Generalized Integral Transforms with the Homotopy Perturbation Method

Article

Abstract

This paper applies He’s homotopy perturbation method to compute a large variety of integral transforms. The Esscher, Fourier, Hankel, Laplace, Mellin and Stieljes integrals transforms are particular cases of our generalized integral transform. Our method is of practical importance in order to derive new integration formulae, to approximate certain difficult integrals as well as to calculate the expectation of certain nonlinear functions of random variable.

Keywords

He’s homotopy method Integral transforms Applied probability Type G and spherical distributions Expected utility 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abbasbandy, S.: Application of He homotopy perturbation method for Laplace transform. Chaos, Solitons Fractals 30, 1212–1223 (2006)CrossRefGoogle Scholar
  2. 2.
    Abbasbandy, S.: Application of He’s homotopy perturbation method to functional integral equations. Chaos, Solitons Fractals 30, 1243–1247 (2007)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Abbasbandy, S.: A numerical solution of Blasius equation by Adomian’s decomposition method and comparison with homotopy perturbation method. Chaos, Solitons Fractals 30, 257–260 (2007)CrossRefGoogle Scholar
  4. 4.
    Abbassy, T.A., El-Tawil, M.A., Saleh, H.K.: The solution of KDV and mKDV equations using Adomian Pade approximation. Int. J. Non-linear Sci. Numer. Simul. 5(2), 105–112 (2004)Google Scholar
  5. 5.
    Babolian, E., Biazar, J., Vahidi, A.R.: A new computational method for Laplace transform by decomposition method. Appl. Math. Comput. 150, 841–846 (2004)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Biazar, J., Ghazvini: Convergence of the homotopy perturbation method for differential equations. Nonlinear Analysis: Real World Applications 10, 2633–2640 (2009)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Bronshtein, I.N., Semendyayev, K.A., Musiol, G., Muehlig, H.: Handbook of Mathematics, 4th edn, pp. 705–706. Springer-Verlag, New York (2004)Google Scholar
  8. 8.
    Fotopoulos, S.B.: Type G and spherical distribuions on ℝd. Stat. Probab. Lett., 72(1), 23–32 (2005)MATHMathSciNetGoogle Scholar
  9. 9.
    Hardar, K., Data, B.K.: Integrations by asymptotic decomposition. Appl. Math. Lett. 9(2), 105–112 (1996)Google Scholar
  10. 10.
    Gabutti, B., Minetti, B.: A new application of the discrete Laguerre polynomials in the numerical evaluation of the Hankel transform of strongly decreasing even functions. J. Comput. Phys. 42, 277–287 (1981)CrossRefMATHGoogle Scholar
  11. 11.
    Glauber, R.J.: Lectures in Theoretical Physics, vol. 1. Interscience, New York (1959)Google Scholar
  12. 12.
    I.S Gradshteyn, Ryzhik, I.M.: Table of Integrals, Series and Products, 6th edn. Academic Press, San Diego, (2000)MATHGoogle Scholar
  13. 13.
    He, J.H.: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Methods Appl. Mech. Eng. 167(1–2), 57–68 (1998)CrossRefMATHGoogle Scholar
  14. 14.
    He, J.H.: Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng. 178(3–4), 257–262 (1999)CrossRefMATHGoogle Scholar
  15. 15.
    He, J.H.: A coupling method of homotopy technique and perturbation technique for nonlinear problems. Int. J. Nonlinear Mech. 35(1), 37–43 (2000)MATHGoogle Scholar
  16. 16.
    He, J.H.: Homotopy perturbation method: a new nonlinear analytical technique. Appl. Math. Comput. 135, 73–79 (2003)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    He, J.H.: Comparison of homotopy perturbtaion method and homotopy analysis method. Appl. Math. Comput. 156: 527–539 (2004)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    He, J.H.: Asymptotology by homotopy perturbation method. Appl. Math. Comput. 156, 591–596 (2004)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    He, J.H.: Homotopy perturbation method for bifurcation of nonlinear problems. Int. J. Nonlinear Sci. Numer. Simul. 6(2), 20–78 (2005)Google Scholar
  20. 20.
    He, J.H.: Some asymptotic methods for strongly nonlinar equations. Int. J. Mod. Phys. B 20(10), 1141–1999 (2006)CrossRefMATHGoogle Scholar
  21. 21.
    He, J.H.: New interpretation of homotopy pertubation method. Int. J. Mod. Phys. B 20(18), 2561–2568 (2006)CrossRefGoogle Scholar
  22. 22.
    He, J.H.: Variational iteration method a kind of non-inear analytical technique: some examples. Int. J. Nonlinear Mech. 34(4), 699–708 (1999)CrossRefMATHGoogle Scholar
  23. 23.
    He, J.H., Wu, X.H.: Construction of solitary solution and compacton-like solution by variational iteration method. Chaos, Solitons Fractals 29(1), 108–113 (2006)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Marcus, M.B.: ξ-radial processes and random Fourier series. Mem. Amer. Math. Soc. 68(368), viii+181 (1987)Google Scholar
  25. 25.
    Momani, S., Abuasad, S.: Application of He’s variational iteration method to Helmhotz equation. Chaos, Solitons Fractals 27(5), 1119–1123 (2006)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Servadio, S.: Calculation of O(1) interferences of double-scattering waves in 3–3 collision. Nuovo Cimento 69, 1–22 (1982)CrossRefGoogle Scholar
  27. 27.
    Sadefo Kamdem, J., Qiao, Z.: Decomposition method for the Camassa–Holm. Chaos, Solitons Fractals 31(2), 437–447 (2007)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Paiano, G., Picca, D.: Integrals transforms as computational tools in quantum mechanics. J. Comput. Appl. Math. 1(2), 93–100 (1975)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Wong, R.: Asymptotic Approximations of Integrals. Academic, New York (1989)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Université de Montpellier 1, LAMETA (CNRS UMR 5474)MontpellierFrance

Personalised recommendations