Partial differential equations/Poisson Equation

Poisson's EquationEdit

DefinitionEdit

 

DescriptionEdit

Appears in almost every field of physics.

Solution to Case with 4 Homogeneous Boundary ConditionsEdit

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 VariablesEdit

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

Step 2: Translate Boundary ConditionsEdit

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

Step 3: Solve Both SLPsEdit

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

 

 

Step 4: Solve Non-homogeneous EquationEdit

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 ConditionsEdit

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 ProblemEdit

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 SubproblemsEdit

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 SolutionsEdit

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 linksEdit