Article ID: MTJPAM-D-21-00079

Title: Numerical solution of Love’s integral equation using Daubechies scale function


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-21-00079; Volume 4 / Issue 2 / Year 2022, Pages 45-59

Document Type: Research Paper

Author(s): Jyotirmoy Mouley a , Birendra Nath Mandal b

aDepartment of Applied Mathematics, University of Calcutta, Kolkata, India

bPhysics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata, India

Received: 23 December 2021, Accepted: 9 June 2022, Published: 26 July 2022.

Corresponding Author: Jyotirmoy Mouley (Email address: jyoti87.cu.wavelet@gmail.com)

Full Text: PDF

Abstract

This paper presents an efficient numerical scheme for solving a special type of quasi-singular integral equation attributed to Eric Russel Love. The proposed scheme involves the use of an expansion technique of the unknown function of the integral equation in terms of Daubechies scale function. This integral equation has an immense importance in the field of electrostatics. After obtaining the unknown function f(x) approximately, numerically the capacity of two parallel coaxial disks of unit radius are calculated for different separating parameter κ and the computed results are compared with Nomura-Cook results available in the literature. The potential at any point outside the disks is determined numerically. Also the convergence of the proposed method is established.

Keywords: Love’s integral equation, Daubechies scale and wavelet function, capacity, potential

References:
  1. J. C. Cooke, The coaxial circular disc problem, ZAMM Z. Angew. Math. Mech. 38 (9-10), 349–356, 1958.
  2. D. Elliot, A Chebyshev series method for the numerical solution of Fredholm integral equation, Comput. J. 6 (1), 102–112, 1963.
  3. B. M. Kessler, G. L. Payne and W. W. Polyzou, Notes on wavelets, ArXiv: nucl-th/0305025v2, 2003.
  4. F.-R. Li and Y.-J. Shi, Preconditioned conjugate gradient method for the solution of Love’s integral equation with very small parameter, J. Comput. Appl. Math. 327, 295–305, 2018.
  5. E. R. Love, The electrostatic field of two equal circular co-axial conducting disks, Quart. J. Mech. Appl. Math. 2 (4), 428–451, 1949.
  6. G. V. Milovanovic and D. Joksimovic, Properties of Boubaker polynomials and an application to Love’s integral equation, Appl. Math. Comput. 224, 74–87, 2013.
  7. G. Monegato and A. P. Orsi, Product formulas for Fredholm integral equations with rational kernel functions, In: Numerical Integration III (Ed. by H. Braß and G. Hämmerlin), International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, 85, Birkhäuser, Basel, 140–156, 1988.
  8. 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. 14 (1), 21–27, 2020.
  9. J. Mouley and B. N. Mandal, Wavelet-based collocation technique for fractional integro-differential equation with weakly singular kernel, Comput. Math. Methods 3 (4), 2021; Article ID: e1158.
  10. J. W. Nicholson, XI. The electrification of two parallel circular discs, Philos. Trans. Roy. Soc. A 224 (616-625), 303–369, 1924.
  11. Y. Nomura, The electrostatic problems of two equal parallel circular plates, Proc. Physico-Math. Soci. Japan. 3rd Series 23, 168–180, 1941.
  12. M. M. Panja and B. N. Mandal, Wavelet based approximation schemes for singular integral equations, CRC Press, Taylor & Francis Group, Boca Raton, London, 2020.
  13. 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. 290, 609–632, 2015.
  14. P. Pastore, The numerical treatment of Love’s integral equation having very small paramrter, J. Comput. Appl. Math. 236 (6), 1267–1281, 2011.
  15. J. L. Philips, The use of collocation as a projection method for solving linear operator equations, SIAM J. Numer. Anal. 9 (1), 14–28, 1972.
  16. M. A. Wolfe, The numerical solution of non-singular and integro-differential equations by iteration with Chebyshev series, Comput. J. 12 (2), 193–196, 1969.
  17. A. Young, The application of approximate product-integration to the numerical solution of integral equations, Philos. Trans. Roy. Soc. A 224 (1159), 561–573, 1954.