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: 18.39 Physical Applications
§18.39(i) Quantum Mechanics
3: 18.38 Mathematical Applications
This process has been generalized to spectral methods for solving partial differential equations. …
4: 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.
  • 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: 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. …
    7: 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. …
    8: Bibliography M
  • A. J. MacLeod (1996b) Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.
  • P. L. Marston (1992) Geometrical and Catastrophe Optics Methods in Scattering. In Physical Acoustics, A. D. Pierce and R. N. Thurston (Eds.), Vol. 21, pp. 1–234.
  • J. M. McNamee (2007) Numerical Methods for Roots of Polynomials. Part I. Studies in Computational Mathematics, Vol. 14, Elsevier, Amsterdam.
  • V. Meden and K. Schönhammer (1992) Spectral functions for the Tomonaga-Luttinger model. Phys. Rev. B 46 (24), pp. 15753–15760.
  • S. L. B. Moshier (1989) Methods and Programs for Mathematical Functions. Ellis Horwood Ltd., Chichester.
  • 9: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
    A survey is given of the formal spectral theory of second order differential operators, typical results being presented in §1.18(i) through §1.18(viii). The various types of spectra and the corresponding eigenfunction expansions are illustrated by examples. These are based on the Liouville normal form of (1.13.29). A more precise mathematical discussion then follows in §1.18(ix). The operator T is called self-adjoint if T * = T , and referred to as symmetric if (1.18.23) holds for v , w in the dense domain 𝒟 ( T ) of T . There is also a notion of self-adjointness for unbounded operators, see §1.18(ix). One then needs a self-adjoint extension of a symmetric operator to carry out its spectral theory in a mathematically rigorous manner.Spectral expansions of T , and of functions F ( T ) of T , these being expansions of T and F ( T ) in terms of the eigenvalues and eigenfunctions summed over the spectrum, then follow:
    Spectral expansions and self-adjoint extensions
    The above results, especially the discussions of deficiency indices and limit point and limit circle boundary conditions, lay the basis for further applications. The reader is referred to Coddington and Levinson (1955), Friedman (1990, Ch. 3), Titchmarsh (1962a), and Everitt (2005b, pp. 45–74) and Everitt (2005a, pp. 272–331), for detailed methods and results.
    10: Bibliography K
  • D. K. Kahaner, C. Moler, and S. Nash (1989) Numerical Methods and Software. Prentice Hall, Englewood Cliffs, N.J..
  • J. P. Keating (1999) Periodic Orbits, Spectral Statistics, and the Riemann Zeros. In Supersymmetry and Trace Formulae: Chaos and Disorder, J. P. Keating, D. E. Khmelnitskii, and I. V. Lerner (Eds.), pp. 1–15.
  • M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
  • M. K. Kerimov (1999) The Rayleigh function: Theory and computational methods. Zh. Vychisl. Mat. Mat. Fiz. 39 (12), pp. 1962–2006.
  • A. D. Kerr (1978) An indirect method for evaluating certain infinite integrals. Z. Angew. Math. Phys. 29 (3), pp. 380–386.