About the Project

spectral methods


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1: Daniel W. Lozier
Then he transferred to NIST (then known as the National Bureau of Standards), where he collaborated for several years with the Building and Fire Research Laboratory developing and applying finite-difference and spectral methods to differential equation models of fire growth. …
2: 18.38 Mathematical Applications
Differential Equations: Spectral Methods
This process has been generalized to spectral methods for solving partial differential equations. …
Quadrature “Extended” to Pseudo-Spectral (DVR) Representations of Operators in One and Many Dimensions
These methods have become known as pseudo-spectral, and are overviewed in Cerjan (1993), and Shizgal (2015). …
3: Bibliography G
  • J. S. Geronimo, O. Bruno, and W. Van Assche (2004) WKB and turning point theory for second-order difference equations. In Spectral Methods for Operators of Mathematical Physics, Oper. Theory Adv. Appl., Vol. 154, pp. 101–138.
  • D. Gottlieb and S. A. Orszag (1977) Numerical Analysis of Spectral Methods: Theory and Applications. Society for Industrial and Applied Mathematics, Philadelphia, PA.
  • B. Guo (1998) Spectral Methods and Their Applications. World Scientific Publishing Co. Inc., River Edge, NJ-Singapore.
  • 4: Bibliography S
  • B. Shizgal (2015) Spectral Methods in Chemistry and Physics. Applications to Kinetic Theory and Quantum Mechanics. Scientific Computation, Springer-Verlag, Dordrecht.
  • 5: Bibliography
  • J. C. Adams and P. N. Swarztrauber (1997) SPHEREPACK 2.0: A Model Development Facility. NCAR Technical Note Technical Report TN-436-STR, National Center for Atmospheric Research.
  • 6: 18.39 Applications in the Physical Sciences
    Shizgal (2015) gives a broad overview of techniques and applications of spectral and pseudo-spectral methods to problems arising in theoretical chemistry, chemical kinetics, transport theory, and astrophysics. …
    7: DLMF Project News
    error generating summary
    8: 31.17 Physical Applications
    §31.17(i) Addition of Three Quantum Spins
    The problem of adding three quantum spins 𝐬 , 𝐭 , and 𝐮 can be solved by the method of separation of variables, and the solution is given in terms of a product of two Heun functions. … Consider the following spectral problem on the sphere S 2 : 𝐱 2 = x s 2 + x t 2 + x u 2 = R 2 . … For more details about the method of separation of variables and relation to special functions see Olevskiĭ (1950), Kalnins et al. (1976), Miller (1977), and Kalnins (1986). … For applications of Heun’s equation and functions in astrophysics see Debosscher (1998) where different spectral problems for Heun’s equation are also considered. …
    9: Brian D. Sleeman
    Sleeman published numerous papers in applied analysis, multiparameter spectral theory, direct and inverse scattering theory, and mathematical medicine. He is author of the book Multiparameter spectral theory in Hilbert space, published by Pitman in 1978, and coauthor (with D. … Sleeman was elected a Fellow of the Royal Society of Edinburgh in 1976 and is the founding editor of the journal Computational and Mathematical Methods in Medicine. …
    10: Errata
    It was decided that much more information should be given in the section on general OP’s, and as a consequence Chapter 1 (Algebraic and Analytic Methods), also required a significant expansion. This especially included updated information on matrix analysis, measure theory, spectral analysis, and a new section on linear second order differential operators and eigenfunction expansions. … We have also completely expanded our discussion on applications of orthogonal polynomials in the physical sciences, and also methods of computation for orthogonal polynomials. … The specific updates to Chapter 1 include the addition of an entirely new subsection §1.18 entitled “Linear Second Order Differential Operators and Eigenfunction Expansions” which is a survey of the formal spectral analysis of second order differential operators. The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions. …