Partial differential equations/Poisson Equation

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Poisson's Equation[edit]

Definition[edit]

Description[edit]

Appears in almost every field of physics.

Solution to Case with 4 Homogeneous Boundary Conditions[edit]

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[edit]

Consider the solution to the Poisson equation as Separating variables as in the solution to the Laplace equation yields:

Step 2: Translate Boundary Conditions[edit]

As in the solution to the Laplace equation, translation of the boundary conditions yields:

Step 3: Solve Both SLPs[edit]

Because all of the boundary conditions are homogeneous, we can solve both SLPs separately.



Step 4: Solve Non-homogeneous Equation[edit]

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[edit]

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[edit]

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.

  1. 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:

  2. 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[edit]

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[edit]

The complete solution to the Poisson equation is the sum of the solution from the Laplace sub-problem and the homogeneous Poisson sub-problem :

External links[edit]