A spectral method for the solution of integral and differential equations is generally understood to be an expansion of the solution in a Fourier series. Chebyshev polynomials are often the preferred basis set for many problems. In kinetic theory, the Sonine polynomials have been used for decades for the calculation of transport coefficients. This talk will focus on the use of nonclassical polynomials orthonormal with respect an appropriate weight function chosen dependent on the problem considered. The associated quadrature rules are also used in the pseudospectral solution of several different problems in kinetic theory and quantum mechanics. The recurrence coefficients in the three term recurrence relation for the nonclassical polynomials are determined numerically with the Gautschi-Stieltjes procedure and define the Jacobi matrix, J. The quadrature points are eigenvalues of J and the weights are the first component of the ith eigenfunction. This methodology is applied to the solution of the Fokker-Planck equation (1), the Schroedinger equation (2), the evaluation of integrals in quantum chemistry (3) and for nuclear reaction rate coefficients (4).
(1) Pseudospectral solution of the Fokker-Planck equation: the eigenvalue spectrum and the approach to equilibirum. J. Stat. Phys. 164, 1379-1393 (2016).
(2) Pseudospectral method of solution of the Schroedinger equation with nonclassical polynomials; the Morse and Poschl-Teller (SUSY) potentials. J. Comput. Theor. Chem. 1084, 51-58 (2016).
(3) A novel Rys quadrature algorithm for use in the calculation of electron repulsion integrals. J. Comput. Theor. Chem. 1074, 178-184 (2015).
(4) An efficient nonclassical quadrature for the calculation of nonresonant nuclear fusion reaction rate coefficients from cross section data. Comp. Phys. Comm. 205, 61-69 (2016).