Appears in almost every field of physics.
Solution to Case with 4 Homogeneous Boundary Conditions
Let's consider the following example, where and the Dirichlet boundary conditions are as follows:
In order to solve this equation, let's consider that the solution to the homogeneous equation will allow us to obtain a system of basis functions that satisfy the given boundary conditions. We start with the Laplace equation:
Step 1: Separate Variables
Consider the solution to the Poisson equation as Separating variables as in the solution to the Laplace equation yields:
Step 2: Translate Boundary Conditions
As in the solution to the Laplace equation, translation of the boundary conditions yields:
Step 3: Solve Both SLPs
Because all of the boundary conditions are homogeneous, we can solve both SLPs separately.
Step 4: Solve Non-homogeneous Equation
Consider the solution to the non-homogeneous equation as follows:
We substitute this into the Poisson equation and solve:
Solution to General Case with 4 Non-homogeneous Boundary Conditions
Let's consider the following example, where and the boundary conditions are as follows:
The boundary conditions can be Dirichlet, Neumann or Robin type.
Step 1: Decompose Problem
For the Poisson equation, we must decompose the problem into 2 sub-problems and use superposition to combine the separate solutions into one complete solution.
- The first sub-problem is the homogeneous Laplace equation with the non-homogeneous boundary conditions. The individual conditions must retain their type (Dirichlet, Neumann or Robin type) in the sub-problem:
- The second sub-problem is the non-homogeneous Poisson equation with all homogeneous boundary conditions. The individual conditions must retain their type (Dirichlet, Neumann or Robin type) in the sub-problem:
Step 2: Solve Subproblems
Depending on how many boundary conditions are non-homogeneous, the Laplace equation problem will have to be subdivided into as many sub-problems. The Poisson sub-problem can be solved just as described above.
Step 3: Combine Solutions
The complete solution to the Poisson equation is the sum of the solution from the Laplace sub-problem and the homogeneous Poisson sub-problem :