\documentclass[twoside]{article} \pagestyle{myheadings} \setcounter{page}{125} \markboth{\hfil Initial/final-value problem for Beechem-Haas equations\hfil}% {\hfil Valentino Anthony Simpao \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc 15th Annual Conference on Applied Mathematics, Univ. of Central Oklahoma}, \newline Electronic Journal of Differential Equations, Conference~02, 1999, pp. 125--131. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ Operational WKB solution to the initial/final-value problem for Beechem-Haas equations \thanks{ {\em 1991 Mathematics Subject Classifications:} 92C45, 34E20, 58J35. \hfil\break\indent {\em Key words and phrases:} operational methods, WKB, Analytic solution. \hfil\break\indent \copyright 2000 Southwest Texas State University and University of North Texas. \hfil\break\indent Published January 21, 2000.} } \date{} \author{Valentino Anthony Simpao} \maketitle \begin{abstract} Operational and Wantzel-Kramers-Brillouin (WKB) methods are utilized to obtain an analytical approximate solution of the Beechem-Haas equation for the initial/final-value problem with system parameters of radial profile. The Beechem-Haas equation herein is expanded from its traditional form, which determines its solutions in a spherical coordinate system as functions of the radial and time coordinates only, to one which includes the angular coordinates as well, thus providing a natural modeling framework for polarization phenomena. \end{abstract} \section{Introduction} Consider the Beechem-Haas equation and its initial/boundary conditions, as given in \cite{b1}, \begin{eqnarray} \label{I} &\partial_t\overline{N}(r,t) = \bigg(\sum_{j=1}^n \frac{\alpha_j}{\tau_j} (1+(R_0/r)^6)\bigg) \overline{N}(r,t) + \frac 1{N_0(r)} \partial_r[N_0(r)D(r)\partial_r\overline{N}(r,t)]\,,& \nonumber \\ &\overline{N}(r,t)\bigg|_{t=0} = 1\,,\quad \overline{N}_{t=0}(r,t) = 1\,, &\\ &\partial_r\overline{N}(r,t)\bigg|_{r=r_{\mbox{\scriptsize min}}} = 0\,, \quad \partial_r\overline{N}(r,t)\bigg|_{r=r_{\mbox{\scriptsize max}}} = 0\,, &\nonumber \end{eqnarray} where \$0