Conformal mappings versus other power series methods for solving ordinary differential equations: illustration on anharmonic oscillators

Details Conformal mappings versus other power series methods for solving ordinary differential equations: illustration on anharmonic oscillators Conformal mappings versus other power series methods for solving ordinary differential equations: illustration on anharmonic oscillators

2008-12-11

arxiv.org/abs/0812.2262v1

J. Phys. A: Math. Theor. 42 (2009) 485202

Physics

Mathematical Physics

32 pages, 7 figures, 8 tables

Scientific paper

10.1088/1751-8113/42/48/485202

The simplicity and the efficiency of a quasi-analytical method for solving nonlinear ordinary differential equations (ODE), is illustrated on the study of anharmonic oscillators (AO) with a potential $V(x) =\beta x^{2}+x^{2m}$ ($m>0$). The method [Nucl. Phys. B801, 296 (2008)], applies a priori to any ODE with two-point boundaries (one being located at infinity), the solution of which has singularities in the complex plane of the independent variable $x$. A conformal mapping of a suitably chosen angular sector of the complex plane of $x$ upon the unit disc centered at the origin makes convergent the transformed Taylor series of the generic solution so that the boundary condition at infinity can be easily imposed. In principle, this constraint, when applied on the logarithmic-derivative of the wave function, determines the eigenvalues to an arbitrary level of accuracy. In practice, for $\beta \geq 0$ or slightly negative, the accuracy of the results obtained is astonishingly large with regards to the modest computing power used. It is explained why the efficiency of the method decreases as $\beta $ is more and more negative. Various aspects of the method and comparisons with some seemingly similar methods, based also on expressing the solution as a Taylor series, are shortly reviewed, presented and discussed.

Affiliated with

Also associated with

No associations

Rating

Conformal mappings versus other power series methods for solving ordinary differential equations: illustration on anharmonic oscillators does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Conformal mappings versus other power series methods for solving ordinary differential equations: illustration on anharmonic oscillators, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Conformal mappings versus other power series methods for solving ordinary differential equations: illustration on anharmonic oscillators will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-609205