Hostname: page-component-5c6d5d7d68-tdptf Total loading time: 0 Render date: 2024-08-06T13:56:30.289Z Has data issue: false hasContentIssue false
  • English
  • Français

Some new methods for the numerical integration of ordinary differential equations

Published online by Cambridge University Press:  24 October 2008

L. Fox  and
E. T. Goodwin
L. Fox
Affiliation:
Mathematics DivisionNational Physical LaboratoryTeddington, Middlesex
E. T. Goodwin
Affiliation:
Mathematics DivisionNational Physical LaboratoryTeddington, Middlesex
Get access Rights & Permissions [Opens in a new window]

Extract

The choice of a numerical method for the solution of ordinary differential equations depends on the associated boundary conditions. When all the boundary conditions are specified at one end of the range of integration, one of the well-known step-by-step methods will generally be used, while the method of relaxation is reserved for the case in which boundary conditions are specified at more than one point (1). In the latter, simple but inaccurate finite-difference formulae are used to provide a first approximation to the required solution; this approximation is then used to give an estimate of the errors involved in the use of the inaccurate formulae, and successive corrections are obtained until the full, accurate finite-difference equations are satisfied (1). The same principle is followed in this paper with regard to step-by-step methods, the main difference being the way in which approximate solutions are obtained. In relaxation methods simultaneous equations are solved, while in the use of the step-by-step methods suggested here successive pivotal values are built up by the use of recurrence relations. All the methods of this paper follow this principle, differing only in the method of obtaining a recurrence relation, and consequently in the form of the correction terms.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 45 , Issue 3 , July 1949 , pp. 373 - 388
Copyright
Copyright © Cambridge Philosophical Society 1949

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)Fox, L.The solution by relaxation methods of ordinary differential equations. Proc. Cambridge Phil. Soc. 45 (1949), 50–68.CrossRefGoogle Scholar
(2)Hartree, D. R. and Womersley, J.A method for the numerical or mechanical solution of certain types of partial differential equations. Proc. Roy. Soc. A, 161 (1937), 353–66.Google Scholar
(3)Crank, J. and Nicholson, P.A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. Proc. Cambridge Phil. Soc. 43 (1947), 50–67.CrossRefGoogle Scholar
(4)Hausman, L. F. and Schwarzchild, M.Automatic integration of linear sixth-order differential equations by means of punched-card machines. Rev. Sci. Instrum. 18 (1947), 877–83.CrossRefGoogle Scholar