Solving Laplace Equation w/ Neumann Boundary Conditions

In summary, the problem is that the condition that the derivative of u on the edge of the domain be zero is sufficient, but not necessary, for a solution to exist. The solution proposed is based on the substitution of x=A and y=B, where A and B are constants. The By is included because it is needed to satisfy the boundary conditions at the origin (0,0) and (pi,pi).
  • #1
maria clara
58
0

Homework Statement



I need to solve Laplace equation in the domain D= 0 < x,y < pi

Neumann boundary conditions are given:

du/dx(0,y)=du/dx(pi,y)=0
du/dy(x,pi)=x^2-pi^2/3+1
du/dy(x,0)=1

2. The attempt at a solution

first, we check that the integral of directional derivative of u on the edge of D is zero. This should be the necessary condition for the existence of a solution to the problem. the condition is satisfied (the integral is zero indeed), but why is this condition sufficient and not only necessary?

Anyway, assuming a solution does exist, I propose a solution of the form u=X(x)Y(t), and solving the appropriate SL system I find that the solution sould be of the form:
u(x,y)= A+sigma{Cos(nx)*[Cosh(ny)+Sinh(ny)]}

but the solution should have the form

u(x,y)= A+By+sigma{Cos(nx)*[Coshny+Cosh[n(y-pi)]}

where does the By come from? and does Cosh[n(y-pi)] equal Sinhny? It doesn't make sense because for example if we have y=pi , Cosh[n(y-pi)] = 1, while Sinh(ny) gives an infinite number of answers, depending on n.

Thanks in advance...:)
 
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  • #2
maria clara said:

Homework Statement



I need to solve Laplace equation in the domain D= 0 < x,y < pi

Neumann boundary conditions are given:

du/dx(0,y)=du/dx(pi,y)=0
du/dy(x,pi)=x^2-pi^2/3+1
du/dy(x,0)=1

2. The attempt at a solution

first, we check that the integral of directional derivative of u on the edge of D is zero. This should be the necessary condition for the existence of a solution to the problem. the condition is satisfied (the integral is zero indeed), but why is this condition sufficient and not only necessary?
I don't understand this. What makes you think this condition is sufficient?

Anyway, assuming a solution does exist, I propose a solution of the form u=X(x)Y(t), and solving the appropriate SL system I find that the solution sould be of the form:
u(x,y)= A+sigma{Cos(nx)*[Cosh(ny)+Sinh(ny)]}

but the solution should have the form

u(x,y)= A+By+sigma{Cos(nx)*[Coshny+Cosh[n(y-pi)]}

where does the By come from? and does Cosh[n(y-pi)] equal Sinhny? It doesn't make sense because for example if we have y=pi , Cosh[n(y-pi)] = 1, while Sinh(ny) gives an infinite number of answers, depending on n.

Thanks in advance...:)
The By is because of that "+1" on the du/dy at both y= 0 and y=[itex]\pi[/itex].
 
  • #3
Well, this is what they write in the answer, that here the condition is "necessary and sufficient"...
I guess it is not merely a mistake, because this is not the first time I encounter this argument.


any suggestions regarding the problem in the equation:
cosh[n(y-pi)]=sinh(ny)
?
 
  • #4
It's to satisfy the boundary conditions at 0 and pi.
solutions are u(x,y)=A exp(ny) + Bexp(-ny)
or =a sinh(ny)+bcosh(ny)
or =cosh(ny)+cosh n(y-pi) this satisfy thr BC at 0 and pi
The laplace equation is invariant under translation
belgium 12
 

1. How do I approach solving a Laplace equation with Neumann boundary conditions?

Solving a Laplace equation with Neumann boundary conditions requires you to first understand the problem at hand and then apply mathematical techniques such as separation of variables, Green's functions, or numerical methods. It is important to carefully define the boundary conditions and choose an appropriate method for solving the equation.

2. What are Neumann boundary conditions and how do they differ from Dirichlet boundary conditions?

Neumann boundary conditions specify the value of the derivative of the solution at the boundary, while Dirichlet boundary conditions specify the value of the solution itself at the boundary. In other words, Neumann boundary conditions involve the flux or flow of the solution at the boundary, while Dirichlet boundary conditions involve the value of the solution itself.

3. Can you provide an example of solving a Laplace equation with Neumann boundary conditions?

For example, consider a 2-dimensional Laplace equation with Neumann boundary conditions in a rectangular region with sides of length a and b. The boundary conditions may specify the normal derivative of the solution at the top and bottom boundaries to be equal to f(x), while the normal derivative at the left and right boundaries is equal to g(y). By applying separation of variables and solving the resulting ODEs, the solution can be found as a series of sine and cosine functions multiplied by coefficients that depend on the boundary conditions.

4. What are some applications of solving Laplace equation with Neumann boundary conditions?

The Laplace equation is a fundamental equation in physics, engineering, and mathematics. Solving it with Neumann boundary conditions can help solve problems related to heat transfer, electrostatics, fluid flow, and other physical phenomena. It can also be used in mathematical models and simulations to understand and predict the behavior of systems with Neumann boundary conditions.

5. Are there any limitations or challenges in solving a Laplace equation with Neumann boundary conditions?

One challenge is that the solution may not always exist or may not be unique, depending on the boundary conditions and the geometry of the region. Additionally, some methods for solving Laplace equation with Neumann boundary conditions may be computationally intensive and may require specialized software or programming skills. It is important to carefully consider the problem and choose an appropriate method to ensure a valid and accurate solution.

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