Simple Sturm-Liouville system resembling Associated-Legendre equation?

  • Thread starter Thread starter omyojj
  • Start date Start date
  • Tags Tags
    System
omyojj
Messages
32
Reaction score
0
I'm trying to solve the following Sturm-Liouville system
<br /> \frac{d}{dx}\left((1-x^2)^2\frac{d}{dx}y\right) + (\lambda - k^2)y=0<br />

defined in an interval -a<x<a (or 0<x<a) with 0<a<=1.
Here, k is a real number and λ is the eigenvalue of the system.
y satisfies boundary conditions
y^{\prime}(a) = y^{\prime}(-a) = 0
plus the parity condition
y(x) = y(-x).
(or y'(a) = 0 and y'(0) = 0)

Can anybody give me any hint on how to obtain ground state(Lower-bound eigenvalue and the corresponding eigenfunction) solution, say y_0 and λ_0?

Of course being able to obtain general solution would be much better.

Thanks
 
Physics news on Phys.org
One thing I tried is to integrate the above equation from x=0 to x=a to get
\lambda_n \int_0^a y_n dx = k^2 \int_0^a y_n dx
(The first term on the left-hand side vanished from the given boundary conditions.
Hence,
\lambda_n = k^2
which is strange because all the eigenvalues are given as λ_n = k^2.
Where have I been wrong?
 
I don't think you got it wrong.

It's \lambda_k
 
No.
Maybe I should explain the background to this problem.
I encountered the above equation while solving the p-mode(acoustic wave) dispersion relation in an horizontally infinite isothermal disk with vertical stratification in the z direction.
Boundary conditions are chosen so that vertical displacement at the disk boundaries become zero.

Vertical density distribution given by
\rho(z) = \rm{sech}^2(z)
or
\rho(x) = (1-x^2)
when we make use of a Lagrangian variable z = tanh(x)

y here is perturbation variable
y = \rho_1(x)/\rho(x)

λ_n is the square of frequency ω_n for horizontal Fourier wavenumber k>0.

Physically, for each horizontal wavenumber k, there would be corresponding infinite number of p-modes, each with increasing frequency ω_n and eigenfunction y_n having n zeros between z=-a and z=a.
I want to find the solution to the fundamental mode (ω_0^2 = λ_0).

Anyway, k should be regarded as a given number (like ν(nu), the index representing the order of Bessel's equation)).
 
Last edited:
If you have not already solved your problem, use the Frobenius method with y(x) as an infinite series polynomial in x. This method is used in Math World's internet info for solving the Legendre differential equation. Also there are many other references available on the net or literature. Best wishes.
 
This particular Sturm-Liouville equation can be solved in terms of associeted Legendre functions (see attachment)
 

Attachments

  • LegendreODE.JPG
    LegendreODE.JPG
    44.3 KB · Views: 519
Thread 'Direction Fields and Isoclines'
I sketched the isoclines for $$ m=-1,0,1,2 $$. Since both $$ \frac{dy}{dx} $$ and $$ D_{y} \frac{dy}{dx} $$ are continuous on the square region R defined by $$ -4\leq x \leq 4, -4 \leq y \leq 4 $$ the existence and uniqueness theorem guarantees that if we pick a point in the interior that lies on an isocline there will be a unique differentiable function (solution) passing through that point. I understand that a solution exists but I unsure how to actually sketch it. For example, consider a...

Similar threads

Replies
8
Views
5K
Replies
5
Views
2K
Replies
1
Views
10K
Replies
45
Views
9K
Replies
11
Views
3K
Replies
13
Views
4K
Back
Top