Skip to main content
SpringerLink
Log in

The Product Midpoint Rule for Abel-Type Integral Equations of the First Kind with Perturbed Data

  • Chapter
  • First Online:
New Trends in Parameter Identification for Mathematical Models

Part of the book series: Trends in Mathematics ((TM))

Abstract

We consider the regularizing properties of the product midpoint rule for the stable solution of Abel-type integral equations of the first kind with perturbed right-hand sides. The impact of continuity and smoothness properties of solutions on the convergence rates is described in detailed manner by using a scale of Hölder spaces. In addition, correcting starting weights are introduced to get rid of undesirable initial conditions. The proof of the inverse stability of the quadrature weights relies on Banach algebra techniques. Finally, numerical results are presented.

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 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.00
Price excludes VAT (USA)
  • Durable hardcover 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

References

  1. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, 10th edn. (National Bureau of Standards, Dover, New York, 1972)

    MATH  Google Scholar 

  2. R.S. Anderssen, Stable procedures for the inversion of Abel’s equation. IMA J. Appl. Math. 17, 329–342 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  3. H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations (Cambridge University Press, Cambridge, 2004)

    Book  MATH  Google Scholar 

  4. H. Brunner, P.J. van der Houwen, The Numerical Solution of Volterra Equations (Elsevier, Amsterdam, 1986)

    MATH  Google Scholar 

  5. A.L. Bughgeim, Volterra Equations and Inverse Problems (VSP/de Gruyter, Zeist/Berlin, 1999)

    Book  MATH  Google Scholar 

  6. R.F. Cameron, S. McKee, High accuracy convergent product integration methods for the generalized Abel equation. J. Integr. Equ. 7, 103–125 (1984)

    MathSciNet  MATH  Google Scholar 

  7. R.F. Cameron, S. McKee, The analysis of product integration methods for Abel’s equation using fractional differentiation. IMA J. Numer. Anal. 5, 339–353 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  8. P.P.B. Eggermont, A new analysis of the Euler-, midpoint- and trapezoidal-discretization methods for the numerical solution of Abel-type integral equations. Technical report, Department of Computer Science, University of New York, Buffalo, 1979

    Google Scholar 

  9. P.P.B. Eggermont, A new analysis of the trapezoidal-discretization method for the numerical solution of Abel-type integral equations. J. Integr. Equ. 3, 317–332 (1981)

    MathSciNet  MATH  Google Scholar 

  10. P.P.B. Eggermont, Special discretization methods for the integral equations of image reconstruction and for Abel-type integral equations. Ph.D. thesis, University of New York, Buffalo, 1981

    Google Scholar 

  11. P. Erdős, W. Feller, H. Pollard, A property of power series with positive coefficients. Bull. Am. Math. Soc. 55, 201–204 (1949)

    Google Scholar 

  12. R. Gorenflo, S. Vessella, Abel Integral Equations (Springer, New York, 1991)

    Book  MATH  Google Scholar 

  13. W. Hackbusch, Integral Equations (Birkhäuser, Basel, 1995)

    Book  MATH  Google Scholar 

  14. G.H. Hardy, Divergent Series (Oxford University Press, Oxford, 1948)

    MATH  Google Scholar 

  15. P. Henrici, Applied and Computational Complex Analysis, vol. 1 (Wiley, New York, 1974)

    MATH  Google Scholar 

  16. B. Kaltenbacher, A convergence analysis of the midpoint rule for first kind Volterra integral equations with noisy data. J. Integr. Equ. 22, 313–339 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. T. Kaluza, Über die Koeffizienten reziproker Funktionen. Math. Z. 28, 161–170 (1928)

    Article  MathSciNet  MATH  Google Scholar 

  18. P. Linz, Analytical and Numerical Methods for Volterra Equations, 1st edn. (SIAM, Philadelphia, 1985)

    Book  MATH  Google Scholar 

  19. Ch. Lubich, Discretized fractional calculus. SIAM J. Math. Anal. 17(3), 704–719 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ch. Lubich, Fractional linear multistep methods for Abel–Volterra integral equations of the first kind. IMA J. Numer. Anal. 7, 97–106 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  21. R. Plato, Fractional multistep methods for weakly singular Volterra equations of the first kind with noisy data. Numer. Funct. Anal. Optim. 26(2), 249–269 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. R. Plato, The regularizing properties of the composite trapezoidal method for weakly singular Volterra integral equations of the first kind. Adv. Comput. Math. 36(2), 331–351 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. R. Plato, The regularizing properties of multistep methods for first kind Volterra integral equations with smooth kernels. Comput. Methods Appl. Math. 17(1), 139–159 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. B.A. Rogozin, Asymptotics of the coefficients in the Levi–Wiener theorems on absolutely convergent trigonometric series. Sib. Math. J. 14, 917–923 (1973)

    Article  MATH  Google Scholar 

  25. B.A. Rogozin, Asymptotic behavior of the coefficients of functions of power series and Fourier series. Sib. Math. J. 17, 492–498 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  26. W. Rudin, Functional Analysis, 2nd edn. (McGraw-Hill, New York, 1991)

    MATH  Google Scholar 

  27. G. Szegö, Bemerkungen zu einer Arbeit von Herrn Fejér über die Legendreschen Polynome. Math. Z. 25, 172–187 (1926)

    Article  MathSciNet  MATH  Google Scholar 

  28. U. Vögeli, K. Nedaiasl, S. Sauter, A fully discrete Galerkin method for Abel-type integral equations (2016). arXiv:1612.01285

    Google Scholar 

  29. R. Weiss, Product integration for the generalized Abel equation. Math. Comput. 26, 177–190 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  30. R. Weiss, R.S. Anderssen, A product integration method for a class of singular first kind Volterra equations. Numer. Math. 18, 442–456 (1972)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Siegen, Walter-Flex-Str. 3, 57068, Siegen, Germany

    Robert Plato

Authors
  1. Robert Plato
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Robert Plato .

Editor information

Editors and Affiliations

  1. Fakultät für Mathematik, Technische Universität Chemnitz, Chemnitz, Germany

    Bernd Hofmann

  2. Department of Mathematics, Federal University of Santa Catarin, Florianopolis, Brazil

    Antonio Leitão

  3. Pura e Aplicada, Instituto Nacional de Matematica, Rio de Janeiro, Rio de Janeiro, Brazil

    Jorge P. Zubelli

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Plato, R. (2018). The Product Midpoint Rule for Abel-Type Integral Equations of the First Kind with Perturbed Data. In: Hofmann, B., Leitão, A., Zubelli, J. (eds) New Trends in Parameter Identification for Mathematical Models. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-70824-9_11

Download citation

Publish with us

Policies and ethics

Search

Navigation