zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Variational iteration method for solving a generalized pantograph equation. (English) Zbl 1189.65172
Summary: The variational iteration method is applied to solve the generalized pantograph equation. This technique provides a sequence of functions which converges to the exact solution of the problem and is based on the use of Lagrange multipliers for identification of optimal value of a parameter in a functional. Employing this technique, it is possible to find the exact solution or an approximate solution of the problem. Some examples are given to demonstrate the validity and applicability of the method and a comparison is made with existing results.

65L99Numerical methods for ODE
Full Text: DOI
[1] Ockendon, J. R.; Tayler, A. B.: The dynamics of a current collection system for an electric locomotive, Proc. R. Soc. lond. Ser. A 322, 447-468 (1971)
[2] Ajello, W. G.; Freedman, H. I.; Wu, J.: A model of stage structured population growth with density depended time delay, SIAM J. Appl. math. 52, 855-869 (1992) · Zbl 0760.92018
[3] Buhmann, M. D.; Iserles, A.: Stability of the discretized pantograph differential equation, J. math. Comput. 60, 575-589 (1993) · Zbl 0774.34057 · doi:10.2307/2153103
[4] Derfel, G. A.; Vogl, F.: On the asymptotics of solutions of a class of linear functional-differential equations, European J. Appl. math. 7, 511-518 (1996) · Zbl 0859.34049 · doi:10.1017/S0956792500002527
[5] Feldstein, A.; Liu, Y.: On neutral functional differential equations with variable time delays, Math. proc. Cambridge philos. Soc. 124, 371-384 (1998) · Zbl 0913.34067 · doi:10.1017/S0305004198002497
[6] Sezer, M.; Yalcinbas, S.; Sahin, N.: Approximate solution of multi-pantograph equation with variable coefficients, J. comput. Appl. math. 214, 406-416 (2008) · Zbl 1135.65345 · doi:10.1016/j.cam.2007.03.024
[7] Sezer, M.; Akyuz-Dascioglu, A.: A Taylor method for numerical solution of generalized pantograph equations with linear functional argument, J. comput. Appl. math. 200, 217-225 (2007) · Zbl 1112.34063 · doi:10.1016/j.cam.2005.12.015
[8] Liu, M. Z.; Li, D.: Properties of analytic solution and numerical solution of multi-pantograph equation, Appl. math. Comput. 155, 853-871 (2004) · Zbl 1059.65060 · doi:10.1016/j.amc.2003.07.017
[9] Li., D.; Liu, M. Z.: Runge--Kutta methods for the multi-pantograph delay equation, Appl. math. Comput. 163, 383-395 (2005) · Zbl 1070.65060 · doi:10.1016/j.amc.2004.02.013
[10] Keskin, Y.; Kurnaz, A.; Kiris, M. E.; Oturanc, G.: Approximate solutions of generalized pantograph equations by the differential transform method, Int. J. Nonlinear sci. Numer. simul. 8, 159-164 (2007)
[11] Derfel, G. A.; Iserles, A.: The pantograph equation in the complex plane, J. math. Anal. appl. 213, 117-132 (1997) · Zbl 0891.34072 · doi:10.1006/jmaa.1997.5483
[12] Shakeri, F.; Dehghan, M.: Solution of the delay differential equations via homotopy perturbation method, Math. comput. Modelling 48, 486-498 (2008) · Zbl 1145.34353 · doi:10.1016/j.mcm.2007.09.016
[13] Guglielmi, N.: Geometric proofs numerical stability for delay equations, IMA J. Anal. 21, 439-450 (2001) · Zbl 0976.65077 · doi:10.1093/imanum/21.1.439
[14] Hout, I.: On the stability of adaptations of Runge--Kutta methods to systems of delay differential equations, Appl. numer. Math. 22, 237-250 (1996) · Zbl 0867.65045 · doi:10.1016/S0168-9274(96)00035-9
[15] Bellen, A.; Guglielmi, N.; Torelli, L.: Asymptotic stability properties of ${\theta}$-methods for pantograph equation, Appl. numer. Math. 24, 279-293 (1997) · Zbl 0878.65064 · doi:10.1016/S0168-9274(97)00026-3
[16] Liu, Y.: Numerical investigation of the pantograph equation, Appl. numer. Math. 24, 309-317 (1997) · Zbl 0878.65065 · doi:10.1016/S0168-9274(97)00028-7
[17] He, J. H.: Variational iteration method for delay differential equations, Commun. nonlinear. Sci. numer. Simul. 2, 235-236 (1997) · Zbl 0924.34063
[18] He, J. H.: Approximate solution of nonlinear differential equations with convolution product non-linearities, Comput. methods. Appl. mech. Engrg. 167, 69-73 (1998) · Zbl 0932.65143 · doi:10.1016/S0045-7825(98)00109-1
[19] He, J. H.: Variational iteration method for autonomous ordinary differential systems, Appl. math. Comput. 114, 115-123 (2000) · Zbl 1027.34009 · doi:10.1016/S0096-3003(99)00104-6
[20] Inokuti, M.; Sekine, H.; Mura, T.: General use of the Lagrange multiplier in non-linear mathematical physics, , 156-162 (1978)
[21] He., J. H.: Variational iteration method-some recent results and new interpretation, J. comput. Appl. math. 207, 3-17 (2007) · Zbl 1119.65049 · doi:10.1016/j.cam.2006.07.009
[22] Tatari, M.; Dehghan, M.: He’s variational iteration method for computing a control parameter in a semi-linear inverse parabolic equation, Choas solitons fractals 33, 671-677 (2007) · Zbl 1131.65084 · doi:10.1016/j.chaos.2006.01.059
[23] Dehghan, M.; Tatari, M.: The use of he’s variational iteration method for solving a Fokker--Planck equation, Phys. scr. 74, 310-316 (2006) · Zbl 1108.82033 · doi:10.1088/0031-8949/74/3/003
[24] Tatari, M.; Dehghan, M.: On the convergence of he’s variational iteration method, J. comput. Appl. math. 207, 121-128 (2007) · Zbl 1120.65112 · doi:10.1016/j.cam.2006.07.017
[25] Abdou, M. A.; Soliman, A. A.: Variational iteration method for solving burger’s and coupled burger’s equations, J. comput. Appl. math. 181, 245-251 (2005) · Zbl 1072.65127 · doi:10.1016/j.cam.2004.11.032
[26] Momani, S.; Abuasad, S.: Application of he’s variational iteration method to Helmholtz equation, Choas solitons fractals 27, 1119-1123 (2006) · Zbl 1086.65113 · doi:10.1016/j.chaos.2005.04.113
[27] Dehghan, M.; Shakeri, F.: Approximate solution of a differential equation arising in astrophysics using the variational iteration method, New astronomy 13, 53-59 (2008)
[28] Shakeri, F.; Dehghan, M.: Numerical solution of the Klein--Gordon equation via he’s variational iteration method, Nonlinear dynam. 51, 89-97 (2008) · Zbl 1179.81064 · doi:10.1007/s11071-006-9194-x
[29] Shakeri, F.; Dehghan, M.: Solution of a model describing biological species living together using the variational iteration method, Math. comput. Modelling 48, 685-699 (2008) · Zbl 1156.92332 · doi:10.1016/j.mcm.2007.11.012
[30] Dehghan, M.; Shakeri, F.: Application of he’s variational iteration method for solving the Cauchy reaction--diffusion problem, J. comput. Appl. math. 214, 435-446 (2008) · Zbl 1135.65381 · doi:10.1016/j.cam.2007.03.006
[31] Dehghan, M.; Shakeri, F.: Numerical solution of a biological population model using he’s variational iteration method, Comput. math. Appl. 54, 1197-1209 (2007) · Zbl 1137.92033 · doi:10.1016/j.camwa.2006.12.076
[32] He, J. H.: Some asymptotic methods for strongly nonlinear equations, Internat. J. Modern phys. B 20, No. 10, 1141-1199 (2006) · Zbl 1102.34039 · doi:10.1142/S0217979206033796
[33] He, J. H.: Non-perturbative methods for strongly nonlinear problems, (2006)
[34] Wazwaz, A. M.: A comparison between the variational iteration method and Adomian decomposition method, J. comput. Appl. math. 207, 129-136 (2007) · Zbl 1119.65103 · doi:10.1016/j.cam.2006.07.018
[35] Dehghan, M.: Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Math. comput. Simul. 71, 16-30 (2006) · Zbl 1089.65085 · doi:10.1016/j.matcom.2005.10.001
[36] He., J. H.; Wu, X. H.: Variational iteration method: new development and applications, Comput. math. Appl. 54, 881-894 (2007) · Zbl 1141.65372 · doi:10.1016/j.camwa.2006.12.083
[37] Evans, D. J.; Raslan, K. R.: The Adomian decomposition method for solving delay differential equation, Int. J. Comput. math. 82, No. 1, 49-54 (2005) · Zbl 1069.65074 · doi:10.1080/00207160412331286815
[38] Muroya, Y.; Ishiwata, E.; Brunner, H.: On the attainable order of collocation methods for pantograph integro-differential equations, J. comput. Appl. math. 152, 347-366 (2003) · Zbl 1023.65146 · doi:10.1016/S0377-0427(02)00716-1
[39] Dehghan, M.; Shakeri, F.: The use of the decomposition procedure of Adomian for solving a delay differential equation arising in electrodynamics, Phys. scripta 78, 111 (2008) · Zbl 1159.78319 · doi:10.1088/0031-8949/78/06/065004
[40] Dehghan, M.; Tatari, M.: Identifying an unknown function in a parabolic equation with overspecified data via he’s variational iteration method, Chaos solitons fractals 36, 157-166 (2008) · Zbl 1152.35390 · doi:10.1016/j.chaos.2006.06.023
[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) · Zbl 1198.65202
[42] Tatari, M.; Dehghan, M.: Solution of problems in calculus of variations via he’s variational iteration method, Phys. lett. A 362, 401-406 (2007) · Zbl 1197.65112 · doi:10.1016/j.physleta.2006.09.101
[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) · Zbl 1192.65158
[44] Yousefi, S. A.; Lotfi, A.; Dehghan, M.: He’s variational iteration method for the nonlinear mixed Volterra--Fredholm integral equations, Comput. math. Appl. 58, No. 11--12, 2172-2176 (2009) · Zbl 1189.65317 · doi:10.1016/j.camwa.2009.03.083
[45] Tatari, M.; Dehghan, M.: Improvement of the he’s variational iteration method for solving system of differential equations, Comput. math. Appl. 58, No. 11--12, 2160-2166 (2009) · Zbl 1189.65178 · doi:10.1016/j.camwa.2009.03.081
[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) · Zbl 1191.65078
[47] M. Dehghan, F. Shakeri, The numerical solution of the second Painleve equation, Numer. Methods Partial Differential Equations (in press) · Zbl 1172.65037