-homogeneous equation and homogeneous boundary conditions.
We end up with 2 separate IBVPs:
Step 2: Solve Steady-State Portion
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
Step 3.1: Solve Associated Homogeneous BVP
The associated homogeneous BVP equation is:
The boundary conditions for v are the ones in the IBVP above.
By similar methods, you obtain the following ODEs:
Translate Boundary Conditions
We have obtained eigenfunctions that we can use to solve the nonhomogeneous IBVP.
Step 3.2: Solve Non-homogeneous IBVP
Just like in the 1-D case, we define v(x,y,t) and q(x,y,t) as infinite sums:
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
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
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.