Variational iteration method for solving a generalized pantograph equation

Computers and Mathematics with Applications

Volumn 58, Issue 11-12, 2009, Pages 2190-2196

  Saadatmandi, Abbas ^a   Dehghan, Mehdi ^b  

^a UNIVERSITY OF KASHAN   (Iran)

^b AMIRKABIR UNIVERSITY OF TECHNOLOGY   (Iran)

Author keywords

Closed form solution; Delay differential equation; Pantograph equation; Symbolic computations; Variational iteration method

Indexed keywords

CLOSED FORM SOLUTION; DELAY DIFFERENTIAL EQUATION; PANTOGRAPH EQUATION; SYMBOLIC COMPUTATIONS; VARIATIONAL ITERATION METHOD;

DIFFERENTIAL EQUATIONS; LAGRANGE MULTIPLIERS; OSCILLATORS (MECHANICAL); PANTOGRAPHS;

ITERATIVE METHODS;

EID: 70350575223     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.camwa.2009.03.017     Document Type: Article

References (47)

1. Ockendon J.R., and Tayler A.B. The dynamics of a current collection system for an electric locomotive. Proc. R. Soc. Lond. Ser. A 322 (1971) 447-468
2. Ajello W.G., Freedman H.I., and Wu J. A model of stage structured population growth with density depended time delay. SIAM J. Appl. Math. 52 (1992) 855-869
3. Buhmann M.D., and Iserles A. Stability of the discretized pantograph differential equation. J. Math. Comput. 60 (1993) 575-589
4. Derfel G.A., and Vogl F. On the asymptotics of solutions of a class of linear functional-differential equations. European J. Appl. Math. 7 (1996) 511-518
5. Feldstein A., and Liu Y. On neutral functional differential equations with variable time delays. Math. Proc. Cambridge Philos. Soc. 124 (1998) 371-384
6. Sezer M., Yalcinbas S., and Sahin N. Approximate solution of multi-pantograph equation with variable coefficients. J. Comput. Appl. Math. 214 (2008) 406-416
7. Sezer M., and Akyuz-Dascioglu A. A Taylor method for numerical solution of generalized pantograph equations with linear functional argument. J. Comput. Appl. Math. 200 (2007) 217-225
8. Liu M.Z., and Li D. Properties of analytic solution and numerical solution of multi-pantograph equation. Appl. Math. Comput. 155 (2004) 853-871
9. Li. D., and Liu M.Z. Runge-Kutta methods for the multi-pantograph delay equation. Appl. Math. Comput. 163 (2005) 383-395
10. Keskin Y., Kurnaz A., Kiris M.E., and Oturanc G. Approximate solutions of generalized pantograph equations by the differential transform method. Int. J. Nonlinear Sci. Numer. Simul. 8 (2007) 159-164
11. Derfel G.A., and Iserles A. The pantograph equation in the complex plane. J. Math. Anal. Appl. 213 (1997) 117-132
12. Shakeri F., and Dehghan M. Solution of the delay differential equations via homotopy perturbation method. Math. Comput. Modelling 48 (2008) 486-498
13. Guglielmi N. Geometric proofs numerical stability for delay equations. IMA J. Anal. 21 (2001) 439-450
14. Hout I. On the stability of adaptations of Runge-Kutta methods to systems of delay differential equations. Appl. Numer. Math. 22 (1996) 237-250
15. Bellen A., Guglielmi N., and Torelli L. Asymptotic stability properties of θ-methods for pantograph equation. Appl. Numer. Math. 24 (1997) 279-293
16. Liu Y. Numerical investigation of the pantograph equation. Appl. Numer. Math. 24 (1997) 309-317
17. He J.H. Variational iteration method for delay differential equations. Commun. Nonlinear. Sci. Numer. Simul. 2 (1997) 235-236
18. He J.H. Approximate solution of nonlinear differential equations with convolution product non-linearities. Comput. Methods. Appl. Mech. Engrg. 167 (1998) 69-73
19. He J.H. Variational iteration method for autonomous ordinary differential systems. Appl. Math. Comput. 114 (2000) 115-123
20. Inokuti M., Sekine H., and Mura T. General use of the Lagrange multiplier in non-linear mathematical physics. Variational Methods in the Mechanics of Solids (1978), Pergamon Press, Oxford 156-162
21. He. J.H. Variational iteration method-Some recent results and new interpretation. J. Comput. Appl. Math. 207 (2007) 3-17
22. Tatari M., and Dehghan M. He's variational iteration method for computing a control parameter in a semi-linear inverse parabolic equation. Choas Solitons Fractals 33 (2007) 671-677
23. Dehghan M., and Tatari M. The use of He's variational iteration method for solving a Fokker-Planck equation. Phys. Scr. 74 (2006) 310-316
24. Tatari M., and Dehghan M. On the convergence of He's variational iteration method. J. Comput. Appl. Math. 207 (2007) 121-128
25. Abdou M.A., and Soliman A.A. Variational iteration method for solving Burger's and coupled Burger's equations. J. Comput. Appl. Math. 181 (2005) 245-251
26. Momani S., and Abuasad S. Application of He's variational iteration method to Helmholtz equation. Choas Solitons Fractals 27 (2006) 1119-1123
27. Dehghan M., and Shakeri F. Approximate solution of a differential equation arising in astrophysics using the variational iteration method. New Astronomy 13 (2008) 53-59
28. Shakeri F., and Dehghan M. Numerical solution of the Klein-Gordon equation via He's variational iteration method. Nonlinear Dynam. 51 (2008) 89-97
29. Shakeri F., and Dehghan M. Solution of a model describing biological species living together using the variational iteration method. Math. Comput. Modelling 48 (2008) 685-699
30. Dehghan M., and Shakeri F. Application of He's variational iteration method for solving the Cauchy reaction-diffusion problem. J. Comput. Appl. Math. 214 (2008) 435-446
31. Dehghan M., and Shakeri F. Numerical solution of a biological population model using He's variational iteration method. Comput. Math. Appl. 54 (2007) 1197-1209
32. He J.H. Some asymptotic methods for strongly nonlinear equations. Internat. J. Modern Phys. B 20 10 (2006) 1141-1199
33. He J.H. Non-Perturbative Methods for Strongly Nonlinear Problems (2006), Dissertation de-Verlag im Internet Gmb H, Berlin
34. Wazwaz A.M. A comparison between the variational iteration method and Adomian decomposition method. J. Comput. Appl. Math. 207 (2007) 129-136
35. Dehghan M. Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Math. Comput. Simul. 71 (2006) 16-30
36. He. J.H., and Wu X.H. Variational iteration method: New development and applications. Comput. Math. Appl. 54 (2007) 881-894
37. Evans D.J., and Raslan K.R. The Adomian decomposition method for solving delay differential equation. Int. J. Comput. Math. 82 1 (2005) 49-54
38. Muroya Y., Ishiwata E., and Brunner H. On the attainable order of collocation methods for pantograph integro-differential equations. J. Comput. Appl. Math. 152 (2003) 347-366
39. Dehghan M., and Shakeri F. The use of the decomposition procedure of Adomian for solving a delay differential equation arising in electrodynamics. Phys. Scripta 78 (2008) 111 Article No. 065004
40. Dehghan M., and Tatari M. Identifying an unknown function in a parabolic equation with overspecified data via He's variational iteration method. Chaos Solitons Fractals 36 (2008) 157-166
41. M. Dehghan, A. Saadatmandi, Variational iteration method for solving the wave equation subject to an integral conservation condition, Chaos Solitons Fractals (2008) (in press)
42. Tatari M., and Dehghan M. Solution of problems in calculus of variations via He's variational iteration method. Phys. Lett. A 362 (2007) 401-406
43. M. Dehghan, F. Shakeri, Solution of parabolic integro-differential equations arising in heat conduction in materials with memory via He's variational iteration technique, Comm. Numer. Methods Engrg. (2008) (in press)
44. Yousefi S.A., Lotfi A., and Dehghan M. He's variational iteration method for the nonlinear mixed Volterra-Fredholm integral equations. Comput. Math. Appl. 58 11-12 (2009) 2172-2176
45. Tatari M., and Dehghan M. Improvement of the He's variational iteration method for solving system of differential equations. Comput. Math. Appl. 58 11-12 (2009) 2160-2166
46. S.A. Yousefi, M. Dehghan, The use of He's variational iteration method for solving variational problems, Int. J. Comput. Math. (2008) (in press)
47. M. Dehghan, F. Shakeri, The numerical solution of the second Painleve equation, Numer. Methods Partial Differential Equations (in press)