Approximate solution of Abel integral equation in Daubechies wavelet basis

  • Jyotirmoy Mouley Department of Applied Mathematics, University of Calcutta, 92, A.P.C. Road, Kolkata-700 009, India.
  • M. M. Panja Department of Mathematics, Visva-Bharati, Santiniketan, West Bengal, 731235, India.
  • B. N. Mandal Physics and Applied Mathematics Unit, Indian Statistical Institute, 203, B.T. Road, Kolkata-700108, India.
Keywords: Abel integral equation, Daubechies scale function, Daubechies wavelet, Gauss-Daubechies quadrature rule


This paper presents a new computational method for solving Abel integral equation (both first kind and second kind). The numerical scheme is based on approximations in Daubechies wavelet basis. The properties of Daubechies scale functions are employed to reduce an integral equation to the solution of a system of algebraic equations. The error analysis associated with the method is given. The method is illustrated with some examples and the present method works nicely for low resolution.


S. B. Healy, J. Haase, O. Lesne, “Abel transform inversion of radio occulation measurement made with a receiver inside the earth’s atmosphere”, Ann. Geophys., vol. 20, no. 8, pp. 1253- 1256, 2002.

R. N. Bracewell, A. C. Riddle, “Inversion of Fan-Beam scans in radio astronomy”, Astrophysical Journal, vol. 150, pp. 427-434, 1967.

Lj. M. Ignjatovic and A. A. Mihajlov, “The realization of Abel’s inversion in the case of discharge with undetermined radius”, Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 72, no. 5, pp. 677-689, 2002.

S. De, B. N. Mandal and A. Chakrabarti, “Water wave scattering by two submerged plane vertical barriers–Abel integral-equation approach”, J. Eng. Math., vol. 65, no. 1, pp. 75-87, 2009.

J. Fourier, Théorie Analytique de la chaleur, Firmin Didot, United Kingdom: Cambridge University Press, ISBN 978-1-108-00180-9, 2009.

A. Graps, “An introduction to wavelets”, IEEE Computing in Science and Engineering, vol. 2, no.2, pp. 50-61, 1995.

A. Grossman and J. Morlet, “Decomposition of Hardy functions into square integrables wavelets of constant shape”, SIAM J. Math. Anal., vol. 15, no. 4, pp. 723-736, 1984.

P. G. Lamarie and Y. Meyer, “Ondelettes et bases hilbertiennes”, Rev. Mat. Iberoam., vol. 2, no. 1, pp. 1-18, 1986.

I. Daubechies, “Orthonormal bases of compactly supported wavelets”, Comm. Pure Appl. Math., vol. 41, no.7, pp. 909-996, 1988.

I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Philadelphia, PA: SIAM, 1992.

G. Beylkin, R. Coifman and V. Rokhlin, “Fast wavelet transforms and numerical algorithms I”, Comm. Pure Appl. Math, vol. 44, no. 2, pp. 141-183, 1991.

S. A. Yousefi, “Numerical solution of Abel’s integral equation by using Legendre wavelets”, Appl. Math. Comput., vol. 175, no.1, pp. 574-580, 2006.

N. Mandal, A. Chakrabarti and B. N. Mandal, “Solution of a system of generalized Abel integral equations using fractional calculus”, Appl. Math. Lett., vol.9, no. 5, pp. 1-4, 1996.

Y. Liu and L. Tao, “Mechanical quadrature methods and their extrapolation for solving first kind Abel integral equations”, J. Comput. Appl. Math, vol. 201, no.1, pp. 300-313, 2007.

H. Derili and S. Sohrabi, “Numerical solution of singular integral equations using orthogonal functions”, Math. Sci. (QJMS), vol. 3, pp. 261-272, 2008.

M. Alipour and D. Rostamy, “Bernstein polynomials for solving Abel’s integral equation”, J. Math. Comput. Sci., vol. 3, no. 4, pp. 403-412, 2011.

A. Shahsavaram, “Haar Wavelet Method to Solve Volterra Integral Equations with Weakly Singular Kernel by Collocation Method”, Appl. Math. Sci., vol. 5, pp. 3201-3210, 2011.

J. Mouley, M. M. Panja and B. N. Mandal, “Numerical solution of an integral equation arising in the problem of cruciform crack using Daubechies scale function”, Math. Sci., vol. 14, no. 1, pp. 21-27, 2020.

M. M. Panja and B. N. Mandal, “Solution of second kind integral equation with Cauchy type kernel using Daubechies scale function”, J. Comput. Appl. Math., vol. 241, pp. 130-142, 2013.

L. J. Curtis, “Concept of the exponential law prior to 1900”, Amer. J. Phys., vol. 46, no. 9, pp. 896-906, 1978.

B. M. Kessler, G. L. Payne, W. W. Polyzou, “Notes on Wavelets”, 2003. .

M. M. Panja and B. N. Mandal, “Gauss-type quadrature rule with complex nodes and Weights for integrals involving Daubechies scale functions and wavelets”, J. Comput. Appl. Math., vol. 290, pp. 609-632, 2015.

E. M. Stein, R. Shakarchi, Functional Analysis: Introduction to Further topics in Analysis’, Princeton Lectures in Analysis, Princeton: Princeton University Press, ISBN-978-0-691-11387- 6, 2011.

A. Wang, “Lebesgue measure and L2 space”, Mathematics department, University of Chicago, 2011.

M. M. Panja and B. N. Mandal, “Evaluation of singular integrals using Daubechies scale function”, Adv. Comput. Math. Appl., vol. 1, pp. 64-75, 2012.

How to Cite
J. Mouley, M. M. Panja, and B. N. Mandal, “Approximate solution of Abel integral equation in Daubechies wavelet basis”, CUBO, vol. 23, no. 2, pp. 245–264, Aug. 2021.