Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-05-14T14:21:29.787Z Has data issue: false hasContentIssue false
  • English
  • Français

The numerical solution of linear differential equations in Chebyshev series

Published online by Cambridge University Press:  24 October 2008

C. W. Clenshaw
C. W. Clenshaw
Affiliation:
National Physical Laboratory Teddington, Middlesex
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper describes a method for computing the coefficients in the Chebyshev expansion of a solution of an ordinary linear differential equation. The method is valid when the solution required is bounded and possesses a finite number of maxima and minima in the finite range of integration. The essence of the method is that an expansion in Chebyshev polynomials is assumed for the highest derivative occurring in the equation; the coefficients are then determined by integrating this series, substituting in the original equation and equating coefficients.

Comparison is made with the Fourier series method of Dennis and Foots, and with the polynomial approximation method of Lanczos. Examples are given of the application of the method to some first and second order equations, including one eigenvalue problem.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 53 , Issue 1 , January 1957 , pp. 134 - 149
Copyright
Copyright © Cambridge Philosophical Society 1957

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Clenshaw, C. W.A note on the summation of Chebyshev series. Math. Tab., Wash., 9 (1955), 118.Google Scholar
(2)Clenshaw, C. W. and Olver, F. W. J.Solution of differential equations by recurrence relations. Math. Tab., Wash., 5 (1951), 34–9.Google Scholar
(3)Dennis, S. C. R. and Poots, G.The solution of linear differential equations. Proc. Comb. Phil. Soc. 51 (1955), 422–32.CrossRefGoogle Scholar
(4)Fox, L.The solution by relaxation methods of ordinary differential equations. Proc. Camb. Phil. Soc. 45 (1949), 50–68.CrossRefGoogle Scholar
(5)Fox, L. and Goodwin, E. T.Some new methods for the numerical integration of ordinary differential equations. Proc. Camb. Phil. Soc. 45 (1949), 373–88.Google Scholar
(6)Fox, L. and Robertson, H. H. The numerical solution of ordinary differential equations. Proceedings of N.P.L. Symposium on automatic digital computation (H.M.S.O., London, 1954), pp. 137–47.Google Scholar
(7)Lanczos, C.Trigonometric interpolation of empirical and analytical functions. J. Math. Phys. 17 (1938), 123–99.CrossRefGoogle Scholar
(8) National Bureau of Standards Appl. Math. Series No. 9. Tables of Chebyshev Polynomials (Washington: Government Printing Office, 1952).Google Scholar