Journal of Applied Mathematics

Error bound analysis and singularly perturbed Abel-Volterra equations

Angelina M. Bijura

Full-text: Open access

Abstract

Asymptotic solutions of nonlinear singularly perturbed Volterra integral equations with kernels possessing integrable singularity are investigated using singular perturbation methods and the Mellin transform technique. In particular, it is demonstrated that the formal approximation is asymptotically valid.

Article information

Source
J. Appl. Math., Volume 2004, Number 6 (2004), 479-494.

Dates
First available in Project Euclid: 13 December 2004

Permanent link to this document
https://projecteuclid.org/euclid.jam/1102957058

Digital Object Identifier
doi:10.1155/S1110757X04305024

Mathematical Reviews number (MathSciNet)
MR2200995

Zentralblatt MATH identifier
1081.45001

Subjects
Primary: 41A60: Asymptotic approximations, asymptotic expansions (steepest descent, etc.) [See also 30E15] 45G05
Secondary: 26A33

Citation

Bijura, Angelina M. Error bound analysis and singularly perturbed Abel-Volterra equations. J. Appl. Math. 2004 (2004), no. 6, 479--494. doi:10.1155/S1110757X04305024. https://projecteuclid.org/euclid.jam/1102957058


Export citation

References

  • J. S. Angell and W. E. Olmstead, Singularly perturbed Volterra integral equations, SIAM J. Appl. Math. 47 (1987), no. 1, 1--14.
  • Yu. I. Babenko, Heat and Mass Transfer, Khimiya, Leningrad, 1986.
  • A. M. Bijura, Singularly perturbed Volterra integral equations with weakly singular kernels, Int. J. Math. Math. Sci. 30 (2002), no. 3, 129--143.
  • --------, Asymptotics of integrodifferential models with integrable kernels, Int. J. Math. Math. Sci. 2003 (2003), no. 25, 1577--1598.
  • --------, Asymptotics of integrodifferential models with integrable kernels. II, Int. J. Math. Math. Sci. 2003 (2003), no. 50, 3153--3169. \CMP2+012+6392 012 639
  • N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Holt, Rinehart and Winston, New York, 1975.
  • K. Diethelm and N. J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002), no. 2, 229--248.
  • S. S. Dragomir, On some nonlinear generalizations of Gronwall's inequality, Makedon. Akad. Nauk. Umet. Oddel. Mat.-Tehn. Nauk. Prilozi 13 (1992), no. 2, 23--28.
  • W. Eckhaus, Asymptotic Analysis of Singular Perturbations, Studies in Mathematics and Its Applications, vol. 9, North-Holland Publishing, Amsterdam, 1979.
  • A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions. Vol. III, McGraw-Hill, New York, 1955.
  • R. Gorenflo and S. Vessella, Abel Integral Equations. Analysis and Applications, Lecture Notes in Mathematics, vol. 1461, Springer-Verlag, Berlin, 1991.
  • G. Gripenberg, S.-O. Londen, and O. Staffans, Volterra Integral and Functional Equations, Encyclopedia of Mathematics and Its Applications, vol. 34, Cambridge University Press, Cambridge, 1990.
  • J.-P. Kauthen, A survey of singularly perturbed Volterra equations, Appl. Numer. Math. 24 (1997), no. 2-3, 95--114.
  • P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM Studies in Applied Mathematics, vol. 7, Society for Industrial and Applied Mathematics, Philadelphia, 1985.
  • R. B. Paris and D. Kaminski, Asymptotics and Mellin-Barnes Integrals, Encyclopedia of Mathematics and Its Applications, vol. 85, Cambridge University Press, Cambridge, 2001.
  • I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, vol. 198, Academic Press, San Diego, 1999.