how can i solve a boundary value numerically on an infinite interval ?? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T14:31:15Z http://mathoverflow.net/feeds/question/103926 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103926/how-can-i-solve-a-boundary-value-numerically-on-an-infinite-interval how can i solve a boundary value numerically on an infinite interval ?? unknown (yahoo) 2012-08-04T09:36:31Z 2012-08-05T21:08:08Z <p>let be the differential equation $-y''(x)+x^{4}y(x)-E_{n}y(x)=0$ with the boundary conditions $y(0)=0=y(\infty)$</p> <p>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 $y(0)=0=y(R)$</p> <p>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 $u=1$</p> http://mathoverflow.net/questions/103926/how-can-i-solve-a-boundary-value-numerically-on-an-infinite-interval/103931#103931 Answer by Igor Khavkine for how can i solve a boundary value numerically on an infinite interval ?? Igor Khavkine 2012-08-04T10:37:36Z 2012-08-04T10:37:36Z <p>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.</p> <p>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 <a href="http://www.math.niu.edu/SL2/" rel="nofollow">SLEIGN2</a> package and its documentation.</p> http://mathoverflow.net/questions/103926/how-can-i-solve-a-boundary-value-numerically-on-an-infinite-interval/104053#104053 Answer by Alexandre Eremenko for how can i solve a boundary value numerically on an infinite interval ?? Alexandre Eremenko 2012-08-05T21:08:08Z 2012-08-05T21:08:08Z <p>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.</p>