What Are the Steps to Solve a Boundary Condition Problem with Three Conditions?

In summary, in order to apply the boundary conditions on the x-axis, you need to find the Fourier transform of f(x), which is A(k).
  • #1
Mazimillion
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0
Hi, this question has been bugging me for weeks and any help would be greatly appreciated.

In lectures we derived a general expression for the potential distribution across the xy half plane (y>0) in terms of a known potential distribution along the boundary defined by the x-axis where the potential,phi, is described by Laplace's equation in 2-D. In the case where the potential along the boundary is: 0 for x greater than equal to a, 0 for x less then equal to -a, phi0 for -a<x<a, deduce an expression for the potential distribution throughout the half plane.

I think what is getting me are the 3 boundary conditions - I'm not sure exactly what i should be doing with them to find values for the coefficients.

i'm sorry that it's not very clear but if you have any ideas they would be greatly appreciated

thanks
 
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  • #2
The solution to these things is always Fourier analysis, or something like it.

There are two boundary conditions here: the potential should go to zero as y gets large, and it should become your specified function of x when y is 0. Because we want the potential to die off, we have a decaying exponential in the y direction, and the general solution is a weighted sum of terms like like Ae^(-ky)e^(ikx). Since you can't restrict the value of k at all, this sum becomes an integral: Int A(k) e^(-ky)e^(ikx) dk. When y is 0, you know that the potential is your given f(x), so f(x) = Int A(k) e^(ikx) dx. So A(k) is just the Fourier transform of your boundary condition f(x)!

So, find A(k) by taking the Fourier transform of f(x), plug it into your original integral with e^(-ky), and you are done!
 
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  • #3
Thanks for that - I've really made some progress. The only problem now is applying the boundary conditions on the x-axis. How do I apply both phi=0 for x<-a, x>a and phi=phi0 for -a<x<a?

I'm probably missing something really simple but I've been pulling my hair out for ages.

Thanks
 
  • #4
The integral I've mentioned is from k = negative infinity to positive infinity. f(x) is how I'm referring to your pulse between -a and a on the x axis.

The key is that A(k) and f(x) are a Fourier transform pair. Did they not discuss this in your class? So if f(x) = Int A(k) e^(ikx) dx, then A(k) = (1/2 pi) Int f(x) e^(-ikx) dx, where this integral is over all space. Since f(x) is zero except between -a and a, you can change the limits to there.
 
  • #5
Got it now

Thanks a million, you've been a great help
 

Related to What Are the Steps to Solve a Boundary Condition Problem with Three Conditions?

1. What is a boundary condition problem?

A boundary condition problem is a mathematical problem that involves solving a differential equation subject to certain conditions on the boundary. These conditions specify the behavior of the solution at the edges of the domain.

2. Why are boundary conditions important in scientific research?

Boundary conditions are important because they allow us to accurately model real-world systems and phenomena. They provide constraints on the solution that reflect physical or mathematical properties of the system being studied.

3. What are some common types of boundary conditions?

Some common types of boundary conditions include Dirichlet boundary conditions, which specify the value of the solution at the boundary, and Neumann boundary conditions, which specify the derivative of the solution at the boundary. Other types include Robin, periodic, and mixed boundary conditions, among others.

4. How do boundary conditions affect the solution of a problem?

Boundary conditions directly affect the solution of a problem by providing information about the behavior of the solution at the edges of the domain. They can significantly alter the properties of the solution, such as its stability, regularity, and convergence.

5. Can boundary conditions be modified or changed?

In some cases, boundary conditions can be modified or changed to investigate different scenarios or to improve the accuracy of the solution. However, in most cases, boundary conditions are determined by physical or mathematical considerations and cannot be arbitrarily altered without changing the problem being studied.

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