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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 763139, 14 pages
doi:10.1155/2012/763139
Research Article
Application of Homotopy Perturbation and Variational Iteration Methods for Fredholm Integrodifferential Equation of Fractional Order
1Faculty of Science and Technology, Universiti Sains Islam Malaysia, 71800 Nilai, Malaysia
2Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, 43400 UPM, Serdang, Selangor, Malaysia
Received 20 May 2012; Revised 4 September 2012; Accepted 4 September 2012
Academic Editor: Dumitru Bǎleanu
Copyright © 2012 Asma Ali Elbeleze et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Linked References
- 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 B.V., Amsterdam, The Netherlands, 2006.
- V. Lakshmikantham, S. Leela, and J. V. Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific, 2009.
- K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993.
- I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
- K. Diethelm and A. D. Freed, “On the solution of nonlinear fractional order differential equations used in the modeling of viscoelasticity,” in Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, F. Keil, W. Mackens, H. Voss, and J. Werther, Eds., pp. 217–224, Springer, Heidelberg, Germany, 1999.
- R. Metzler, W. Schick, H.-G. Kilian, and T. F. Nonnenmacher, “Relaxation in filled polymers: a fractional calculus approach,” Journal of Chemical Physics, vol. 103, no. 16, pp. 7180–7186, 1995. View at Scopus
- L. Gaul, P. Klein, and S. Kemple, “Damping description involving fractional operators,” Mechanical Systems and Signal Processing, vol. 5, no. 2, pp. 81–88, 1991. View at Scopus
- W. G. Glockle and T. F. Nonnenmacher, “A fractional calculus approach of self-similar protein dynamics,” Biophysical Journal, vol. 68, pp. 46–53, 1995.
- R. Hilfert, Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA, 2000. View at Publisher · View at Google Scholar
- R. P. Agarwal, M. Benchohra, and S. Hamani, “A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions,” Acta Applicandae Mathematicae, vol. 109, no. 3, pp. 973–1033, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- Z. Bai and H. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495–505, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus, vol. 3 of Series on Complexity, Nonlinearity and Chaos, World Scientific, Hackensack, NJ, USA, 2012. View at Publisher · View at Google Scholar
- P. K. Kythe and P. Puri, Computational Methods for Linear Integral Equations, Birkhäuser, Boston, Mass, USA, 2002. View at Publisher · View at Google Scholar
- Z.-B. Li and J.-H. He, “Fractional complex transform for fractional differential equations,” Mathematical & Computational Applications, vol. 15, no. 5, pp. 970–973, 2010. View at Zentralblatt MATH
- F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” in Fractals and Fractional Calculus in Continuum Mechanics, vol. 378 of CISM Courses and Lectures, pp. 291–348, Springer, Vienna, Austria, 1997. View at Zentralblatt MATH
- A.-M. Wazwaz, “A comparison study between the modified decomposition method and the traditional methods for solving nonlinear integral equations,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 1703–1712, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- 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. View at Scopus
- J.-H. He, “Homotopy perturbation method: a new nonlinear analytical technique,” Applied Mathematics and Computation, vol. 135, no. 1, pp. 73–79, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- J.-H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257–262, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- 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. View at Publisher · View at Google Scholar
- J. H. He, Non-Pertubation methods for strongly nonlinear problems [dissertation], Internet Gmbh, Berlin, Germany, 2006.
- J.-H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porous media,” Computer Methods in Applied Mechanics and Engineering, vol. 167, no. 1-2, pp. 57–68, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- J.-H. He, “A short remark on fractional variational iteration method,” Physics Letters A, vol. 375, no. 38, pp. 3362–3364, 2011. View at Publisher · View at Google Scholar
- J.-H. He, “Homotopy perturbation method with an auxiliary term,” Abstract and Applied Analysis, vol. 2012, Article ID 857612, 7 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- H. E. Ji-Huan, “A Note on the homotopy perturbation method,” Thermal Science, vol. 14, no. 2, pp. 565–568, 2010. View at Scopus
- 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. View at Publisher · View at Google Scholar
- S. Abbasbandy, “An approximation solution of a nonlinear equation with Riemann-Liouville's fractional derivatives by He's variational iteration method,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 53–58, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- O. Abdulaziz, I. Hashim, and S. Momani, “Application of homotopy-perturbation method to fractional IVPs,” Journal of Computational and Applied Mathematics, vol. 216, no. 2, pp. 574–584, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- A. Le Méhauté, L. Nivanen, and A. El Kaabouchi, “Contribution of Non Integer Integro-differential operators (NIDO) to the geometrical understanding of Riemann's conjecture-(I),” in Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and Its Applications (FDA '06), pp. 230–233, 2006. View at Scopus
- B. Ross, Fractional Calculus and Its Applications, vol. 457 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1975.
- U. Sumita, “The matrix Laguerre transform,” Applied Mathematics and Computation, vol. 15, no. 1, pp. 1–28, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- A. Kilicman 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- S.-J. Liao, “Boundary element method for general nonlinear differential operators,” Engineering Analysis with Boundary Elements, vol. 20, no. 2, pp. 91–99, 1997. View at Scopus
- A. Yıldırım, “Solution of BVPs for fourth-order integro-differential equations by using homotopy perturbation method,” Computers & Mathematics with Applications, vol. 56, no. 12, pp. 3175–3180, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- H. Jafari, A. Kadem, D. Baleanu, and T. Yilmaz, “Solutions of the fractional Davey-Stewartson Equations with variational iteration method,” Romanian Reports in Physics, vol. 64, no. 2, pp. 337–346, 2012.
- A. Kadem and A. Kiliçman, “The approximate solution of fractional Fredholm integrodifferential equations by variational iteration and homotopy perturbation methods,” Abstract and Applied Analysis, vol. 2012, Article ID 486193, 10 pages, 2012. View at Publisher · View at Google Scholar
- Y. Nawaz, “Variational iteration method and homotopy perturbation method for fourth-order fractional integro-differential equations,” Computers & Mathematics with Applications, vol. 61, no. 8, pp. 2330–2341, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- Z. Odibat and S. Momani, “The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics,” Computers & Mathematics with Applications, vol. 58, no. 11-12, pp. 2199–2208, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- N. H. Sweilam, “Fourth order integro-differential equations using variational iteration method,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 1086–1091, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH