Skip to main content

Solving the nonlinear Schlomilch’s integral equation arising in ionospheric problems

Afrika Matematika volume 28, pages 459–480 (2017)Cite this article

Abstract

In this paper, the linear and nonlinear Schlomilch’s integral equations and their generalized forms are studied. The Schlomilch’s integral equations are used for many ionospheric problems, atmospheric and terrestrial physics. The generalized fractional order of the Chebyshev orthogonal functions (GFCF) collocation method is used to handle many forms of Schlomilch’s integral equations. The GFCF method can be used in the applied physics, applied mathematics, and engineering applications. The reliability of the GFCF method is justified through illustrative examples.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

References

  1. Wazwaz, A.M.: Solving Schlomilch’s integral equation by the regularization Adomian method. Rom. J. Phys. 60, 56–71 (2015)

    Google Scholar 

  2. Unz, H.: Schlomilch’s integral equation for oblique incidence. J. Atmos. Terr. Phys. 28, 315–316 (1966)

    Article  Google Scholar 

  3. Unz, H.: Schlomilch’s integral equation. J. Atmos. Terr. Phys. 25, 101–102 (1963)

    Article  Google Scholar 

  4. Gething, P.J., Maliphant, R.G.: Unz’s application of Schlomilch’s integral equation to oblique incidence observations. J. Atmos. Terr. Phys. 29, 599–600 (1967)

    Article  Google Scholar 

  5. De, S., Sarkar, B., Mal, M., De, M., Ghosh, B., Adhikari, S.: On Schlomilch’s Integral Equation for the Ionospheric Plasma. Jpn. J. Appl. Phys. 33, 4154–4156 (1994)

    Article  Google Scholar 

  6. Gullberg, G.T., Budinger, T.F.: The use of filtering methods to compensate for constant attenuation in single-photon emission computed tomography. IEEE Trans. Biomed. Eng. 28(2), 142–157 (1981)

    Article  Google Scholar 

  7. Bougoffa, L., Al-Haqbani, M., Rach, R.: A convenient technique for solving integral equations of the first kind by the Adomian decomposition method. Kybernetes 41(1), 145–156 (2012)

    Article  MathSciNet  Google Scholar 

  8. Boyd, J.P.: Chebyshev spectral methods and the Lane-Emden problem. Numer. Math. Theor. Methods Appl. 4, 142–157 (2011)

    MathSciNet  MATH  Google Scholar 

  9. Parand, K., Yousefi, H., Delkhosh, M., Ghaderi, A.: A novel numerical technique to obtain an accurate solution to the Thomas-Fermi equation. Eur. Phys. J. Plus 131, 228 (2016)

    Article  Google Scholar 

  10. Kazem, S., Rad, J.A., Parand, K., Abbasbandy, S.: A new method for solving steady flow of a third-grade fluid in a porous half space based on radial basis functions. Z. Naturforsch. A 66, 591–598 (2011)

    Article  Google Scholar 

  11. Shen, J., Tang, T.: High order numerical methods and algorithms. Chinese Science Press, Chinese (2005)

    Google Scholar 

  12. Rad, J.A., Parand, K., Ballestra, L.V.: Pricing European and American options by radial basis point interpolation. Appl. Math. Comput. 251, 363–377 (2015)

    MathSciNet  MATH  Google Scholar 

  13. Parand, K., Dehghan, M., Taghavi, A.: Modified generalized Laguerre function Tau method for solving laminar viscous flow: The Blasius equation. Int. J. Numer. Method. H. 20(7), 728–743 (2010)

    Article  MathSciNet  Google Scholar 

  14. Kazem, S., Abbasbandy, S., Kumar, S.: Fractional-order Legendre functions for solving fractional-order differential equations. Appl. Math. Model. 37, 54985510 (2013)

    MathSciNet  Google Scholar 

  15. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. Elsevier, San Diego (2006)

    MATH  Google Scholar 

  16. Craven, B.D.: Stone’s theorem and completeness of orthogonal systems. J. Aust. Math. Soc. 12(2), 211–223 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  17. Odibat, Z., Momani, S.: An algorithm for the numerical solution of differential equations of fractional order. J. Appl. Math. Inf. 26, 15–27 (2008)

    MATH  Google Scholar 

  18. Eslahchi, M.R., Dehghan, M., Amani, S.: Chebyshev polynomials and best approximation of some classes of functions. J. Numer. Math. 23(1), 41–50 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Doha, E.H., Bhrawy, A.H., Ezz-Eldien, S.S.: A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order. Comput. Math. Appl. 62, 2364–2373 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Parand, K., Delkhosh, M.: Solving Volterra’s population growth model of arbitrary order using the generalized fractional order of the Chebyshev functions. Ricerche Mat. 65, 307–328 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Nkwanta, A., Barnes, E.R.: Two Catalan-type Riordan arrays and their connections to the Chebyshev polynomials of the first kind. J. Integer Seq. 15, 1–19 (2012)

    MathSciNet  MATH  Google Scholar 

  22. Saadatmandi, A., Dehghan, M.: Numerical solution of hyperbolic telegraph equation using the Chebyshev tau method. Numer. Meth. Part. D. E. 26(1), 239–252 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Boyd, J.P.: Chebyshev and Fourier spectral methods, 2nd edn. DOVER Publications, Mineola (2000)

    Google Scholar 

  24. Darani, M.A., Nasiri, M.: A fractional type of the Chebyshev polynomials for approximation of solution of linear fractional differential equations. Comput. Methods Differ. Equ. 1, 96–107 (2013)

    MATH  Google Scholar 

  25. Parand, K., Khaleqi, S.: The rational Chebyshev of second kind collocation method for solving a class of astrophysics problems. Eur. Phys. J. Plus 131, 1–24 (2016)

    Article  Google Scholar 

  26. Parand, K., Shahini, M., Dehghan, M.: Solution of a laminar boundary layer flow via a numerical method. Commun. Nonlinear Sci. Num. Simul. 15(2), 360–367 (2010)

    Article  MATH  Google Scholar 

  27. Parand, K., Abbasbandy, S., Kazem, S., Rezaei, A.R.: An improved numerical method for a class of astrophysics problems based on radial basis functions. Phys. Scripta 83(1), 015011 (2011)

    Article  MATH  Google Scholar 

  28. Adomian, G.: Solving frontier problems of physics: the decomposition method, Kluwer Academic Publishers (1994)

  29. Liao, S.J.: The proposed homotopy analysis technique for the solution of nonlinear problems, PhD thesis, Shanghai Jiao Tong University (1992)

  30. Tatari, M., Dehghan, M.: On the convergence of He’s variational iteration method. J. Comput. Appl. Math. 207(1), 121–128 (2007)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors are very grateful to reviewers and editor for carefully reading the paper and for their comments and suggestions which have improved the paper.

Author information

Authors and Affiliations

  1. Department of Computer Sciences, Shahid Beheshti University, G.C., Tehran, Iran

    Kourosh Parand & Mehdi Delkhosh

  2. Department of Cognitive Modelling, Institute for Cognitive and Brain Sciences, Shahid Beheshti University, G.C., Tehran, Iran

    Kourosh Parand

Authors
  1. Kourosh Parand
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mehdi Delkhosh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kourosh Parand.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Parand, K., Delkhosh, M. Solving the nonlinear Schlomilch’s integral equation arising in ionospheric problems. Afr. Mat. 28, 459–480 (2017). https://doi.org/10.1007/s13370-016-0459-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13370-016-0459-3

Keywords

  • Fractional order of the Chebyshev functions
  • Schlomilch’s integral equations
  • Terrestrial physics
  • Collocation method
  • Integral equations

Mathematics Subject Classification

  • 45Gxx
  • 45Bxx
  • 33C45