Sturm-Liouville and Rayleigh Quotient Problem

In summary, by comparing the given equation to the Sturm-Liouville form, it can be reduced to a ratio of two integrals with both integrals bounded from 0 to L. We also know that the eigenfunction cannot be identically 0. It follows from a proposition that both the top and bottom functions must be greater than 0.
  • #1
FAS1998
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Homework Statement
I’ve included an image of the problem below. This isn’t technically a homework problem. It’s a practice test question that I have the solutions for. I’m stuck at part 2. I have to show that eigenvalues are positive with the Rayleigh quotient.
Relevant Equations
d/dx(p(x)d*phi(x)/dx) + q(x)*phi(x) + lamda*sigma(x)*phi(x)
By comparing the given equation to the equation for the Sturm-Liouville form, I see that p(x) must equal 1, q(x) must equal 0 and sigma(x) must equal 1.

After this, I have no idea what I should be doing for part 2.
 

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  • #2
I see now that by plugging in values of p(x), q(x), and σ(x) into the equation, and then applying boundary conditions, it can be reduced to (∫(dΦ(x)/dx)^2)/(∫Φ(x)^2), with both integrals bounded from 0 to L.

I also understand that Φ(x) cannot be identically 0 because it's an eigenfunction.

I still don't understand how we know that both the top and bottom function are not less than 0.
 
  • #3
FAS1998 said:
I see now that by plugging in values of p(x), q(x), and σ(x) into the equation, and then applying boundary conditions, it can be reduced to (∫(dΦ(x)/dx)^2)/(∫Φ(x)^2), with both integrals bounded from 0 to L.

I also understand that Φ(x) cannot be identically 0 because it's an eigenfunction.

I still don't understand how we know that both the top and bottom function are not less than 0.

This follows from the following straightforward proposition:

Let [itex]f : [a,b] \to \mathbb{R}[/itex] be continuous, non-negative and not identically zero. Then [tex]\int_a^b f(x)\,dx > 0.[/tex]
Proof: As [itex]f[/itex] is non-negative and not identically zero, there exists [itex]x_0 \in [a,b][/itex] such that [itex]f(x_0) > 0[/itex]. By continuity of [itex]f[/itex], there exists an interval [itex][c,d] \subseteq [a,b][/itex] of strictly positive width containing [itex]x_0[/itex] such that if [itex]x \in [c,d][/itex] then [itex]f(x) \geq \frac12 f(x_0)[/itex]. But then [tex]
\int_a^b f(x)\,dx \geq \int_c^d f(x)\,dx \geq \int_c^d \tfrac12 f(x_0)\,dx = \tfrac12 (d - c)f(x_0) > 0[/tex] as required.
 
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Related to Sturm-Liouville and Rayleigh Quotient Problem

1. What is the Sturm-Liouville problem?

The Sturm-Liouville problem is a mathematical problem that involves finding the eigenvalues and eigenfunctions of a second-order linear differential equation. It is commonly used in physics and engineering to model physical systems such as vibrating strings, heat transfer, and quantum mechanics.

2. What is the Rayleigh Quotient problem?

The Rayleigh Quotient problem is a mathematical problem that involves finding the minimum or maximum value of a ratio of quadratic forms. It is commonly used in optimization and to find the eigenvalues of a symmetric matrix.

3. How are the Sturm-Liouville and Rayleigh Quotient problems related?

The Sturm-Liouville and Rayleigh Quotient problems are closely related because the eigenvalues of the Sturm-Liouville problem can be found by solving the Rayleigh Quotient problem. The Rayleigh Quotient is also used to find the eigenvalues of a symmetric matrix, which is a common form of the Sturm-Liouville problem.

4. What are the applications of the Sturm-Liouville and Rayleigh Quotient problems?

The Sturm-Liouville and Rayleigh Quotient problems have many applications in physics, engineering, and mathematics. They are used to model and analyze physical systems, such as vibrating strings, heat transfer, and quantum mechanics. They are also used in optimization and to find eigenvalues of symmetric matrices.

5. What are some techniques for solving the Sturm-Liouville and Rayleigh Quotient problems?

There are several techniques for solving the Sturm-Liouville and Rayleigh Quotient problems, including the method of separation of variables, the Frobenius method, and the variational method. These techniques involve manipulating the differential equation or quadratic form to solve for the eigenvalues and eigenfunctions. Numerical methods, such as finite difference or finite element methods, can also be used to solve these problems.

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