Finding the lowest eigenvalue, Rayleigh-Ritz method, Calculus

[omyojj]

I have a Sturm-Liouville system

[tex] \frac{d}{dx}p(x)\frac{du}{dx} - q(x)u(x)+\lambda \rho(x) u(x) = 0 [/tex]

with

[tex] p(x) = (1-x^2)^{2p} [/tex]

[tex] q(x) = k^2 [/tex]

[tex] \rho(x) = (1-x^2)^{p-1} [/tex]

(p,k^2 are positive real)
u(x) is defined on the interval (-A,A) where 0<A<=1.
Boundary condition that u(x) satisfies is u'(A)=u'(-A)=0
and I want it to be symmetric with respect to zero, i.e., u(x) = u(-x).

I don't think that this equation is solvable in a closed form for general k, p values.

However, I want to find or estimate the lowest eigenvalue λ together with a trial eigenfunction that gives good description of true solution.
So I considered a functional

[tex] K[u(x)] = \dfrac{\int_{-A}^{A} pu^{\prime 2} + qu^2 dx}{\int_{-A}^{A} \rho u^2 dx} [/tex]
or
[tex] K[u(x)] = \dfrac{\int_{-A}^{A} (1-x^2)^{2p}u^{\prime 2} - k^2u^2 dx}{\int_{-A}^{A} (1-x^2)^{p-1} u^2 dx} [/tex]

As a trial function I took

[tex] u(x;\alpha) = 1 - \frac{\alpha}{2}\left( x^2 - \frac{x^4}{2A^2} \right) [/tex]

as an approximation to fourth-order in x.

Note that u(x;a) satisfies the boundary conditions.

According to variational principle, the absolute minimum of K is the lowest eigenvalue λ.

The problem is that I want to find a good trial function,

[tex] \alpha = \alpha(p, k^2) [/tex]

that gives absolute minimum(stationary) of K

My question is

1. How do you think I can find the value of α as a function of p, k that gives minimum of K[u(x)].

I think that procedures like
[tex] \frac{d}{d\alpha} K = 0 [/tex]
is needed.

2. Is it better to do it by differentiating K before performing integration?
When integrated first, Denominator and Numerator are expressed in Gaussian hypergeometric functions. (Confirmed it with the help of Wolfram alpha)

 

[jvicens]

I appreciate your interest in finding the lowest eigenvalue and a good trial function for the Sturm-Liouville system you have described. However, I must mention that it is not possible to find a closed form solution for the general case of p and k values. This is because the Sturm-Liouville system is a second-order differential equation, and it is known that closed form solutions only exist for specific values of p and k.

Regarding your question about finding the value of alpha as a function of p and k, I suggest using numerical methods such as the Newton-Raphson method or the gradient descent method to find the stationary point of K. These methods involve iteratively updating the value of alpha until the derivative of K with respect to alpha becomes zero. However, it is important to note that the stationary point found using these methods may not necessarily be the absolute minimum of K, but it can serve as a good approximation.

Alternatively, you can also try using a series expansion method to find the value of alpha. This involves expanding K in a power series in terms of alpha and then solving for the coefficients that make the derivative of K with respect to alpha equal to zero. This method may give a more accurate result, but it can be computationally intensive.

As for your second question, it is generally better to differentiate K before performing integration. This is because it can simplify the integration process and may also lead to a simpler expression for the derivative. However, if the integration can be done analytically, it may not make a significant difference. I suggest trying both approaches and comparing the results to see which one gives a better estimate of the lowest eigenvalue.

In conclusion, finding a good trial function and the corresponding value of alpha for the Sturm-Liouville system is a challenging task, and it may require a combination of numerical and analytical methods. I hope this helps you in your research, and I wish you all the best in your endeavors.