MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

let be the differential equation $ -y''(x)+x^{4}y(x)-E_{n}y(x)=0 $ with the boundary conditions $ y(0)=0=y(\infty) $

how could i use the shooting method or other numerical method to solve this equation ? , my only idea is to set $ R=10000 $ for example and solve $ y(0)=0=y(R) $

of course i also could made the substitution $ u= \frac{x}{x-1} $ so the new boundary conditions could become $ y(0)=0=y(1) $ but know the differential equation would be singular at the point $ u=1 $

share|cite|improve this question
up vote 2 down vote accepted

You're facing a singular Sturm-Liouville eigenvalue problem. One way you may be able to use the shooting method is to get an asymptotic expansion of your solution at the $x=\infty$ (Frobenius or WKB methods should work). Using the asymptotics to evaluate $y(x)$ and $y'(x)$ at some finite $x$ and then shoot from there to $x=0$ to find the eigenvalue.

On the other hand, there exists a fairly well developed numerical library, based also on a sound mathematical analysis of the Sturm-Liouville problem, that is already available. It should be able to handle your problem directly. Look up the SLEIGN2 package and its documentation.

share|cite|improve this answer

See the recent preprints of Andre Voros on the arXiv. He proposed an iterative algorithm, and Artur Avila recently proved convergence. The literature on this problem is enormous; it usually goes under the title "anharmonic oscillator" or "quartic oscillator". The rate of convergence of Voros algorithm is geometric.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.