Numerical Solution of Linear Ordinary Differential Equations in Quantum Chemistry by Clenshaw Method

Authors: M. Saravi, F. Ashrafi, S.R. Mirrajei

Abstract:

As we know, most differential equations concerning physical phenomenon could not be solved by analytical method. Even if we use Series Method, some times we need an appropriate change of variable, and even when we can, their closed form solution may be so complicated that using it to obtain an image or to examine the structure of the system is impossible. For example, if we consider Schrodinger equation, i.e., We come to a three-term recursion relations, which work with it takes, at least, a little bit time to get a series solution[6]. For this reason we use a change of variable such as or when we consider the orbital angular momentum[1], it will be necessary to solve. As we can observe, working with this equation is tedious. In this paper, after introducing Clenshaw method, which is a kind of Spectral method, we try to solve some of such equations.

Keywords: Chebyshev polynomials, Clenshaw method, ODEs, Spectral methods

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1075994

References:

[1] C. Canuto, M. Hussaini, A. Quarteroni, T. Zang, Spectral Methods in Fluid Dynamics, Springer, Berlin, 1988.
[2] C. Lanczos, Trigonometric interpolation of empirical and analytical functions, J. Math. Phys. 17 (1938) 123-129.
[3] D. Gottlieb, S. Orszag, Numerical Analysis of Spectral Methods, Theory and Applications, SIAM, Philadelphia, PA, 1977.
[4] L. M. Delves and J. L. Mohamed, Computational methods for integral equations, Cambridge University Press, 1985.
[5] E. Babolian and L .M. Delves, A fast Galerkin scheme for linear integro-differential equations, IMAJ. Numer. Anal, Vol.1, pp. 193-213,1981.
[6] Ira N. Levine, Quantum Chemistry, Fifth Edition ,City University of New York Prentice-Hall Publications.
[7] E. Babolian, M. Bromilow, R. England, M. Saravi, ÔÇÿA modification of pseudo-spectral method for solving linear ODEs with singularity-, AMC 188 (2007) 1260-1266.