Comparison of Legendre polynomial approximation and variational iteration method for the solutions of general linear Fredholm integro-differential equations. (English) Zbl 1189.65307

Bildik, Necdet; Konuralp, Ali; Yalçınbaş, Salih

Comput. Math. Appl. 59, No. 6, 1909-1917 (2010).

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:

65R20	Numerical methods for integral equations
45J05	Integro-ordinary differential equations

Keywords:

variational iteration method; Legendre polynomial approximation; Fredholm integro-differential equation; integral equation

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, 2, 213-225, (2000) · Zbl 1018.65152
[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
[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
[4]	Yalçınbaş, S., Taylor polynomial solutions of nonlinear volterra – fredholm integral equations, Applied mathematics and computation, 127, 195-206, (2002) · Zbl 1025.45003
[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
[6]	Maleknejad, K.; Tavassoli Kajani, M., 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
[7]	Wang, S.-Q.; He, J.H., Variational iteration method for solving integro-differential equations, Physics letters A, 367, 3, 188-191, (2007) · Zbl 1209.65152
[8]	Xu, Lan, Variational iteration method for solving integral equations, Computers and mathematics with applications, 54, 7-8, 1071-1078, (2007) · Zbl 1141.65400
[9]	Ramos, J.I., On the variational iteration method and other iterative techniques for nonlinear differential equations, Applied mathematics and computation, 199, 1, 39-69, (2008) · Zbl 1142.65082
[10]	Shu-Qiang, Wang; Ji-Huan, He, Variational iteration method for solving integro-differential equations, Physics letters A, 367, 3, 188-191, (2007) · Zbl 1209.65152
[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
[12]	Yildirim, A., Applying he’s variational iteration method for solving differential-difference equation, Mathematical problems in engineering, 2008, 1-7, (2008), Article ID 869614 · Zbl 1155.65384
[13]	Yildirim, A., Variational iteration method for modified camassa – holm and degasperis – procesi equations, International journal for numerical methods in biomedical engineering, 26, 2, 266-272, (2010) · Zbl 1185.65193
[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, 2, 105-112, (2004) · Zbl 1401.65149
[16]	Bildik, N.; İ nç, M., Modified decomposition method for nonlinear volterra – fredholm integral equations, Chaos, solitons and fractals, 33, 308-313, (2007) · Zbl 1152.45301
[17]	Yalçınbaş, S.; Sezer, M.; Hilmi Sorkun, H., Legendre polynomial solutions of higher-order linear Fredholm integro-differential equations, Applied mathematics and computation, (2009) · Zbl 1162.65420
[18]	Yildirim, A., Solution of BVPs for fourth-order integro-differential equations by using homotopy perturbation method, Computers and mathematics with applications, 56, 12, 3175-3180, (2008) · Zbl 1165.65377
[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
[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
[21]	Hesaaraki, M.; Jalilian, Y., A numerical method for solving \(n\)th-order boundary-value problems, Applied mathematics and computation, 196, 2, 889-897, (2008) · Zbl 1135.65031