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)
The approximate solution of fractional Fredholm integrodifferential equations by variational iteration and homotopy perturbation methods. (English) Zbl 1242.65284
Summary: Variational iteration method and homotopy perturbation method are used to solve the fractional Fredholm integrodifferential equations with constant coefficients. The obtained results indicate that the method is efficient and also accurate.

65R20Integral equations (numerical methods)
65L99Numerical methods for ODE
45J05Integro-ordinary differential equations
Full Text: DOI
[1] R. Hilfert, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. · doi:10.1142/9789812817747
[2] K. B. Oldham and J. Spanier, The Fractional Calculus, vol. 198 of Mathematics in Science and Engineering, Academic Press, 1974. · Zbl 0292.26011
[3] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. · Zbl 0924.34008
[4] A. Kadem and D. Baleanu, “Fractional radiative transfer equation within Chebyshev spectral approach,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1865-1873, 2010. · Zbl 1189.35359 · doi:10.1016/j.camwa.2009.08.030
[5] A. Kadem and D. Baleanu, “Analytical method based on Walsh function combined with orthogonal polynomial for fractional transport equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 3, pp. 491-501, 2010. · Zbl 1221.45008 · doi:10.1016/j.cnsns.2009.05.024
[6] J. H. He, “Variational iteration method for delay differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 2, no. 4, pp. 235-236, 1997. · Zbl 0924.34063 · doi:10.1016/S1007-5704(97)90008-3
[7] S. J. Liao, “An approximate solution technique not depending on small parameters: a special example,” International Journal of Non-Linear Mechanics, vol. 30, no. 3, pp. 371-380, 1995. · Zbl 0837.76073 · doi:10.1016/0020-7462(94)00054-E
[8] S. J. Liao, “Boundary element method for general nonlinear differential operators,” Engineering Analysis with Boundary Elements, vol. 20, no. 2, pp. 91-99, 1997.
[9] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, vol. 149, Cambridge University Press, New York, NY, USA, 1991. · Zbl 0900.65350 · doi:10.1016/0010-4655(91)90165-H
[10] G. Adomian, Nonlinear Stochastic Systems Theory and Applications to Physics, vol. 46, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989. · Zbl 0754.60066 · doi:10.1016/0893-9659(89)90076-1
[11] Y. Chen and Z. Yan, “Weierstrass semi-rational expansion method and new doubly periodic solutions of the generalized Hirota-Satsuma coupled KdV system,” Applied Mathematics and Computation, vol. 177, no. 1, pp. 85-91, 2006. · Zbl 1094.65104 · doi:10.1016/j.amc.2005.10.037
[12] E. Fan, “Extended tanh-function method and its applications to nonlinear equations,” Physics Letters A, vol. 277, no. 4-5, pp. 212-218, 2000. · Zbl 1167.35331 · doi:10.1016/S0375-9601(00)00725-8
[13] Y. Luchko and R. Gorenflo, “An operational method for solving fractional differential equations with the Caputo derivatives,” Acta Mathematica Vietnamica, vol. 24, no. 2, pp. 207-233, 1999. · Zbl 0931.44003
[14] S. Momani and Z. Odibat, “Numerical approach to differential equations of fractional order,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 96-110, 2007. · Zbl 1119.65127 · doi:10.1016/j.cam.2006.07.015
[15] S. Kempfle and H. Beyer, “Global and causal solutions of fractional differential equations,” in Proceedings of the 2nd International Workshop on Transform Methods and Special Functions, pp. 210-216, Science Culture Technology Publishing, Varna, Bulgaria, 1996. · Zbl 0923.34007
[16] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993. · Zbl 0818.26003
[17] Yu. F. Luchko and H. M. Srivastava, “The exact solution of certain differential equations of fractional order by using operational calculus,” Computers & Mathematics with Applications, vol. 29, no. 8, pp. 73-85, 1995. · Zbl 0824.44011 · doi:10.1016/0898-1221(95)00031-S
[18] B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Springer, New York, NY, USA, 2003.
[19] J. H. He, “Variational iteration method-a kind of non-linear analytical technique: some examples,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 699-708, 1999. · Zbl 05137891
[20] J. H. He., S. K. Elagan, and Z. B. Li, “Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus,” Physics Letters A, vol. 376, no. 4, pp. 257-259, 2012. · Zbl 1255.26002
[21] A. Le Méhauté, A. El Kaabouchi, and L. Nivanen, “Contribution of non integer Integro-differential operators(NIDO) to the geometrical undersanding of Riemann’s conjecture-(I),” in Proceedings of the 2nd IFAC workshop on Fractional Differentiation and Its Applications, vol. 2, pp. 230-233, 2006. · Zbl 1210.11129
[22] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006. · Zbl 1092.45003
[23] A. Kılı\ccman and Z. A. A. Al Zhour, “Kronecker operational matrices for fractional calculus and some applications,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 250-265, 2007. · Zbl 1123.65063 · doi:10.1016/j.amc.2006.08.122
[24] J. H. He, “Homotopy perturbation method: a new nonlinear analytical technique,” Applied Mathematics and Computation, vol. 135, no. 1, pp. 73-79, 2003. · Zbl 1030.34013 · doi:10.1016/S0096-3003(01)00312-5
[25] J. H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257-262, 1999. · Zbl 0956.70017 · doi:10.1016/S0045-7825(99)00018-3
[26] J. H. He, “A coupling method of a homotopy technique and a perturbation technique for non-linear problems,” International Journal of Non-Linear Mechanics, vol. 35, no. 1, pp. 37-43, 2000. · Zbl 1068.74618 · doi:10.1016/S0020-7462(98)00085-7
[27] J. H. He, “Some asymptotic methods for strongly nonlinear equations,” International Journal of Modern Physics B, vol. 20, no. 10, pp. 1141-1199, 2006. · Zbl 1102.34039 · doi:10.1142/S0217979206033796
[28] J. H. He, Non-pertubative methods for strongly nonlinear problems, dissertation, GmbH, Berlin, Germany, 2006.
[29] Y. Nawaz, “Variational iteration method and homotopy perturbation method for fourth-order fractional integro-differential equations,” Computers and Mathematics with Applications, vol. 61, no. 8, pp. 2330-2341, 2011. · Zbl 1219.65081 · doi:10.1016/j.camwa.2010.10.004
[30] J. H. He, “A short remark on fractional variational iteration method,” Physics Letters A, vol. 375, no. 38, pp. 3362-3364, 2011. · Zbl 1252.49027 · doi:10.1016/j.physleta.2011.07.033
[31] H. E. Ji-Huan, “A Note on the homotopy perturbation method,” Thermal Science, vol. 14, no. 2, pp. 565-568, 2010.
[32] A. Yildirim, S. A. Sezer, and Y. Kaplan, “Numerical solutions of fourth-order fractional integro-differential equations,” Zeitschrift fur Naturforschung A, vol. 65, no. 12, pp. 1027-1032, 2010.
[33] S. Yüzbasi, N. Sahin, and A. Yildirim, “A collocation approach for solving high-order linear Fredholm-Volterra integro-differential equations,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 547-563, 2012.