How to solve the Laplace equation in a half disk with given boundary conditions?

In summary, the conversation is about solving the Laplace equation within a half disk with given boundary conditions. The solution has been attempted, but the person is stuck at calculating the unknown constants. The conversation includes a link to a picture of the work done so far.
  • #1
EsponV
15
0

Homework Statement


Solve the Laplace equation: delta u = d2u/dx2+d2u/dy2
inside the half disk 0<r<R, 0<phi<pi
Temperature on the bottom side of the disk is zero, u(x,y=0)=0. Temperature on the upper side of the disk is u(r=R, theta) = u0(phi), 0<phi<pi



Homework Equations


I'm assuming the Laplace equation and the given bc conditions


The Attempt at a Solution


I attempted the solution, which I think up until this point is correct, but I've gotten stuck trying to figure out the constants.

http://img708.imageshack.us/img708/707/p2150074.jpg

Thank you all for your help.
 
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  • #2
I just got a email telling me I had a response, however I don't see it in the thread. Anyhow, the picture that only shows half the page of paper because that's all the work I had done on it. I got stuck at that point and wasn't sure how to go about calculating the unknown constants (if I had done everything correctly to that point).
 

What is the LaPlace equation in polar?

The LaPlace equation in polar coordinates is a partial differential equation that describes the relationship between the potential or electric field at any point in space and the distribution of sources or charges in the surrounding region. It is a fundamental equation in electromagnetics and is used to solve a variety of problems involving electric and magnetic fields.

Where is the LaPlace equation in polar commonly used?

The LaPlace equation in polar is commonly used in problems involving cylindrical or spherical symmetry, such as in the study of electric fields of charged cylinders or spherical conductors. It is also used in problems involving boundary value conditions, where the potential or electric field is known at certain points in space and needs to be solved for at other points.

What are the advantages of using the LaPlace equation in polar?

One of the main advantages of using the LaPlace equation in polar is that it allows for the simplification of complex problems involving cylindrical or spherical symmetry. It also provides a more intuitive understanding of the relationship between the potential or electric field and the sources or charges in the surrounding region.

What are the limitations of the LaPlace equation in polar?

The LaPlace equation in polar is limited to problems with cylindrical or spherical symmetry and cannot be applied to more complex geometries. It also assumes a linear relationship between the potential or electric field and the sources or charges, which may not hold true in all cases.

How is the LaPlace equation in polar solved?

The LaPlace equation in polar is typically solved using separation of variables, where the solution is expressed as a product of two functions, one depending solely on the radial coordinate and the other solely on the angular coordinate. Boundary value conditions are then applied to determine the specific form of these functions, resulting in a unique solution for the potential or electric field. Numerical methods can also be used to solve the equation for more complex problems.

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