Spectral Methods in Chemistry and Physics

1. Front Matter

   Pages i-xvii

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2. Introduction to Spectral/Pseudospectral Methods

   • Bernard Shizgal

   Pages 1-27

3. Polynomial Basis Functions and Quadratures

   • Bernard Shizgal

   Pages 29-107

4. Numerical Evaluation of Integrals and Derivatives

   • Bernard Shizgal

   Pages 109-186

5. Representation of Functions in Basis Sets

   • Bernard Shizgal

   Pages 187-246

6. Integral Equations in the Kinetic Theory of Gases and Related Topics

   • Bernard Shizgal

   Pages 247-330

7. Spectral and Pseudospectral Methods of Solution of the Fokker-Planck and Schrödinger Equations

   • Bernard Shizgal

   Pages 331-410

8. Back Matter

   Pages 411-415

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This book is a pedagogical presentation of the application of spectral and pseudospectral methods to kinetic theory and quantum mechanics. There are additional applications to astrophysics, engineering, biology and many other fields. The main objective of this book is to provide the basic concepts to enable the use of spectral and pseudospectral methods to solve problems in diverse fields of interest and to a wide audience. While spectral methods are generally based on Fourier Series or Chebychev polynomials, non-classical polynomials and associated quadratures are used for many of the applications presented in the book. Fourier series methods are summarized with a discussion of the resolution of the Gibbs phenomenon. Classical and non-classical quadratures are used for the evaluation of integrals in reaction dynamics including nuclear fusion, radial integrals in density functional theory, in elastic scattering theory and other applications. The subject matter includes the calculation of transport coefficients in gases and other gas dynamical problems based on spectral and pseudospectral solutions of the Boltzmann equation. Radiative transfer in astrophysics and atmospheric science, and applications to space physics are discussed.  The relaxation of initial non-equilibrium distributions to equilibrium for several different systems is studied with the Boltzmann and Fokker-Planck equations.
The eigenvalue spectra of the linear operators in the Boltzmann, Fokker-Planck and Schrödinger equations are studied with spectral and pseudospectral methods based on non-classical orthogonal polynomials.
The numerical methods referred to as the Discrete Ordinate Method, Differential Quadrature, the Quadrature Discretization Method, the Discrete Variable Representation, the Lagrange Mesh Method, and others are discussed and compared.
MATLAB codes are provided for most of the numerical results reported in the book - see Link under 'Additional Information' on the the right-hand column.

• Eigenvalue problems
• Gibbs phenomenon
• Pseudospectral methods explained
• Quantum chemistry book
• Quantum mechanics
• Spectral quadratures
• Statistical mechanics

“I recently had the pleasure of looking at Shizgal’s book "Spectral Methods in Chemistry and Physics" and was pleased to see the vast coverage given to the use of classical and nonclassical orthogonal polynomials in chemistry and physics.” (Prof. emer. Walter Gautschi, Purdue University, West Lafayette)



“This is a masterful book. Eigenvalue methods have been a backbone of applied mathematics for a very long time.  In this book Shizgal simultaneously honors the past - with superb treatments of classical topics -- and looks towards the future showing how spectral methods are important and relevant in many branches of science.  This is the kind of book that one will give to students (I have given it to mine), recommend to colleagues, and continue to open for both specific advice on particular problems and for general understanding and appreciation of the methods.” (Prof. Marc Mangel, University of California Santa Cruz)

• University of British Columbia, Vancouver, Canada

  Bernard Shizgal

Bernard Shizgal was born in 1942 in Montreal, Canada.  He received a B.Sc. in Honours Chemistry and Physics from McGill University in 1964, and a Ph.D. in Chemical Physics from Columbia University in 1968 under the supervision of Martin Karplus. He was a postdoctoral fellow in Physics at Leiden University in the Netherlands with Peter Mazur, and in Chemistry at the University of British Columbia with Bob Snider before his appointment there as an Assistant Professor in 1970. His interests in kinetic theory naturally lead to research projects in space physics and the implementation of pseudospectral methods.  On numerous occasions he was a Fellow of the Japanese Society for the Promotion of Science and a visiting Professor at the Japan Aerospace Exploration Agency, the National Institute of Information and Communications Technology, the Solar Terrestrial Environment Laboratory of Nagoya University and other institutes in Japan. His interests in mathematics and spectral methods were pursued as a visiting Professor in the Department of Mathematics, at the Université de Nice Sophia-Antipolis  and at the Institut Nonlinèare de Nice,  France.  Professor Shizgal has approximately 140 publications in peer-reviewed journals. He is currently Professor Emeritus at the University of British Columbia and remains active with research projects in kinetic theory and quantum mechanics, with applications to atmospheric and space science, stochastic processes, rarefied gas dynamics and pseudospectral methods.

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