zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Comparison of Legendre polynomial approximation and variational iteration method for the solutions of general linear Fredholm integro-differential equations. (English) Zbl 1189.65307
Summary: We show that the numerical solutions of linear Fredholm integro-differential equations obtained by using Legendre polynomials can also be found by using the variational iteration method. Furthermore the numerical solutions of the given problems which are solved by the variational iteration method obviously converge rapidly to exact solutions better than the Legendre polynomial technique. Additionally, although the powerful effect of the applied processes in Legendre polynomial approach arises in the situations where the initial approximation value is unknown, it is shown by the examples that the variational iteration method produces more certain solutions where the first initial function approximation value is estimated. In this paper, the Legendre polynomial approximation (LPA) and the variational iteration method (VIM) are implemented to obtain the solutions of the linear Fredholm integro-differential equations and the numerical solutions with respect to these methods are compared.
MSC:
65R20Integral equations (numerical methods)
45J05Integro-ordinary differential equations
References:
[1]Nas, S.; Yalçınbaş, S.; Sezer, M.: A Taylor polynomial approach for solving high-order linear Fredholm integro-differential equations, International journal for mathematical education in science and technology 31, No. 2, 213-225 (2000) · Zbl 1018.65152 · doi:10.1080/002073900287273
[2]Yalçınbaş, S.; Sezer, M.: The approximate solution of high-order linear Volterra–Fredholm integro-differential equations in terms of Taylor polynomials, Applied mathematics and computation 112, 291-308 (2000) · Zbl 1023.65147 · doi:10.1016/S0096-3003(99)00059-4
[3]Akyüz, A.; Sezer, M.: A Taylor polynomial approach for solving high-order linear Fredholm integro-differential equations in the most general form, International journal of computer mathematics, 527-539 (2007) · Zbl 1118.65129 · doi:10.1080/00207160701227848
[4]Yalçınbaş, S.: Taylor polynomial solutions of nonlinear Volterra–Fredholm integral equations, Applied mathematics and computation 127, 195-206 (2002) · Zbl 1025.45003 · doi:10.1016/S0096-3003(00)00165-X
[5]Streltsov, I. P.: Approximation of Chebyshev and Legendre polynomials on discrete point set to function interpolation and solving Fredholm integral equations, Computer physics communications 126, 178-181 (2000) · Zbl 0963.65143 · doi:10.1016/S0010-4655(99)00520-2
[6]Maleknejad, K.; Kajani, M. Tavassoli: Solving second kind integral equation by Galerkin methods with hybrid Legendre and block-pulse functions, Applied mathematics and computation 145, 623-629 (2003) · Zbl 1101.65323 · doi:10.1016/S0096-3003(03)00139-5
[7]Wang, S. -Q.; He, J. H.: Variational iteration method for solving integro-differential equations, Physics letters A 367, No. 3, 188-191 (2007) · Zbl 1209.65152 · doi:10.1016/j.physleta.2007.02.049
[8]Xu, Lan: Variational iteration method for solving integral equations, Computers and mathematics with applications 54, No. 7–8, 1071-1078 (2007) · Zbl 1141.65400 · doi:10.1016/j.camwa.2006.12.053
[9]Ramos, J. I.: On the variational iteration method and other iterative techniques for nonlinear differential equations, Applied mathematics and computation 199, No. 1, 39-69 (2008) · Zbl 1142.65082 · doi:10.1016/j.amc.2007.09.024
[10]Shu-Qiang, Wang; Ji-Huan, He: Variational iteration method for solving integro-differential equations, Physics letters A 367, No. 3, 188-191 (2007) · Zbl 1209.65152 · doi:10.1016/j.physleta.2007.02.049
[11]He, Ji-Huan; Wu, Xu-Hong: Variational iteration method: new development and applications, Computers and mathematics with applications 54, 881-894 (2007) · Zbl 1141.65372 · doi:10.1016/j.camwa.2006.12.083
[12]Yildirim, A.: Applying he’s variational iteration method for solving differential-difference equation, Mathematical problems in engineering 2008, 1-7 (2008) · Zbl 1155.65384 · doi:10.1155/2008/869614
[13]Yildirim, A.: Variational iteration method for modified Camassa–Holm and Degasperis–Procesi equations, International journal for numerical methods in biomedical engineering 26, No. 2, 266-272 (2010) · Zbl 1185.65193 · doi:10.1002/cnm.1154
[14]Yildirim, A.: Variational iteration method for inverse problem of diffusion equation, Communications in numerical methods in engineering (2009)
[15]El-Sayed, S. M.; Kaya, D.; Zarea, S.: The decomposition method applied to solve high-order linear Volterra–Fredholm integro-differential equations, International journal of nonlinear sciences and numerical simulation 5, No. 2, 105-112 (2004)
[16]Bildik, N.; Nç, M. İ: Modified decomposition method for nonlinear Volterra–Fredholm integral equations, Chaos, solitons and fractals 33, 308-313 (2007) · Zbl 1152.45301 · doi:10.1016/j.chaos.2005.12.058
[17]Yalçınbaş, S.; Sezer, M.; Sorkun, H. Hilmi: Legendre polynomial solutions of higher-order linear Fredholm integro-differential equations, Applied mathematics and computation (2009)
[18]Yildirim, A.: Solution of BVPs for fourth-order integro-differential equations by using homotopy perturbation method, Computers and mathematics with applications 56, No. 12, 3175-3180 (2008) · Zbl 1165.65377 · doi:10.1016/j.camwa.2008.07.020
[19]El-Mikkawy, M. E. A.; Cheon, G. S.: Combinatorial and hypergeometric identities via the Legendre polynomials-A computational approach, Applied mathematics and computation 166, 181-195 (2005) · Zbl 1073.65019 · doi:10.1016/j.amc.2004.04.066
[20]Elbarbary, E. M. E.: Legendre expansion method for the solution of the second-and fourth-order elliptic equations, Mathematics and computers in simulation 59, 389-399 (2002) · Zbl 1004.65120 · doi:10.1016/S0378-4754(01)00421-9
[21]Hesaaraki, M.; Jalilian, Y.: A numerical method for solving nth-order boundary-value problems, Applied mathematics and computation 196, No. 2, 889-897 (2008) · Zbl 1135.65031 · doi:10.1016/j.amc.2007.07.023