Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-20T01:03:55.906Z Has data issue: false hasContentIssue false
  • English
  • Français

XL.—The Linear Difference-differential Equation with Constant Coefficients

Published online by Cambridge University Press:  14 February 2012

E. M. Wright
E. M. Wright
Affiliation:
University of Aberdeen
Get access Rights & Permissions [Opens in a new window]

Summary

Under the condition that one at least of the leading coefficients amn, a0n differs from zero, the equation

has as solution a series convergent for all x greater (or all x less) than a fixed number. The coefficients of the various terms in the series are expressed in terms of the arbitrary values of the solution and its first n derivatives in an initial interval of appropriate length.

This paper was assisted in publication by a grant from the Carnegie Trust for the Universities of Scotland.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 62 , Issue 4 , 1949 , pp. 387 - 393
Copyright
Copyright © Royal Society of Edinburgh 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 to Literature

Bochner, S., 1932. Vorlesungen Ueber Fouriersche Integrale, Leipzig.Google Scholar
Callender, A., Hartree, D. R., and Porter, A., 1936. “Time-lag in a control system”, Phil. Trans. Roy. Soc. London, A, ccxxxv, 415–444.Google Scholar
Hartree, D. R., Porter, A., Callender, A., and Stevenson, A. B., 1937. “Time-lag in a control system. II”, Proc Roy. Soc. London, A, CLXI, 460–476.Google Scholar
Hilb, E., 1918. “Zur Theorie der linearen funktionalen Differentialgleichungen”, Math. Ann., LXXVIII, 137–170.Google Scholar
Langer, R. E., 1929. “The asymptotic location of the roots of a certain transcendental equation”, Trans. Anter. Math. Soc., xxxi, 837–844.CrossRefGoogle Scholar
Pitt, H. R., 1944. “On a class of integro-differential equations”, Proc. Camb. Phil. Soc., XL, 199–211.CrossRefGoogle Scholar
Pitt, H. R., 1947. “On a class of linear integro-differential equations”, Proc. Camb. Phil. Soc., XLIII, 153–163.CrossRefGoogle Scholar
Schmidt, E., 1911. “Ueber eine Klasse linearer funktionaler Differentialgleichungen”, Math. Ann., LXX, 499–524.CrossRefGoogle Scholar
Sievert, R. M., 1941. “Zur theoretischer-mathematischen Behandlung des Problems der biologischen Strahlenwirkung”, Acta Radiologica, XXII, 237–251.Google Scholar
Titchmarsh, E. C., 1937. Theory of Fourier Integrals, Oxford.Google Scholar
Titchmarsh, E. C., 1939. “Solutions of some functional equations”, Journ. London Math. Soc., XIV, 118–124.CrossRefGoogle Scholar
Van Der Werff, J. Th., 1942. “Die mathematische Theorie der biologischen Reaktionserscheinungen, besonders nach Roentgenbestrahlung”, Acta Radiologica, XXIII, 603–621.Google Scholar
Widder, D. V., 1941. The Laplace Transform, Princeton.Google Scholar
Wilder, C. E., 1917. “Expansion problems of ordinary linear differential equations, etc.”, Trans. Amer. Math. Soc., XVIII, 415–442.CrossRefGoogle Scholar
Wright, E. M.;, 1948a. “Linear difference-differential equations”, Proc. Camb. Phil. Soc., XLIV, 179–185.CrossRefGoogle Scholar
Wright, E. M.;, 1948b. “The linear difference-differential equation with asymptotically constant coefficients”, Amer. Journ. Math., LXX, 221–238.CrossRefGoogle Scholar