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)
Introducing an efficient modification of the homotopy perturbation method by using Chebyshev polynomials. (English) Zbl 1236.65079
Summary: An efficient modification of the homotopy perturbation method is presented by using Chebyshev polynomials. Special attention is given to prove the convergence of the method. Some examples are given to verify the convergence hypothesis, and illustrate the efficiency and simplicity of the method. We compare our numerical results against the conventional numerical method, fourth-order Runge-Kutta method (RK4). From the numerical results obtained from these two methods we find that the proposed technique and RK4 are in excellent conformance. From the presented examples, we find that the proposed method can be applied to a wide class of linear and non-linear ordinary differential equations.

65L05Initial value problems for ODE (numerical methods)
34A34Nonlinear ODE and systems, general
65L06Multistep, Runge-Kutta, and extrapolation methods
65L20Stability and convergence of numerical methods for ODE
Full Text: DOI
[1] Babolian, E.; Hosseini, M. M.: A modified spectral method for numerical solution of odes with non-analytic solution, Appl math comput 132, 341-351 (2002) · Zbl 1024.65071 · doi:10.1016/S0096-3003(01)00197-7
[2] Bell WW. Special functions for scientists and engineers, New York, Toronto, Melbourne, 1967.
[3] Biazar, J.; Ghazvini, H.: Convergence of the HPM for partial differential equations, Nonlinear anal: real world appl 10, 2633-2640 (2009) · Zbl 1173.35395
[4] Biazar, J.; Ghazvini, H.: Numerical solution for special non-linear Fredholm integral equation by HPM, Appl math comput 195, 681-687 (2008) · Zbl 1132.65115 · doi:10.1016/j.amc.2007.05.015
[5] Ganji, D. D.: The application of he’s homotopy perturbation method to nonlinear equations arising in heat transfer, Phys lett A 355, 337-341 (2006) · Zbl 1255.80026
[6] Ghoreishi, M.; Ismail, A. I. B.M.: The HPM for nonlinear parabolic equation with non- local boundary conditions, Appl math sci 5, No. 3, 113-123 (2011) · Zbl 1235.35157
[7] He, J. H.: Homotopy perturbation method for solving boundary value problems, Phys lett A 350, 87-88 (2006) · Zbl 1195.65207 · doi:10.1016/j.physleta.2005.10.005
[8] He, J. H.: Application of homotopy perturbation method to nonlinear wave equations, Chaos, solitons fractals 26, 695-700 (2005) · Zbl 1072.35502 · doi:10.1016/j.chaos.2005.03.006
[9] He, J. H.: Homotopy perturbation technique, Comput methods appl mech eng 178, No. 3 -- 4, 257-262 (1999)
[10] Hosseini, M. M.: Numerical solution of ordinary differential equations with impulse solution, Appl math comput 163, 373-381 (2005) · Zbl 1060.65627 · doi:10.1016/j.amc.2004.02.017
[11] Hossein, A.; Milad, H.: An analytical technique for solving nonlinear heat transfer equations, Appl appl math: int J 5, No. 10, 1389-1399 (2010) · Zbl 1205.65282 · http://www.pvamu.edu/pages/7187.asp
[12] Khader, M. M.: On the numerical solutions for the fractional diffusion equation, Commun nonlinear sci numer simul 16, 2535-2542 (2011) · Zbl 1221.65263 · doi:10.1016/j.cnsns.2010.09.007
[13] Nemati, H.; Eskandari, Z.; Noori, F.; Ghorbanzadeh, M.: Application of the homotopy perturbation method to seven-order Sawada -- kotara equations, J eng sci technol rev 4, No. 1, 101-104 (2011)
[14] Sweilam, N. H.; Khader, M. M.: Exact solutions of some coupled nonlinear partial differential equations using the homotopy perturbation method, Comput math appl 58, 2134-2141 (2009) · Zbl 1189.65259 · doi:10.1016/j.camwa.2009.03.059
[15] Sweilam, N. H.; Khader, M. M.; Al-Bar, R. F.: Numerical studies for a multi-order fractional differential equation, Phys lett A 371, 26-33 (2007) · Zbl 1209.65116 · doi:10.1016/j.physleta.2007.06.016
[16] Sweilam, N. H.; Khader, M. M.: A Chebyshev pseudo-spectral method for solving fractional order integro-differential equations, Anzim 51, 464-475 (2010) · Zbl 1216.65187 · doi:10.1017/S1446181110000830
[17] Taghipour, R.: Application of homotopy perturbation method on some linear and nonlinear periodic equations, World appl sci J 10, No. 10, 1232-1235 (2010)