Skip to main content
SpringerLink
Log in
Book cover

Modelling Electroanalytical Experiments by the Integral Equation Method pp 49–60Cite as

Mathematical Preliminaries

  • Chapter
  • First Online:

  • 789 Accesses

Part of the book series: Monographs in Electrochemistry ((MOEC))

Abstract

For a proper understanding of the integral equation method, several mathematical concepts have to be defined. The concepts include: integral and integro-differential equations; integral operators and their kernels; linearity and nonlinearity of the integral operators and integral equations; convolution kernels; kinds of integral equations (first and second kind); distinction between Fredholm and Volterra integral equations; regular, singular and weakly singular kernels (and integral equations). Techniques such as the Laplace, Fourier and Hankel transformations have to be recalled, as they are used for the conversion of initial boundary value problems into integral equations.

This is a preview of subscription content, 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   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
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

Learn about institutional subscriptions

References

  1. Abramowitz M, Stegun IA (1972) Handbook of mathematical functions. Dover Publications, New York

    Google Scholar 

  2. Atkinson KE (1997) The numerical solution of integral equations of the second kind. Cambridge University Press, Cambridge

    Book  Google Scholar 

  3. Baker CTH (1978) The numerical treatment of integral equations. Clarendon Press, Oxford

    Google Scholar 

  4. Brunner H (2004) Collocation methods for Volterra integral and related functional differential equations. Cambridge University Press, Cambridge

    Book  Google Scholar 

  5. Brunner H, Van der Houwen PJ (1986) The numerical solution of Volterra equations. North-Holland, Amsterdam

    Google Scholar 

  6. Carslaw HS, Jaeger JC (1949) Operational methods in applied mathematics. Oxford University Press, London

    Google Scholar 

  7. Churchill RV (1972) Operational mathematics. McGraw-Hill, New York

    Google Scholar 

  8. Cohen AM (2007) Numerical methods for Laplace transform inversion. Springer, New York

    Google Scholar 

  9. Davies B, Martin B (1979) Numerical inversion of the Laplace transform: a survey and comparison of methods. J Comput Phys 33:1–32

    Article  Google Scholar 

  10. Delves LM, Mohamed JL (1985) Computational methods for integral equations. Cambridge University Press, Cambridge

    Book  Google Scholar 

  11. Doetsch G (1974) Introduction to the theory and application of the Laplace transformation. Springer, Berlin

    Book  Google Scholar 

  12. Encyclopedia of Mathematics (2014). http://www.encyclopediaofmath.org. Accessed 10 June 2014

  13. Gorenflo R, Vessella S (1991) Abel integral equations, analysis and applications. Springer, Berlin

    Google Scholar 

  14. GWR (2014). http://library.wolfram.com/infocenter/MathSource/4738. Accessed 10 June 2014

  15. Hackbusch W (1995) Integral equations, theory and numerical treatment. Birkhäuser, Basel

    Google Scholar 

  16. Hammerstein A (1930) Nichtlineare Integralgleichungen nebst Anwendungen. Acta Math 54:117–176

    Article  Google Scholar 

  17. Kythe PK, Puri P (2002) Computational methods for linear integral equations. Birkhäuser, Boston

    Book  Google Scholar 

  18. LePage WR (1980) Complex variables and the Laplace transform for engineers. Dover Publications, New York

    Google Scholar 

  19. Linz P (1985) Analytical and numerical methods for Volterra equations. SIAM, Philadelphia

    Book  Google Scholar 

  20. Mandal BN, Chakrabarti A (2011) Applied singular integral equations. CRC Press, Boca Raton, and Science Publishers, Enfield

    Google Scholar 

  21. MAPLE (2014). http://www.maplesoft.com/products/maple. Accessed 10 June 2014

  22. MATHEMATICA (2014) Wolfram Res. Inc., Champaigne, IL. http://www.wolfram.com. Accessed 10 June 2014

  23. MAXIMA (2014). http://maxima.sourceforge.net. Accessed 10 June 2014

  24. Miles JW (1971) Integral transforms in applied mathematics. Cambridge University Press, Cambridge

    Book  Google Scholar 

  25. Montella C (2008) LSV modelling of electrochemical systems through numerical inversion of Laplace transforms. I—The GS–LSV algorithm. J Electroanal Chem 614:121–130

    Article  CAS  Google Scholar 

  26. Montella C (2009) Re-examination of the potential-step chronoamperometry method through numerical inversion of Laplace transforms. I. General formulation and numerical solution. J Electroanal Chem 633:35–44

    CAS  Google Scholar 

  27. Montella C (2009) Re-examination of the potential-step chronoamperometry method through numerical inversion of Laplace transforms. II. Application examples. J Electroanal Chem 633:45–56

    Article  CAS  Google Scholar 

  28. Montella C, Diard JP (2008) New approach of electrochemical systems dynamics in the time-domain under small-signal conditions. I. A family of algorithms based on numerical inversion of Laplace transforms. J Electroanal Chem 623:29–40

    CAS  Google Scholar 

  29. Montella C, Diard JP (2009) New approach of electrochemical systems dynamics in the time-domain under small-signal conditions. II. Modelling the responses of electrochemical systems by numerical inversion of Laplace transforms. J Electroanal Chem 625:156–164

    CAS  Google Scholar 

  30. Montella C, Michel R (2009) New approach of electrochemical systems dynamics in the time-domain under small-signal conditions. III. Discrimination between nine candidate models for analysis of PITT experimental data from LixCo O2 film electrodes. J Electroanal Chem 628:97–112

    Article  CAS  Google Scholar 

  31. Montella C, Michel R, Diard JP (2007) Numerical inversion of Laplace transforms. A useful tool for evaluation of chemical diffusion coefficients in ion-insertion electrodes investigated by PITT. J Electroanal Chem 608:37–46

    CAS  Google Scholar 

  32. Oldham K, Myland J, Spanier J (2009) An atlas of functions. Springer, New York

    Book  Google Scholar 

  33. Polyanin AD, Manzhirov AV (2008) Handbook of integral equations, 2nd edn. Chapman & Hall, Boca Raton

    Book  Google Scholar 

  34. Rahman M (2007) Integral equations and their applications. WIT Press, Southampton

    Google Scholar 

  35. Roberts GE, Kaufman H (1966) Table of Laplace transforms. Saunders, Philadelphia

    Google Scholar 

  36. Schmidt E (1908) Zur Theorie der Linearen und Nichtlinearen Integralgleichungen. III Teil. Über die Auflösung der nichtlinearen Integralgleichung und die Verzweigung ihrer Lösungen. Math Ann 65:370–399

    Article  Google Scholar 

  37. Spiegel MR (1965) Schaum’s outline of the theory and problems of Laplace Transforms. McGraw-Hill, New York

    Google Scholar 

  38. Tricomi FG (1957) Integral equations. Interscience, New York

    Google Scholar 

  39. Vainikko G (1993) Multidimensional weakly singular integral equations. Springer, Berlin

    Google Scholar 

  40. Valkó PP, Abate J (2004) Comparison of sequence accelerators for the Gaver method of numerical Laplace transform inversion. Comput Math Appl 48:629–636

    Article  Google Scholar 

  41. Wazwaz AM (2011) Linear and nonlinear integral equations. Higher Education Press, Beijing, and Springer, Berlin

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Cracow University of Technology, Cracow, Poland

    Lesław K. Bieniasz

Authors
  1. Lesław K. Bieniasz
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Bieniasz, L.K. (2015). Mathematical Preliminaries. In: Modelling Electroanalytical Experiments by the Integral Equation Method. Monographs in Electrochemistry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44882-3_3

Download citation

Publish with us

Policies and ethics

Search

Navigation