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Hybrid Functions for Nonlinear Differential Equations with Applications to Physical Problems

  • M. Razzaghi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8236)

Abstract

A numerical method for solving nonlinear initial-value problems is proposed. The Lane-Emden type equations which have many applications in mathematical physics are then considered. The method is based upon hybrid function approximations. The properties of hybrid functions of block-pulse functions and Bernoulli polynomials are presented and are utilized to reduce the computation of nonlinear initial-value problems to a system of equations. The method is easy to implement and yields very accurate results.

Keywords

Block-pulse functions Bernoulli polynomials hybrid nonlinear initial-value problems Lane-Emden equation 

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References

  1. 1.
    Razzaghi, M., Elnagar, G.: Linear quadratic optimal control problems via shifted Legendre state parameterization. Int. J. Syst. Sci. 25, 393–399 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Razzaghi, M., Razzaghi, M.: Instabilities in the solution of a heat conduction problem using Taylor series and alternative approaches. J. Frank. Instit. 326, 683–690 (1989)CrossRefzbMATHGoogle Scholar
  3. 3.
    Razzaghi, M., Marzban, H.R.: Direct method for variational problems via hybrid of block-pulse and Chebyshev functions. Math. Prob. Eng. 6, 85–97 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Wang, X.T., Li, Y.M.: Numerical solutions of integro differential systems by hybrid of general block-pulse functions and the second Chebyshev polynomials. Appl. Math. Comput. 209, 266–272 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Marzban, H.R., Razzaghi, M.: Optimal control of linear delay systems via hybrid of block-pulse and Legendre polynomials. J. Frank. Instit. 341, 279–293 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Singh, V.K., Pandey, R.K., Singh, S.: A stable algorithm for Hankel transforms using hybrid of block-pulse and Legendre polynomials. Comput. Phys. Communications. 181, 1–10 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Marzban, H.R., Razzaghi, M.: Analysis of time-delay systems via hybrid of block-pulse functions and Taylor series. J. Vibra. Contr. 11, 1455–1468 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Marzban, H.R., Razzaghi, M.: Solution of multi-delay systems using hybrid of block-pulse functions and Taylor series. J. Sound. Vibra. 292, 954–963 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Davis, H.T.: Introduction to Nonlinear Differential and Integral Equations. Dover Publications, New York (1962)zbMATHGoogle Scholar
  10. 10.
    Chandrasekhar, S.: Introduction to the Study of Stellar Structure. Dover Publications, New York (1967)Google Scholar
  11. 11.
    Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration. Academic Press, New York (1975)zbMATHGoogle Scholar
  12. 12.
    Bender, C.M., Pinsky, K.S., Simmons, L.M.: A new perturbative approach to nonlinear problems. J. Math. Phys. 30, 1447–1455 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Wazwaz, A.M.: A new algorithm for solving differential equations of Lane-Emden type. Appl. Math. Comput. 118, 287–310 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    He, J.H.: Variational approach to the Lane-Emden equation. Appl. Math. Comput. 143, 539–541 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ramos, J.I.: Linearization techniques for singular initial-value problems of ordinary differential equations. Appl. Math. Comput. 161, 525–542 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Marzban, H.R., Tabrizidooz, H.R., Razzaghi, M.: Hybrid functions for nonlinear initial-value problems with applications to Lane-Emden type equations. Phys. Let. A. 372, 5883–5886 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Costabile, F., Dellaccio, F., Gualtieri, M.I.: A new approach to Bernoulli polynomials. Rendiconti di Matematica, Serie VII 26, 1–12 (2006)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Arfken, G.: Mathematical Methods for Physicists, 3rd edn. Academic Press, San Diego (1985)Google Scholar
  19. 19.
    Kreyszig, E.: Introductory Functional Analysis with Applications. John Wiley and Sons Press, New York (1978)zbMATHGoogle Scholar
  20. 20.
    Mashayekhi, S., Ordokhani, Y., Razzaghi, M.: Hybrid functions approach for nonlinear constrained optimal control problems. Commun. Nonli. Sci. Numer. Simul. 17, 1831–1843 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration. Academic Press (1975)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • M. Razzaghi
    • 1
    • 2
  1. 1.Department of Mathematics and StatisticsMississippi State UniversityUSA
  2. 2.Department of Mathematics and Computer SciencesTechnical University of Civil EngineeringBucharestRomania

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