@article {4131,
	title = {The quadrature discretization method in the solution of the Fokker-Planck equation with nonclassical basis functions},
	journal = {Journal of Chemical Physics},
	volume = {107},
	number = {19},
	year = {1997},
	note = {ISI Document Delivery No.: YG655Times Cited: 12Cited Reference Count: 122},
	month = {Nov},
	pages = {8051-8063},
	type = {Review},
	abstract = {Fokker-Planck equations are used extensively to study a variety of problems in nonequilibrium statistical mechanics. A discretization method referred to as the quadrature discretization method (QDM) is introduced for the time-dependent solution of Fokker-Planck equations. The QDM is based on the discretization of the probability density function on a grid of points that coincide with the points of a quadrature. The quadrature is based on a set of nonclassical polynomials orthogonal with respect to some weight function. For the Fokker-Planck equation, the weight functions that have often provided rapid convergence of the eigenvalues of the Fokker-Planck operator are the steady distributions at infinite time. Calculations are carried out for several systems with bistable potentials that arise in the study of optical bistability. reactive systems and climate models. The rate of convergence of the eigenvalues and the eigenfunctions of the Fokker-Planck equation is very rapid with this approach. The time evolution is determined in terms of the expansion of the distribution function in the eigenfunctions. (C) 1997 American Institute of Physics.},
	keywords = {CLIMATIC TRANSITIONS, COLORED NOISE, DISCRETE-ORDINATE METHOD, INITIAL-VALUE PROBLEM, ONE-DIMENSIONAL, OPTICAL BISTABILITY, ORTHOGONAL POLYNOMIALS, SCHRODINGER-EQUATION, STOCHASTIC RESONANCE, SYSTEMS, TIME-DEPENDENT NUCLEATION},
	isbn = {0021-9606},
	url = {<Go to ISI>://A1997YG65500053},
	author = {Shizgal, B. D. and Chen, H. L.}
}
@article {3811,
	title = {The quadrature discretization method (QDM) in the solution of the Schrodinger equation with nonclassical basis functions},
	journal = {Journal of Chemical Physics},
	volume = {104},
	number = {11},
	year = {1996},
	note = {ISI Document Delivery No.: TY728Times Cited: 34Cited Reference Count: 66},
	month = {Mar},
	pages = {4137-4150},
	type = {Article},
	abstract = {A discretization method referred to as the Quadrature Discretization Method (QDM) is introduced for the solution of the Schrodinger equation. The method has been used previously for the solution of Fokker-Planck equations. The Fokker-Planck equation can be transformed to a Schrodinger equation with a potential of the form that occurs in supersymmetric quantum mechanics, For this class of potentials, the groundstate wave function is known. The QDM is based on the discretization of the wave function on a grid of points that coincide with the points of a quadrature. The quadrature is based on a set of nonclassical polynomials orthogonal with respect to a weight function determined by the potential function in the Schrodinger equation. For the Fokker-Planck operator, the weight function that provides rapid convergence of the eigenvalues are the steady distributions at infinite time, that is, the ground state wave functions. In the present paper, the weight functions used in an analogous solution of the Schrodinger equation are related to the ground state wave functions if known, or some approximate form. Calculations are carried out for a model systems, the Morse potential, and for the vibrational levels of O-2 and Ar-Xe with realistic pair potentials. For O-2, the wave functions are used to calculate the vibrationally inelastic transition amplitudes for a Morse potential and compared with exact analytic results. The eigenvalues of a two-dimensional Schrodinger equation with the Henon-Heiles potential are also calculated. The rate of convergence of the eigenvalues and the eigenfunctions of the Schrodinger equation is very rapid with this approach. (C) 1996 American Institute of Physics.},
	keywords = {BOUND-STATES, COEFFICIENTS, DISCRETE-ORDINATE METHOD, EIGENVALUE PROBLEMS, FOKKER-PLANCK EQUATION, ORTHOGONAL POLYNOMIALS, POTENTIALS, QUANTUM-MECHANICS, RECURRENCE, ROVIBRATIONAL STATES},
	isbn = {0021-9606},
	url = {<Go to ISI>://A1996TY72800030},
	author = {Shizgal, B. D. and Chen, H.}
}
@article {2760,
	title = {ON THE GENERATION OF ORTHOGONAL POLYNOMIALS USING ASYMPTOTIC METHODS FOR RECURRENCE COEFFICIENTS},
	journal = {Journal of Computational Physics},
	volume = {104},
	number = {1},
	year = {1993},
	note = {ISI Document Delivery No.: KF519Times Cited: 7Cited Reference Count: 29},
	month = {Jan},
	pages = {140-149},
	type = {Article},
	keywords = {APPROXIMATION, DISCRETE-ORDINATE METHOD, equations, EXPANSIONS, EXPONENTIAL WEIGHTS},
	isbn = {0021-9991},
	url = {<Go to ISI>://A1993KF51900016},
	author = {Clarke, A. S. and Shizgal, B.}
}
@article {7109,
	title = {COMPARISON OF WKB (WENTZEL-KRAMERS-BRILLOUIN) AND SWKB SOLUTIONS OF FOKKER-PLANCK EQUATIONS WITH EXACT RESULTS - APPLICATION TO ELECTRON THERMALIZATION},
	journal = {Canadian Journal of Physics},
	volume = {69},
	number = {6},
	year = {1991},
	note = {ISI Document Delivery No.: FZ294Times Cited: 5Cited Reference Count: 41},
	month = {Jun},
	pages = {712-719},
	type = {Article},
	abstract = {A comparison of WKB (Wentzel-Kramers-Brillouin) and SWKB eigenfunctions of the Schrodinger equation for potentials in the class encountered in supersymmetric quantum mechanics is presented. The potentials that are studied are those that result from the transformation of a Fokker-Planck eigenvalue problem to a Schrodinger equation. Linear Fokker-Planck equations of the type considered in this paper give the probability distribution function for a large number of physical situations. The time-dependent solutions can be expressed as a sum of exponential terms with each term characterized by an eigenvalue of the Fokker-Planck operator. The specific Fokker-Planck operator considered is the one that describes the thermalization of electrons in the inert gases. The WKB and SWKB semiclassical approximations are compared with exact numerical results. Although the eigenvalues can be very close to the exact values, we find significant departures for the eigenfunctions.},
	keywords = {ACTIVATED, APPROXIMATION, DISCRETE-ORDINATE METHOD, EIGENVALUES, INVARIANCE, ISOMERIZATION, NUCLEATION, RARE-GAS MODERATORS, RATE-PROCESSES, SHAPE, SOLVABLE POTENTIALS, SUPERSYMMETRIC QUANTUM-MECHANICS},
	isbn = {0008-4204},
	url = {<Go to ISI>://A1991FZ29400011},
	author = {Shizgal, B. and Demeio, L.}
}
