Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-12T22:27:20.876Z Has data issue: false hasContentIssue false
  • English
  • Français

A distributional theory of fractional calculus

Published online by Cambridge University Press:  14 November 2011

W. Lamb
W. Lamb
Affiliation:
Department of Mathematics, University of Strathclyde, Glasgow
Get access Rights & Permissions [Opens in a new window]

Synopsis

In this paper, a theory of fractional calculus is developed for certain spaces D′p,μ of generalised functions. The theory is based on the construction of fractionalpowers of certain simple differential and integral operators. With the parameter μ suitably restricted, these fractional powers are shown to coincide with the Riemann-Liouville and Weyl operators of fractional integration and differentiation. Standard properties associated with fractional integrals and derivatives follow immediately from results obtained previously by the author on fractional powers of operators; see [6], [7]. Some spectral properties are also obtained.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 99 , Issue 3-4 , 1985 , pp. 347 - 357
Copyright
Copyright © Royal Society of Edinburgh 1985

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

1Buschman, R. G.. Decomposition of an integral operator by use of Mikusinski's calculus. SIAM J. Math. Anal. 3 (1972), 83–85.CrossRefGoogle Scholar
2Erdélyi, A. et al. Higher transcendental functions (vol. 1) (New York: McGraw-Hill, 1953).Google Scholar
3Gel'fand, I. M. and Shilov, G. E.. Generalised functions (vol. 1) (New York: Academic Press, 1964).Google Scholar
4Hille, E. and Phillips, R. S.. Functional analysis and semigroups, (rev. edn.). (Colloq. Publ. Amer. Math. Soc, Providence, R.I., 1957).Google Scholar
5Hövel, H. W. and Westphal, U.. Fractional powers of closed operators. Studia Math. 42 (1972), 177–194.CrossRefGoogle Scholar
6Lamb, W.. Fractional powers of operators on Fréchet spaces with applications (Strathclyde Univ. Ph.D. Thesis, 1980).Google Scholar
7Lamb, W.. Fractional powers of operators defined on a Fréchet space. Proc. Edinburgh Math. Soc. 27 (1984), 165–181.CrossRefGoogle Scholar
8Love, E. R.. Two index laws for fractional integrals and derivatives. J. Austral. Math. Soc. 14 (1972), 385–410.CrossRefGoogle Scholar
9McBride, A. C.. Fractional calculus and integral transforms of generalized functions. Research Notes in Mathematics 31 (London: Pitman, 1979).Google Scholar
10Okikiolu, G. O.. Aspects of the theory of bounded integral operators in Lp-spaces (London: Academic Press, 1971).Google Scholar
11Prabhakar, T. R.. Two singular integral equations involving confluent hypergeometric functions. Proc. Cambridge Philos. Soc. 66 (1969), 71–89.CrossRefGoogle Scholar
12Ross, B. (Ed.). Fractional calculus and its applications. Lecture Notes in Mathematics 457 (Berlin: Springer, 1975).Google Scholar
13Schwartz, L.. Théorie des distributions (Paris: Hermann, 1966).Google Scholar
14Zemanian, A. H.. Generalized integral transformations (New York: Interscience, 1968).Google Scholar