Solving Poisson/Dirichlet PDE with Boundary Conditions and 2u Variable

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In summary, the given problem must be broken into two problems in order to solve the PDE uxx+uyy=2u+f(x,y) with the shown boundary conditions. The first problem would involve solving the homogeneous equation uxx+uyy=2u using separation of variables, while the second problem would involve solving the inhomogeneous equation uxx+uyy-2u=f(x,y) with a forcing function f(x,y). The 2u term in the PDE is incorporated into the final solution.
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erok81
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Homework Statement



Show the two problems (i.e. give the boundary conditions and PDE's) that the given problem must be broken into in order to solve the PDE uxx+uyy=2u+f(x,y) satisfying the shown boundary conditions.

Homework Equations



See attachment.

The Attempt at a Solution



We have a midterm coming up in a couple days and of course were given a practice midterm. This problem came up. I can solve these if I have uxx=f(x) or probably even uxx+uyy=f(x,y). I have never seen a problem that also contains a 2u. I checked my notes and my text but cannot find anything regarding how to solve with a u in the PDE.

From the image I am not sure why it gets broken up that way either. If it was just f(x) alone I'd set one problem with all zero boundary conditions equal to f(x) and the non-zero boundary conditions would just have ∇2=0.

What is this 2u (or really any u variable here) and how does one incorporate it into the final solution?
 
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  • #2
Try to solve

uxx+uyy=2u

using separation of variables. This is a homogeneous equation. Then f(x,y) is a forcing function for the inhomogeneous equation

uxx+uyy - 2u = f(x,y)
 

Related to Solving Poisson/Dirichlet PDE with Boundary Conditions and 2u Variable

1. What is a Poisson PDE?

A Poisson partial differential equation (PDE) is a mathematical equation that relates an unknown function to its second derivatives. It is commonly used in physics and engineering to describe the behavior of physical phenomena, such as heat diffusion or electrostatics.

2. How is a Poisson PDE typically solved?

Poisson PDEs are typically solved using numerical methods, such as finite difference or finite element methods. These methods approximate the solution to the PDE by dividing the domain into a grid and solving the equation at discrete points.

3. What is a Dirichlet PDE?

A Dirichlet PDE is a type of boundary value problem where the solution is specified on the boundary of the domain. This means that the value of the function at the boundary is known, and the PDE is used to determine the values inside the domain.

4. How does a Dirichlet PDE differ from a Neumann PDE?

A Neumann PDE is another type of boundary value problem where the derivative of the solution is specified on the boundary of the domain. This means that the gradient of the function at the boundary is known, and the PDE is used to determine the values inside the domain. In contrast, a Dirichlet PDE specifies the function values at the boundary, rather than its derivative.

5. What are some real-world applications of Poisson/Dirichlet PDEs?

Poisson/Dirichlet PDEs have many applications in physics and engineering, such as in heat transfer, electromagnetism, fluid dynamics, and elasticity. They are also used in finance, biology, and image processing for modeling and analyzing various phenomena.

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