Skip to main content
SpringerLink
Log in

The use in mathematical physics of Erdélyi-Kober operators and of some of their generalizations

  • Chapter
  • First Online:
Fractional Calculus and Its Applications

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 457))

Abstract

A brief survey of the properties of the Erdélyi-Kober operators of fractional integration and their generalizations (due to Cooke, Lowndes and others) is given. The use of these operators to solve the dual (and triple) integral equations, to which mixed boundary value problems of mathematical physics may be reduced, is illustrated by reference to specific problems. The emphasis throughout is on those results of the fractional calculus which arise frequently in applications but some indication is also given of possible theoretical developments.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ahmad, M.I., "Quadruple integral equations and operators of fractional integration," Glasgow Math. J. 12 (1971), 60.

    Article  MathSciNet  MATH  Google Scholar 

  2. Belward, J.A., "Solutions of some Fredholm integral equations using fractional integration with an application to a forced convection problem," ZAMP 23 (1972), 901.

    Article  MathSciNet  MATH  Google Scholar 

  3. Burlak, J., "A pair of dual integral equations occurring in diffraction theory," Proc. Edinburgh Math. Soc. (2), 13 (1962), 179.

    Article  MathSciNet  MATH  Google Scholar 

  4. Buschman, R.G., "Fractional integration," Math. Japon 9 (1964), 99.

    MathSciNet  MATH  Google Scholar 

  5. Cooke, J.C., "Triple integral equations," Quart. J. Mech. Appl. Math. 16 (1963), 193.

    Article  MathSciNet  MATH  Google Scholar 

  6. Cooke, J.C., "Some further triple integral equations," Proc. Edinburgh Math. Soc. 13 (1963), 303.

    Article  MathSciNet  MATH  Google Scholar 

  7. Cooke, J.C., "The solution of triple integral equations in operational form," Quart. J. Mech. Appl. Math. 18 (1965), 57.

    Article  MathSciNet  MATH  Google Scholar 

  8. Cooke, J.C., "The solution of triple and quadruple integral equations and Fourier-Bessel series," Quart. J. Mech. Appl. Math. 25 (1972), 247.

    Article  MathSciNet  MATH  Google Scholar 

  9. Copson, E.T., "On certain dual integral equations," Proc. Edinburgh Math. Assoc. 5 (1961), 21.

    Article  MathSciNet  MATH  Google Scholar 

  10. Erdélyi, A., "On fractional integration and its application to the theory of Hankel transforms," Quart. J. Math. (Oxford) 11 (1940), 293.

    Article  MathSciNet  MATH  Google Scholar 

  11. Erdélyi, A., "An integral equation involving Legendre functions," J. SIAM 12 (1964), 15.

    MathSciNet  MATH  Google Scholar 

  12. Erdélyi, A., "Axially symmetric potentials and fractional integration," J. SIAM 13 (1965), 216.

    MathSciNet  MATH  Google Scholar 

  13. Erdélyi, A., "An application of fractional integrals," J. Analyse Math. 14 (1965), 113.

    Article  MathSciNet  MATH  Google Scholar 

  14. Erdélyi, A., "Some dual integral equations," SIAM J. Appl. Math. 16 (1968), 1338.

    Article  MathSciNet  MATH  Google Scholar 

  15. Erdélyi, A. and Kober, H., "Some remarks on Hankel transforms," Quart. J. Math. (Oxford), 11 (1940), 212.

    Article  MathSciNet  MATH  Google Scholar 

  16. Erdélyi, A. and Sneddon, I.N., "Fractional integration and dual integration equations," Canad. J. Math. 14 (1962), 685.

    Article  MathSciNet  MATH  Google Scholar 

  17. Gordon, A.N., "Dual integral equations," J. London Math. Soc. 29 (1954), 360.

    Article  MathSciNet  MATH  Google Scholar 

  18. Hardy, G.H. and Littlewood, J.E., "Some properties of fractional integrals: I," Math. Z. 27 (1928), 565.

    Article  MathSciNet  MATH  Google Scholar 

  19. Kesarwani, R.N., "Fractional integration and certain dual integral equations," Math. Z. 98 (1967), 83.

    Article  MathSciNet  MATH  Google Scholar 

  20. Kober, H., "On fractional integrals and derivatives," Quart. J. Math. (Oxford) 11 (1940), 193.

    Article  MathSciNet  MATH  Google Scholar 

  21. Kober, H., "On a theorem of Schur and on fractional integrals of purely imaginary order," Trans. Amer. Math. Soc. 50 (1941), 160.

    MathSciNet  MATH  Google Scholar 

  22. Kober, H., "On Dirichlet's singular integral and Fourier transforms," Quart. J. Math. (Oxford) 12 (1941), 78.

    Article  MathSciNet  MATH  Google Scholar 

  23. Linz, P., "The numerical solution of certain dual integral equations," J. Engng. Math. 3 (1969), 245.

    Article  MathSciNet  MATH  Google Scholar 

  24. Love, E.R., "The electrostatic field of two equal circular coaxial conducting disks," Quart. J. Mech. Appl. Math. 2 (1949), 428.

    Article  MathSciNet  MATH  Google Scholar 

  25. Love, E.R., "Some integral equations involving hypergeometric functions," Proc. Edinburgh Math. Soc. 15 (1967), 169.

    Article  MathSciNet  MATH  Google Scholar 

  26. Love, E.R. and Young, L.C., "On fractional integration by parts," Proc. London Math. Soc. (2) 44 (1938), 1.

    Article  MathSciNet  MATH  Google Scholar 

  27. Lowengrub, M., "The solution of certain simultaneous pairs of dual integral equations," Glasgow Math. J. 9 (1968), 92.

    Article  MathSciNet  MATH  Google Scholar 

  28. Lowengrub, M. and Sneddon, I.N., "An axisymmetric boundary value problem of mixed type for a half-space," Proc. Edinburgh Math. Soc. 13 (1962), 39.

    Article  MathSciNet  MATH  Google Scholar 

  29. Lowengrub, M. and Sneddon, I.N., "The solution of a pair of dual integral equations," Proc. Glasgow Math. Assoc. 6 (1963), 14.

    Article  MathSciNet  MATH  Google Scholar 

  30. Lowndes, J.S., "Simultaneous dual integral equations," Glasgow Math. J. 10 (1969), 73.

    Article  MathSciNet  MATH  Google Scholar 

  31. Lowndes, J.S., "A generalization of the Erdélyi-Kober operators," Proc. Edinburgh Math. Soc. (2) 17 (1970), 139.

    Article  MathSciNet  MATH  Google Scholar 

  32. Lowndes, J.S., "Some triple integral equations," Pacific J. Math. 38 (1971), 515.

    Article  MathSciNet  MATH  Google Scholar 

  33. Noble, B., "Certain dual integral equations," J. Math. Phys. 37, (1958), 128.

    Article  MathSciNet  MATH  Google Scholar 

  34. Peters, A.S., "Certain dual integral equations and Sonine's integrals," N.Y.U. Inst. of Math. Sci. Report No. 285 (1961).

    Google Scholar 

  35. Riesz, M., "L'intégrals de Riemann-Liouville et le problème de Cauchy," Acta Math. 81 (1949), 1.

    Article  MathSciNet  MATH  Google Scholar 

  36. Rooney, P.G., "On the ranges of certain fractional integrals," Canad. J. Math. 24 (1972), 1198.

    Article  MathSciNet  MATH  Google Scholar 

  37. Sneddon, I.N., "The elementary solution of dual integral equations," Proc. Glasgow Math. Assoc. 4 (1960), 108.

    Article  MathSciNet  MATH  Google Scholar 

  38. Sneddon, I.N., "A relation involving Hankel transforms with applications to boundary value problems in potential theory," J. Math. Mech. 14 (1965), 33.

    MathSciNet  MATH  Google Scholar 

  39. Sneddon, I.N., Mixed Boundary Value Problems in Potential Theory, (North Holland Publ. Co., Amsterdam, 1966).

    MATH  Google Scholar 

  40. Srivastav, R.P., "On certain integral equations of convolution type with Bessel function kernels," Proc. Edinburgh Math. Soc. (2), 15 (1955), 111.

    Article  MathSciNet  MATH  Google Scholar 

  41. Srivastav, R.P. and Parihar, K.S., "Dual and triple integral equations involving inverse Mellin transforms," SIAM J. Appl. Math. 16 (1968), 126.

    Article  MathSciNet  MATH  Google Scholar 

  42. Walton, J.R., "A distributional approach to dual integral equations of Titchmarsh type," Ph.D. Thesis, Indiana University, 1973.

    Google Scholar 

  43. Watson, G.N., A Treatise on the Theory of Bessel Functions, 2nd edn., Cambridge University Press, 1944.

    Google Scholar 

  44. Weinstein, A., "Generalized axially symmetric potential theory," Bull. Amer. Math. Soc. 59 (1953), 20.

    Article  MathSciNet  MATH  Google Scholar 

  45. Weyl, H., "Bemerkungen zum Begriff des Differentialquotienten gebrochener Ordnung," Vier. Mat. Ges. Zurich 62 (1917), 296.

    MATH  Google Scholar 

  46. Williams, W.E., "The solution of certain dual integral equations," Proc. Edinburgh Math. Soc. 12 (1961), 213.

    Article  MathSciNet  MATH  Google Scholar 

  47. Williams, W.E., "Note on the electrostatic problem for a circular annulus," Quart. J. Mech. Appl. Math. 16 (1963), 205.

    Article  MathSciNet  MATH  Google Scholar 

  48. Young, A., "Approximate product integration," Proc. Roy. Soc. A 224 (1954), 552.

    Article  MathSciNet  MATH  Google Scholar 

  49. Zygmund, A., Trigonometric Series, 2nd edn., Cambridge University Press, 1959.

    Google Scholar 

Download references

Authors
  1. Ian Naismith Sneddon
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Bertram Ross

Rights and permissions

Reprints and permissions

Copyright information

© 1975 Springer-Verlag

About this chapter

Cite this chapter

Sneddon, I.N. (1975). The use in mathematical physics of Erdélyi-Kober operators and of some of their generalizations. In: Ross, B. (eds) Fractional Calculus and Its Applications. Lecture Notes in Mathematics, vol 457. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0067097

Download citation

  • DOI: https://doi.org/10.1007/BFb0067097

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-07161-7

  • Online ISBN: 978-3-540-69975-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation