Eigenvalue problem with nonlocal condition

In summary, the conversation discusses simplifying or expressing the positive and negative parts of a function in an eigenvalue problem, specifically when ##\lambda>0##. The solution is found to be of the form ##u(x) = C_1 \sin(\sqrt{\lambda}x)## and determining ##C_1## involves finding the number of zeros and intervals where the function is positive or negative. The conversation also notes that any value of ##C_1## will satisfy the last condition, which is not surprising.
  • #1
kajzlik
3
0
Hello guys, suppose we have an eigenvalue problem
[tex]
\left\{
\begin{array}{ll}
u'' + λu = 0, \quad x \in (0,\pi) \\
u(0)=0 \quad \\
\left( \int_0^\pi \! {(u^+)}^2 \, \mathrm{d}x \right)^{\frac{1}{2}} = \left( \int_0^\pi \! {(u^-)}^4 \, \mathrm{d}x \right)^\frac {1}{4}

\end{array}
\right.

[/tex]
where [tex] u^+, u^-[/tex] is positive, negative part of function respectively.
I'm having troubles with case when λ> 0. Is there any way how to simplify or express these parts of function ? I've tried some (analytic) brute force methods, tried to simplify that first but still no valuable result.
Thanks
 
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  • #2
What if you use that the general solution to

[tex]u^{\prime\prime} + \lambda u =0[/tex]

is always of the form

[tex]u(x) = C_1\sin(\sqrt{\lambda}x) + C_2\cos(\sqrt{\lambda}x)[/tex]

for ##\lambda>0##. Then from ##u(0) = 0##, you can deduce that

[tex]C_2=0[/tex]

So your function has the form

[tex]u(x) = C_1 \sin(\sqrt{\lambda}x)[/tex]

Now figuring out what ##C_1## is from the last condition seems tedious, but not all too difficult.
 
  • #3
Hmm, thinking about it, any value of ##C_1## will satisfy the last condition, at least if ##f(x) = \sin(\sqrt{\lambda}x)## satisfies it. So your only job is to find out for which ##\lambda## this is true.
 
  • #4
Thanks for your response.
The formulation of my problem was quite confusing. First condition is ok, its just straight forward process that leads to sine function, but I'm completely lost with the second one.As you said it's obvious that second condition will hold for any "multiplying" constant.

These positive / negative parts of function are bit tricky for me. Since number of zeros depends on λ( and then number of positive and negative parts of sine too), I have to split that function somehow and do piecewise integration.Thats the problem. Still seems something is "hidden" for me and I'm sure that will be obvious, but I can't figure it out now.
I'm not asking for solution, but I would really appreciate some hint ;)
Thanks
 
Last edited:
  • #5
You have already determined, from the equation and the condition that y(0)= 0, that [itex]y= Csin(\sqrt{\lambda} x)[/itex]. Yes, the number of zeros and the number of intervals where y is positive or negative depend upon [itex]\lambda[/itex] but it is not all that difficult to determine. For a given [itex]\lambda[/itex], [itex]cos(\sqrt{\lambda}x)[/itex] will be 0 at [itex](n\pi)/\sqrt{\lambda}[/itex]. It will be positive on the interval from [itex](2n\pi)/\sqrt{\lambda}[/itex] to [itex]((2n+1)\pi)/\sqrt{\lambda}[/itex] and negative on the interval from [itex]((2n+1)\pi)/\sqrt{\lambda}[/itex] to [itex](2(n+1)\pi)/\sqrt{\lambda}[/itex] for any positive integer n.
 
  • #6
Well, not that difficult... Thank you, it really helped.
 
  • #7
micromass said:
Hmm, thinking about it, any value of ##C_1## will satisfy the last condition

That shouldn't be a surprise. If the magnitude of an eigenvector is NOT arbitrary, there is something very strange going on!
 

1. What is an eigenvalue problem with nonlocal condition?

An eigenvalue problem with nonlocal condition is a type of mathematical problem that involves finding the values (eigenvalues) and corresponding functions (eigenvectors) that satisfy a specific nonlocal condition. This condition typically involves a function evaluated at a point on the domain being equal to a weighted average of the function values at other points on the domain.

2. What are some applications of eigenvalue problems with nonlocal condition?

Eigenvalue problems with nonlocal condition have various applications in physics, engineering, and other fields. They are commonly used to model diffusion processes, as well as to study the behavior of waves and oscillations in continuous media. They also have applications in image processing and pattern recognition.

3. How do you solve an eigenvalue problem with nonlocal condition?

The solution to an eigenvalue problem with nonlocal condition involves finding the eigenvalues and eigenvectors that satisfy the nonlocal condition. This can be done by expressing the problem as a system of equations and solving for the eigenvalues and eigenvectors using analytical or numerical methods.

4. What are the challenges of solving eigenvalue problems with nonlocal condition?

Solving eigenvalue problems with nonlocal condition can be challenging due to the nonlocal nature of the condition. This can make it difficult to find analytical solutions, and numerical methods may require a large number of computations. Additionally, there may be multiple solutions or complex eigenvalues and eigenvectors that need to be carefully interpreted.

5. Are there any alternative methods for solving eigenvalue problems with nonlocal condition?

Yes, there are alternative methods for solving eigenvalue problems with nonlocal condition, such as the method of integral equations. This method involves transforming the nonlocal condition into an integral equation, which can then be solved using numerical methods. Other approaches include using finite difference or finite element methods to approximate the nonlocal condition.

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