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Analytical Approximation Solution for Differential Equations with Piecewise Constant Arguments of Alternately Advanced and Retarded Type

  • Qi Wang
  • Cui Guo
  • Ruixiang Zeng
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 392)

Abstract

In this work, the variational iteration method is used for analytic treatment of differential equations with piecewise constant arguments of alternately advanced and retarded type. In order to prove the precision of the results, some comparisons are also made between the exact solutions and the results of the numerical method and the variational iteration method. The obtained results reveal that the method is very effective and convenient for constructing differential equations with piecewise constant arguments.

Keywords

Variational iteration method Piecewise constant arguments Analytical approximation solution Lagrange multiplier 

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References

  1. 1.
    Wiener, J., Debnath, L.: Partial Differential Equations with Piecewise Constant Delay. J. Comput. Int. J. Math. Math. Sci. 14, 485–496 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Nieto, J.J., Rodríguez-López, R.: Study of Solutions to Some Functional Differential Equations with Piecewise Constant Arguments. Abstr. Appl. Anal. 2012, 1–25 (2012)CrossRefGoogle Scholar
  3. 3.
    Akhmet, M.U.: On the Reduction Principle for Differential Equations with Piecewise Constant Argument of Generalized Type. J. Math. Anal. Appl. 336, 646–663 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Seifert, G.: Second Order Neutral Delay Differential Equations with Piecewise Constant Time Dependence. J. Math. Anal. Appl. 281, 1–9 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Gurcan, F., Bozkurt, F.: Global Stability in a Population Model with Piecewise Constant Arguments. J. Math. Anal. Appl. 360, 334–342 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Shah, S.M., Wiener, J.: Advanced Differential Equations with Piecewise Constant Argument Deviations. Int. J. Math. Math. Sci. 6, 671–703 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Cooke, K.L., Wiener, J.: Retarded Differential Equations with Piecewise Constant Delays. J. Math. Anal. Appl. 99, 265–297 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Wiener, J.: Generalized Solutions of Functional Differential Equations. World Scientific, Singapore (1993)Google Scholar
  9. 9.
    Ivanov, A.F.: Global Dynamics of a Differential Equation with Piecewise Constant Argument. Nonlinear Anal. 71, e2384–e2389 (2009)Google Scholar
  10. 10.
    Akhmet, M.U., Yılmaz, E.: Impulsive Hopfield-Type Neural Network System with Piecewise Constant Argument. Nonlinear Anal.: RWA 11, 2584–2593 (2010)CrossRefzbMATHGoogle Scholar
  11. 11.
    Yuan, R.: On Favard’s Theorems. J. Differential Equations 249, 1884–1916 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Lv, W.J., Yang, Z.W., Liu, M.Z.: Stability of Runge-Kutta Methods for the Alternately Advanced and Retarded Differential Equations with Piecewise Continuous Arguments. Comput. Math. Appl. 54, 326–335 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Liu, M.Z., Ma, S.F., Yang, Z.W.: Stability Analysis of Runge-Kutta Methods for Unbounded Retarded Differential Equations with Piecewise Continuous Arguments. Appl. Math. Comput. 191, 57–66 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Wang, Q., Zhu, Q.Y., Liu, M.Z.: Stability and Oscillations of Numerical Solutions of Differential Equations with Piecewise Continuous Arguments of Alternately Advanced and Retarded Type. J. Comput. Appl. Math. 235, 1542–1552 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Wang, Q., Zhu, Q.Y.: Stability Analysis of Runge-Kutta Methods for Differential Equations with Piecewise Continuous Arguments of Mixed Type. Int. J. Comput. Math. 88, 1052–1066 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Allan, F.M.: Construction of Analytic Solution to Chaotic Dynamical Systems Using the Homotopy Analysis Method. Chaos Solitons and Fractals 39, 1744–1752 (2009)CrossRefzbMATHGoogle Scholar
  17. 17.
    Liao, S.J.: Beyond Perturbation: Introduction to the Homotopy Analysis Method. CRC Press, Boca Raton (2004)Google Scholar
  18. 18.
    He, J.H., Wu, X.H.: Variational Iteration Method: New Development and Applications. Comput. Math. Appl. 54, 881–894 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    He, J.H., Wu, G.C., Austin, F.: The Variational Iteration Method which Should be Followed. Nonlinear Sci. Lett. A 1, 1–30 (2010)Google Scholar
  20. 20.
    Al-Sawalha, M.M., Noorani, M.S.F., Hashim, I.: On Accuracy of Adomian Decomposition Method for Hyperchaotic Rössler System. Chaos Solitons and Fractals 40, 1801–1807 (2009)CrossRefzbMATHGoogle Scholar
  21. 21.
    He, J.H.: Homotopy Perturbation Method for Bifurcation of Nonlinear Problems. Int. J. Nonlinear Sci. Numer. Simu. 6, 207–208 (2005)Google Scholar
  22. 22.
    He, J.H.: Approximate Analytical Solution for Seepage Flow with Fractional Derivatives Inporous Media. Comput. Method Appl. Mech. Eng. 167, 57–68 (1998)CrossRefzbMATHGoogle Scholar
  23. 23.
    He, J.H.: Variational Iteration Method–A Kind of Non-linear Analytical Technique: Some Examples. Int. J. Nonlinear Mech. 34, 699–708 (1999)CrossRefzbMATHGoogle Scholar
  24. 24.
    He, J.H.: Variational Iteration Method for Autonomous Ordinary Differential Systems. Appl. Math. Comput. 114, 115–123 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    He, J.H.: Variational Iteration Method-Some Recent Results and New Interpretations. J. Comput. Appl. Math. 207, 3–17 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Wazwaz, A.M.: The Variational Iteration Method for Analytic Treatment for Linear and Nonlinear ODEs. Appl. Math. Comput. 212, 120–134 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Ali, A.H.A., Raslan, K.R.: Variational Iteration Method for Solving Partial Differential Equations with Variable Coefficients. Chaos Solitons and Fractals 40, 1520–1529 (2009)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Qi Wang
    • 1
  • Cui Guo
    • 2
  • Ruixiang Zeng
    • 1
  1. 1.School of Applied MathematicsGuangdong University of TechnologyGuangzhouChina
  2. 2.College of ScienceHarbin Engineering UniversityHarbinChina

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