ISSN:
0020-7608
Keywords:
Computational Chemistry and Molecular Modeling
;
Atomic, Molecular and Optical Physics
Source:
Wiley InterScience Backfile Collection 1832-2000
Topics:
Chemistry and Pharmacology
Notes:
A transformation exists which allows the general Riccati equation \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$$ \begin{array}{*{20}c}{{dy\left( r \right)} \mathord{\left/ {\vphantom {{dy\left( r \right)} {dr = A\left( r \right) + }}} \right. \kern-\nulldelimiterspace} {dr = A\left( r \right) + }}B\left( r \right)y\left( r \right) + C\left( r \right)y\left( r \right)^2 \hfill & 0\leqq r 〈 b \end{array}$$\end{document} to be written in a simpler form: \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$$ d\beta (r)/dr\, = \,P(r)\, + \,R(r)\beta (r)^2 \quad 0\buildrel{〈}\over{=} r 〈 b $$\end{document} The transformed equation has the equivalent nonlinear Hammerstein integral equation \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$$ \begin{array}{*{20}c}\beta (r) = K\int_{r^{\prime} = 0}^b P(r^{\prime}) N(r, r^{\prime})dr^{\prime} \quad 0\buildrel{〈}\over{=} r 〈 b \end{array}$$\end{document} if the kernel N(r, r′) satisfies three conditions: \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$$ \begin{array}{*{20}c} {({\rm i})} & {\{ d/dr - R(r)\beta (r)\} N(r,r)} \\ \end{array}\, = \,\delta (r,r)/K $$\end{document} and \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$$ \begin{array}{*{20}c} {({\rm ii})} & {\{ d/dr'\, + \,R(r')\beta (r')\} N(r,r')} \\ \end{array}\, = \, - \delta (r,r')/K $$\end{document} and \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$$ \begin{array}{*{20}c} {({\rm iii})} & {{\rm [}\beta (r')N(r,r'){\rm ]}_{r' = 0}^b } \\ \end{array} = 0 $$\end{document}A solution of the nonlinear integral equation is devised by repeatedly integrating the Hammerstein equation. During this procedure the kernel generates an equation that contains only coefficients of β(r)0 and β(r)1. As a result, after truncating at the end of the nth cycle, it is a simple matter to write down a Padé-type approximation: all coefficients in this approximation are capable of being evaluated in terms of simple algebraic formulations of P(r), R(r), and integrals over P(r).The zeroes of the denominator of the Padé-type approximation define the points where singularities occur in β(r).
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1002/qua.560080504
Permalink
