Isaac Scientific Publishing

Advances in Analysis

The Abel Integral Equations in Distribution

Download PDF (562.4 KB) PP. 88 - 104 Pub. Date: March 9, 2017

DOI: 10.22606/aan.2017.22003

Author(s)

  • Chenkuan Li*
    Department of Mathematics and Computer Science, Brandon University, Brandon, Manitoba, Canada R7A 6A9
  • Changpin Li

    Department of Mathematics, Shanghai University, Shanghai 200444, China
  • Bailey Kacsmar

    Department of Mathematics and Computer Science, Brandon University, Brandon, Manitoba, Canada R7A 6A9
  • Roque Lacroix

    Department of Mathematics and Computer Science, Brandon University, Brandon, Manitoba, Canada R7A 6A9
  • Kyle Tilbury

    Department of Mathematics and Computer Science, Brandon University, Brandon, Manitoba, Canada R7A 6A9

Abstract

Keywords

Distribution, convolution, delta function, Abel’s equation, Gamma function, Caputo derivative and Riemann-Liouville derivative.

References

[1] C. J. Cremers and R. C. Birkebak, “Applications of the abel’s integral equation to spectrographic data,” Appl. Opt., vol. 5, no. 6, pp. 1057–1064, 1966.

[2] R. Gorenflo and S. Vessella, Abel Integral Equations: Analysis and Applications. Springer-Verlag, 1991.

[3] V. Mirceski and Z. Tomovski, “Analytical solutions of integral equations for modeling of reversible electrode processes under volumetric conditions,” J. Electroanal. Chem., vol. 619–620, pp. 164–168, 2008.

[4] N. H. Abel, “Solution de quelques problèmes à i’aide d’intégrales définies,” Magaziu for Naturvidenskaberue, Alu-gang I, vol. Bind 2, Christiania, pp. 11–18, 1823.

[5] J. D. Tamarkin, “On integrable solutions of abel’s integral equation,” Annals of Mathematics, vol. 31, pp. 219–229, 1930.

[6] D. B. Sumnar, “Abel’s integral equation as a convolution of transform,” Proceedings of the American Mathematical Society, vol. 7, pp. 82–86, 1956.

[7] G. N. Minerbo and M. E. Levy, “Inversion of abel’s integral equation by means of orthogonal polynomials,” SIAM Journal on Numerical Analysis, vol. 6, pp. 598–616, 1969.

[8] J. R. Hatcher, “A nonlinear boundary problem,” Proceedings of the American Mathematical Society, vol. 95, no. 1, pp. 441–448, 1985.

[9] O. P. Singh, V. K. Singh, and R. K. Pandey, “A stable numerical inversion of abel’s integral equation using almost bernstein operational matrix,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 111, pp. 245–252, 2010.

[10] M. Li and W. Zhao, “Solving abel’s type integral equation with mikusinski’s operator of fractional order,” Advances in Mathematical Physics, vol. 2013, Article ID 806984, pp. 1–4, 2013.

[11] S. Jahanshahi, E. Babolian, D. Torres, and A. Vahidi, “Solving abel integral equations of first kind via fractional calculus,” Journal of King Saud University - Science, vol. 27, pp. 161–167, 2015.

[12] M. H. Saleh, S. M. Amer, D. S. Mohamed, and A. E. Mahdy, “Identification of common molecular subsequences,” International Journal of Computer Applications, vol. 100, pp. 19–23, 2014.

[13] I. M. Gel’fand and G. E. Shilov, Generalized Functions. Vol I, Academic Press, New York, 1964.

[14] C. Li, “Several results of fractional derivatives in D0(R+),” Fractional Calculus and Applied Analysis, vol. 18, pp. 192–207, 2015.

[15] C. P. Li, D. Qian, and Y. Chen, “On riemann-liouville and caputo derivatives,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 562494, pp. 1–15, 2011.

[16] C. P. Li and Z. Zhao, “Introduction to fractional integrability and differentiability,” The European Physical Journal-Special Topics, vol. 193, pp. 5–26, 2011.

[17] C. P. Li and W. Deng, “Remarks on fractional derivatives,” Appl. Math. Comput., vol. 187, pp. 777–784, 2007.

[18] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, New York, 2006.

[19] C. K. Li and C. P. Li, “On defining the distributions ()k and (0)k by fractional derivatives,” Applied Mathematics and Computation, vol. 248, pp. 502–513, 2014.

[20] I. Podlubny, Fractional Differential Equations. Academic Press, New Yor, 1999.

[21] H. M. Srivastava and R. G. Buschman, Theory and Applications of Convolution Integral Equations. Kluwer Academic Publishers, Dordrecht-Boston-London, 1992.

[22] X. J. Yang, D. Baleanu, and H. M. Srivastava, Local Fractional Integral Transforms and Their Applications. Academic Press (Elsevier Science Publishers), Amsterdam, Heidelberg, London and New York, 2016.

[23] I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products. Academic Press, 1980.

[24] C. K. Li and C. P. Li, “Remarks on fractional derivatives of distributions,” to appear in Tbilisi Mathematical Journal.