Transforming Non-Homogeneous Boundary Conditions in 2D PDEs

In summary, the conversation discusses a PDE with non-homogeneous boundary conditions and the possibility of transforming it into a PDE with homogeneous boundary conditions through linear transformation. The speaker suggests using a method similar to the one used in the 1D case, but is unsure of how to apply it in 2D.
  • #1
sigh1342
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Homework Statement


now I have a PDE
$$u_{xx}+u_{yy}=0,for 0<x,y<1$$
$$u(x,0)=x,u(0,y)=y^2,u(x,1)=0,u(1,y)=y$$
Then I want to know whether there are some method to make the PDE become homogeneous boundary condition.
$$i.e. u|_{\partialΩ}=0$$
 
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  • #2
sigh1342 said:

Homework Statement


now I have a PDE
$$u_{xx}+u_{yy}=0,for 0<x,y<1$$
$$u(x,0)=x,u(0,y)=y^2,u(x,1)=0,u(1,y)=y$$
Then I want to know whether there are some method to make the PDE become homogeneous boundary condition.
$$i.e. u|_{\partialΩ}=0$$

Could you solve the PDE if the boundary conditions had zero along three sides and a function along the fourth side? For example ##u(x,0)=x,u(0,y)=0,u(x,1)=0,u(1,y)=0##? Say you do that and call the solution ##u_1(x,y)##. Treat the other sides similarly. Then if you sum your solutions they will solve the homogeneous DE and the sum will satisfy all your BC's.
 
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  • #3
Actually the method I want to use need the boundary condition be all zero.
like the 1D case , $$u_{xx}=0 ,a<x<b, u(a)=\alpha, u(b)=\beta$$
the we can use $$u'=u-\beta+\frac{(x-b)(\beta-\alpha)}{a-b}$$
so $$u'_{xx}=0 , a<x<b , u'(a)=0, u'(b)=0$$
I would want to find whether there are similar method can work in 2D case.
Actually my prof. has told us that we can use some linear transform to do it.
But I have no idea.
Thank you :)
 

1. What is a boundary condition in PDE?

A boundary condition in PDE (partial differential equation) is a mathematical statement that describes the behavior of a solution at the edges or boundaries of the domain where the equation is being solved. It specifies the values or derivatives of the solution at these boundaries and is necessary to uniquely determine a solution to the PDE.

2. How many types of boundary conditions are there in PDE?

There are three types of boundary conditions in PDE: Dirichlet boundary conditions, Neumann boundary conditions, and Robin boundary conditions. Dirichlet boundary conditions specify the values of the solution at the boundary, Neumann boundary conditions specify the normal derivative of the solution at the boundary, and Robin boundary conditions specify a linear combination of the solution and its normal derivative at the boundary.

3. What is the importance of boundary conditions in solving PDEs?

Boundary conditions are crucial in solving PDEs because they determine the behavior of the solution at the boundaries of the domain. Without boundary conditions, the solution to a PDE would not be unique, and there would be an infinite number of possible solutions. Boundary conditions also allow us to apply the PDE to a specific physical situation or problem.

4. Can boundary conditions change with time in PDEs?

Yes, boundary conditions can change with time in PDEs. This type of boundary condition is known as a time-dependent boundary condition. It is commonly used in problems involving diffusion or wave propagation, where the boundary conditions may change over time due to external factors.

5. How do boundary conditions affect the stability and accuracy of a numerical solution to a PDE?

Boundary conditions can greatly affect the stability and accuracy of a numerical solution to a PDE. If the boundary conditions are not properly accounted for, it can lead to an unstable or inaccurate solution. It is important to carefully choose and implement appropriate boundary conditions in order to obtain a reliable and accurate numerical solution to a PDE.

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