Dirichlet problem

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In summary, when solving the equation u_{xx}(x,y) + u_{yy}(x,y) = xu(x,y) on the square [0,1]x[0,1] with boundary conditions u(x,0) = x, u(x,1) = x, u(0,y) = 0, u(1,y) = 1 and a mesh size of h = 1/3, the correct system of equations for the point (1/3, 1/3) is -12u_{1,1} + 3u_{2,1} + 3u_{1,2} = -1. It is important to carefully consider the boundary conditions and mesh
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leopard
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Homework Statement



We want to solve the equation

[tex]u_{xx}(x,y) + u_{yy}(x,y) = xu(x,y) [/tex]

on the square [0,1]x[0,1] with boundary conditions

u(x,0) = x
u(x,1) = x
u(0,y) = 0
u(1,y) = 1

Let h = 1/3 be the mesh. Set up a system of equations for [tex]u_{1,1}, u_{2,1}, u_{1,2}[/tex] and [tex]u_{2,2}[/tex]

2. The attempt at a solution

First point (1/3, 1/3):

-4[tex]u_{1,1}[/tex] + [tex]u_{2,1}[/tex] + [tex]u_{1,2}[/tex] = -x + x[tex]u_{1,1}[/tex]

This can be written

-13/3[tex]u_{1,1}[/tex] + [tex]u_{2,1}[/tex] + [tex]u_{1,2}[/tex] = -1/3

The correct equation for this point is

-109[tex]u_{1,1}[/tex] + 27[tex]u_{2,1}[/tex] + 27[tex]u_{1,2}[/tex] = -9

What is wrong?
 
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  • #2


There are a few errors in this attempt. First, the equation should not have an "x" term on the right side, as it is only present in the original equation to show that it is a function of both x and y. Additionally, the coefficients in the correct equation are not correct, as they should be based on the values of the mesh size and the boundary conditions. Finally, the equation should be set up for the point (1/3, 1/3), not (1,1).

The correct equation for this point should be:

-4u_{1,1} + u_{2,1} + u_{1,2} = -1/9

This can be written as:

-12u_{1,1} + 3u_{2,1} + 3u_{1,2} = -1

It is important to carefully consider the boundary conditions and mesh size when setting up a system of equations for solving partial differential equations. Double check your calculations and make sure they are based on the correct values and equations.
 

1. What is the Dirichlet problem?

The Dirichlet problem is a mathematical concept in potential theory that involves finding a solution to a partial differential equation (PDE) with given boundary conditions. It was first introduced by mathematician Peter Gustav Lejeune Dirichlet in the 19th century.

2. What is the significance of the Dirichlet problem?

The Dirichlet problem is significant because it provides a general framework for solving PDEs with specified boundary conditions. It has applications in various fields such as physics, engineering, and finance.

3. How is the Dirichlet problem solved?

The Dirichlet problem is typically solved using the method of separation of variables, which involves decomposing the PDE into simpler equations that can be solved individually. Other methods such as integral transforms and numerical techniques can also be used.

4. What are some real-world examples of the Dirichlet problem?

The Dirichlet problem can be applied to various physical phenomena, such as heat conduction, fluid flow, and electrostatics. It can also be used in financial modeling to determine the fair price of options and other derivatives.

5. Are there any limitations to the Dirichlet problem?

One limitation of the Dirichlet problem is that it assumes the boundary conditions to be known and fixed. In reality, boundary conditions may vary or be uncertain, which can affect the accuracy of the solution. Additionally, the method of separation of variables may not work for all types of PDEs, requiring alternative approaches to be used.

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