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)
Solutions of singular IVPs of Lane-Emden type by the variational iteration method. (English) Zbl 1162.34005
The authors consider the following general nonlinear differential equation $Lu(x)+ Nu(x)= g(x)$, where $L$ is a linear operator, $N$ is a nonlinear operator and $g(x)$ is a known analytic function. They employ the variational iteration method obtaining the successive approximation $u_{n+1}$ by the formula $$u_{n+1}= u_n(x)+ \int^x_0 \lambda(Lu_n(\xi)+ N\widetilde u_n(\xi)- g(\xi))\,d\xi,$$ $n\ge 0$, where $\lambda$ is a general Lagrange multiplier and $\widetilde u_n$ is considered as a restricted variation, namely $\delta\widetilde u_n= 0$.

34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
34A45Theoretical approximation of solutions of ODE
Full Text: DOI
[1] Chandrasekhar, S.: Introduction to the study of stellar structure. (1967) · Zbl 0149.24301
[2] Davis, H. T.: Introduction to nonlinear differential and integral equations. (1962) · Zbl 0106.28904
[3] Richardson, O. U.: The emission of electricity from hot bodies. (1921) · Zbl 48.0118.05
[4] Bender, C. M.; Milton, K. A.; Pinsky, S. S.; Simmons, L. M.: A new perturbation approach to nonlinear problems. J. math. Phys. 30, 1447-1455 (1989) · Zbl 0684.34008
[5] Shawagfeh, N. T.: Non-perturbative approximate solution for Lane--Emden equation. J. math. Phys. 34, No. 9, 4364-4369 (1993) · Zbl 0780.34007
[6] Wazwaz, A-M.: A new algorithm for solving differential equations of Lane--Emden type. Appl. math. Comput. 118, 287-310 (2001) · Zbl 1023.65067
[7] Wazwaz, A-M: A new method for solving singular value problems in the second order ordinary differential equations. Appl. math. Comput. 128, 45-57 (2001)
[8] Nouh, M. I.: Accelerated power series solution of polytropic and isothermal gas spheres. New astron. 9, 467-473 (2004)
[9] Mandelzweig, V. B.; Tabakin, F.: Quasilinearization approach to nonlinear problems in physics with application to nonlinear odes. Comput. phys. Commun. 141, 268-281 (2001) · Zbl 0991.65065
[10] Ramos, J. I.: Linearization method in classical and quantum mechanics. Comput phys. Commun. 153, 199-208 (2003) · Zbl 1196.81114
[11] Bozkhov, Y.; Martins, A. C. G.: Lie point symmetries and exact solutions of quasilinear differential equations with critical exponents. Nonlinear anal. 57, 773-793 (2004) · Zbl 1061.34030
[12] Momoniat, E.; Harley, C.: Approximate implicit solution of a Lane--Emden equation. New astron. 11, 520-526 (2006)
[13] Goenner, H.; Havas, P.: Exact solutions of the generalized Lane--Emden equation. J. math. Phys. 41, 7029-7042 (2000) · Zbl 1009.34002
[14] Liao, S.: A new analytic algorithm of Lane--Emden type equations. Appl. math. Comput. 142, 1-16 (2003) · Zbl 1022.65078
[15] He, J. H.: Variational approach to the Lane--Emden equation. Appl. math. Comput. 143, 539-541 (2003) · Zbl 1022.65076
[16] Ramos, J. I.: Series approach to the Lane--Emden equation and comparison with the homotopy perturbation method. Chaos solitons fractals (2006)
[17] Öziş, T.; Yıldırım, A.: Solutions of singular ivp’s of Lane--Emden type by homotopy pertutbation method. Phys. lett. A 369, 70-76 (2007) · Zbl 1209.65120
[18] Chowdhury, M. S. H.; Hashim, I.: Solutions of a class of singular second-order ivps by homotopy-perturbation method. Phys. lett. A. (2007) · Zbl 1203.65124
[19] Dehghan, M.; Shakeri, F.: Approximate solution of a differential equation arising in astrophysics using the variational iteration method. New astron. 13, 53-59 (2008)
[20] He, J. H.: Variational iteration method -- A kind of non-linear analytical technique: some examples. Int. J. Nonlinear mech. 34, No. 4, 699-708 (1999) · Zbl 05137891
[21] He, J. H.: Some asymptotic methods for strongly nonlinear equations. Int. J. Mod. phys. B 20, No. 10, 1141-1199 (2006) · Zbl 1102.34039
[22] J.H. He, Variational iteration method--Some recent results and new interpretations, J. Comput. Appl. Math. doi:10.1016/j.cam.2006.07.009 · Zbl 1119.65049
[23] J.H He, Non-Perturbative Methods for Strongly nonlinear Problems, Dissertation, de-verlag im Internet GmbH, Berlin, 2006
[24] He, J. H.; Wu, X. H.: Construction of solitary solution and Compton-like solution by variational iteration method. Chaos solitons fractals 29, 108-113 (2006) · Zbl 1147.35338
[25] Bildik, N.; Konuralp, A.: The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations. Int. J. Nonlinear sci. Numer. simul. 7, 65-70 (2006) · Zbl 1115.65365
[26] Odibat, Z. M.; Momani, S.: Application of variational iteration method to nonlinear differential equations of fractional order. Int. J. Nonlinear sci. Numer. simul. 7, 27-34 (2006) · Zbl 05675858
[27] Momani, S.; Abuasad, S.: Application of he’s variational iteration method to helmotz equation. Chaos solitons fractals 27, 1119-1123 (2006) · Zbl 1086.65113
[28] Öziş, T.; Yıldırım, A.: A study of nonlinear oscillators with u 1/3 force by he’s variational iteration method. J. sound vib. 306, No. 1--2, 372-376 (2007) · Zbl 1242.74214
[29] Momani, S.; Odibat, Z. M.: Numerical comparison of methods for solving linear differential equations of fractional order. Chaos solitons fractals 31, 1248-1255 (2007) · Zbl 1137.65450
[30] Tari, H.; Ganji, D. D.; Rostamian, M.: Approximate solutions of $K(2,2)$, KdV and modified KdV equations by variational iteration method, homotopy perturbation method and homotopy analysis method. Int. J. Nonlinear sci. Numer. simul. 8, No. 2, 203-210 (2007)
[31] Sweilam, N. H.; Khader, M. M.: Variational iteration method for one dimensional nonlinear thermoelasticity. Chaos solitons fractals 32, 145-149 (2007) · Zbl 1131.74018
[32] Yusufoğlu, E.: Variational iteration method for construction of some compact and noncompact structures of Klein-Gordon equations. Int. J. Nonlinear sci. Numer. 8, No. 2, 153-158 (2007)
[33] Özer, H.: Application of the variational iteration method to thin circular plates. Int. J. Nonlinear sci. Numer. simul. 9, No. 1, 27-32 (2008)