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Numerical Evaluation of Integrals and Derivatives

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Spectral Methods in Chemistry and Physics

Part of the book series: Scientific Computation ((SCIENTCOMP))

Abstract

The numerical evaluation of integrals referred to as a quadrature is an important aspect of a large number of applied problems in science and engineering. In Chap. 2, we derived several different methods for the numerical evaluation of integrals. These include the trapezoidal and Simpson’s rules, the higher order Newton-Cotes algorithms, the Clenshaw-Curtis scheme and the Gauss quadrature methods based on classical and nonclassical polynomials. In this chapter, general principles for the accurate and efficient numerical evaluation of integrals that occur in the modeling of physical systems are provided. This is the basis for an efficient numerical method of solution of integral equations discussed in Chap. 5. The physical systems considered vary considerably from section to section and a brief introduction is provided in each case with numerous references to textbooks and current research publications. We consider radial integrals that occur in density functional theory, integrals for chemical and nuclear fusion rate coefficients and also for the solution of the Boltzmann equation. The numerical evaluation of matrix elements in kinetic theory and quantum mechanics is also presented with important implications for pseudospectral methods. The latter section of the chapter is devoted to the pseudospectral method for numerical differentiation based on the Lagrange and Sinc interpolants. The numerical solution of Sturm-Liouville differential eigenvalue problems for the classical polynomials is also presented.

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Notes

  1. 1.

    Niels Henrik David Bohr (1885–1962) was a Danish physicist who made fundamental contributions to quantum theory and in particular to the Bohr model of the hydrogen atom. He received the Nobel Prize in Physics in 1922.

  2. 2.

    Eugene Paul Wigner (1902–1995), was an Hungarian American theoretical physicist and mathematician who was awarded the Nobel Prize in Physics in 1963 for his fundamental work on the quantum mechanics of elementary particles and symmetries.

  3. 3.

    Jesse Ernest Wilkins, Jr. (1923–2011) was an African American nuclear physicist and mathematician who contributed to the Manhattan project and nuclear fission reactions.

  4. 4.

    James Clerk Maxwell (1831–1879) was a Scottish mathematical physicist who made a large number of fundamental contributions to electromagnetic theory, kinetic theory and thermodynamics.

  5. 5.

    Max Born (1882–1970) was a German-British physicist and mathematician who made significant contributions to quantum mechanics, solid-state physics and optics, and won the 1954 Nobel Prize in Physics for the statistical interpretation of wavefunctions.

  6. 6.

    Julius Robert Oppenheimer (1904–1967) was an American theoretical physicist and played a prominent role in the Manhattan Project for which he became known as the “father of the atomic bomb”.

  7. 7.

    John Clarke Slater (1900–1976) was an American physicist who pioneered theoretical methods in atomic and molecular electronic structure.

  8. 8.

    Louis Napoleon George Filon (1875–1937) was an English mathematician and worked in classical mechanics, elasticity and continuous media.

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    Bernard Shizgal

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Shizgal, B. (2015). Numerical Evaluation of Integrals and Derivatives. In: Spectral Methods in Chemistry and Physics. Scientific Computation. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9454-1_3

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