Desperate trying to solve a simple Sturm-Liouville equation

  • Thread starter omyojj
  • Start date
  • Tags
    Desperate
In summary, the conversation revolves around a wave problem with a specific equation and boundary conditions. The goal is to obtain the solution for the ground state with the lowest eigenvalue, for general values of k and A. The suggested solution from a physical perspective is a constant for A approaching 1 and cosh(kx) for A approaching 0. A numerical approach in Mathematica is also discussed.
  • #1
omyojj
37
0
While dealing with a wave problem,
I encountered the following equation

[tex] \frac{d}{dx}\left[(1-x^2)^2\frac{d}{dx}y\right] - k^2y = -\omega^2y [/tex]

with x ∈ [0,A], (0<A<=1)

where k is a real number.

Thus it has eigenvalue ω^2 and weight unity.

Boundary conditions are

[tex] \frac{dy}{dx} = 0 [/tex]

at x = 0 and

[tex] y=2A[/tex]

at x= A.


I only need to obtain the solution for the ground state (the one with lowest eigenvalue).

for general values of k>0, 0<A<1.

I find from the physical point of view that the solution should look like

y_0=constant for A->1,

and y_0 = cosh(kx) for A -> 0


Can anybody give me a hint on how to solve this equation?
 
Last edited:
Physics news on Phys.org
  • #2
Here's a numeric approach in Mathematica:

Code:
k = 1.21; 
\[Omega] = 3.25; 
a = 0.825; 

sols = 
   (First[NDSolve[{(1 - x^2)^2*Derivative[2][y][
            x] - 4*x*(1 - x^2)*Derivative[1][y][
            x] + (\[Omega]^2 - k^2)*y[x] == 0, 
        Derivative[1][y][0] == 0, y[a] == 2*a}, y, 
       x, Method -> {"Shooting", 
         "StartingInitialConditions" -> 
          {y[0] == #1, Derivative[1][y][0] == 
            0}}]] & ) /@ {-2, 0, 2}; 

Plot[Evaluate[y[x] /. sols], {x, 0, a}, 
  PlotStyle -> {Black, Blue, Green}]
 

1. What is a Sturm-Liouville equation?

A Sturm-Liouville equation is a type of differential equation that is used to model physical phenomena such as heat transfer, fluid flow, and wave motion. It is named after the mathematicians Jacques Charles François Sturm and Joseph Liouville who first studied these equations in the 19th century.

2. Why is it important to solve Sturm-Liouville equations?

Sturm-Liouville equations are important because they have many real-world applications and can help us better understand and predict the behavior of physical systems. They are also used extensively in mathematical physics and engineering.

3. What makes solving a Sturm-Liouville equation challenging?

Solving a Sturm-Liouville equation can be challenging because they are often nonlinear and may involve complex mathematical techniques such as eigenvalue analysis and separation of variables. Additionally, the boundary conditions for these equations can be difficult to determine.

4. How can I approach solving a Sturm-Liouville equation?

There are several approaches to solving a Sturm-Liouville equation, including using analytical methods such as separation of variables or numerical methods such as finite difference or finite element methods. It is important to carefully analyze the equation and choose the most appropriate approach based on the type of equation and the available resources.

5. Are there any resources available to help with solving Sturm-Liouville equations?

Yes, there are many resources available to help with solving Sturm-Liouville equations. These include textbooks, online tutorials, and software programs specifically designed for solving these types of equations. Additionally, seeking guidance from a mentor or collaborating with other researchers in the field can also be helpful in solving these equations.

Similar threads

  • Differential Equations
Replies
2
Views
917
Replies
7
Views
2K
  • Differential Equations
Replies
1
Views
1K
  • Differential Equations
Replies
8
Views
4K
Replies
7
Views
2K
  • Differential Equations
Replies
1
Views
707
  • Differential Equations
Replies
11
Views
2K
  • Differential Equations
Replies
10
Views
1K
Replies
2
Views
986
  • Differential Equations
Replies
2
Views
1K
Back
Top