Differential Equations and Nonlinear Mechanics
Volume 2009 (2009), Article ID 842094, 15 pages
doi:10.1155/2009/842094
Research Article
Direct Solution of nth-Order IVPs by Homotopy Analysis Method
1Department of Mathematics, Irbid National University, Irbid 2600, Jordan
2Centre for Modelling and Data Analysis, School of Mathematical Sciences, Universiti Kebangsaan Malaysia (National University of Malaysia), 43600 Bangi Selangor, Malaysia
Received 3 February 2009; Accepted 4 June 2009
Academic Editor: Tasawar Hayat
Copyright © 2009 A. Sami Bataineh 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
- S.-J. Liao, The proposed homotopy analysis techniques for the solution of nonlinear problems, Ph.D. dissertation, Shanghai Jiao Tong University, Shanghai, China, 1992.
- S.-J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, vol. 2 of CRC Series: Modern Mechanics and Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2004. View at MathSciNet
- 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 · View at MathSciNet
- S.-J. Liao, “A kind of approximate solution technique which does not depend upon small parameters. II. An application in fluid mechanics,” International Journal of Non-Linear Mechanics, vol. 32, no. 5, pp. 815–822, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- S.-J. Liao, “An explicit, totally analytic approximate solution for Blasius' viscous flow problems,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 759–778, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
- S.-J. Liao, “On the homotopy analysis method for nonlinear problems,” Applied Mathematics and Computation, vol. 147, no. 2, pp. 499–513, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- S.-J. Liao and I. Pop, “Explicit analytic solution for similarity boundary layer equations,” International Journal of Heat and Mass Transfer, vol. 47, no. 1, pp. 75–85, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- S.-J. Liao, “Comparison between the homotopy analysis method and homotopy perturbation method,” Applied Mathematics and Computation, vol. 169, no. 2, pp. 1186–1194, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- S.-J. Liao, “A new branch of solutions of boundary-layer flows over an impermeable stretched plate,” International Journal of Heat and Mass Transfer, vol. 48, no. 12, pp. 2529–2539, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- M. Ayub, A. Rasheed, and T. Hayat, “Exact flow of a third grade fluid past a porous plate using homotopy analysis method,” International Journal of Engineering Science, vol. 41, no. 18, pp. 2091–2103, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
- T. Hayat, M. Khan, and S. Asghar, “Homotopy analysis of MHD flows of an Oldroyd 8-constant fluid,” Acta Mechanica, vol. 168, no. 3-4, pp. 213–232, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- T. Hayat and M. Khan, “Homotopy solutions for a generalized second-grade fluid past a porous plate,” Nonlinear Dynamics, vol. 42, no. 4, pp. 395–405, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- Y. Tan and S. Abbasbandy, “Homotopy analysis method for quadratic Riccati differential equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 3, pp. 539–546, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- S. Abbasbandy, “The application of homotopy analysis method to nonlinear equations arising in heat transfer,” Physics Letters A, vol. 360, no. 1, pp. 109–113, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- S. Abbasbandy, “The application of homotopy analysis method to solve a generalized Hirota-Satsuma coupled KdV equation,” Physics Letters A, vol. 15, pp. 1–6, 2006.
- S. Abbasbandy, “Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by means of the homotopy analysis method,” Chemical Engineering Journal, vol. 136, no. 2-3, pp. 144–150, 2008. View at Publisher · View at Google Scholar
- S. Abbasbandy and S.-J. Liao, “A new modification of false position method based on homotopy analysis method,” Applied Mathematics and Mechanics, vol. 29, no. 2, pp. 223–228, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
- A. S. Bataineh, M. S. M. Noorani, and I. Hashim, “Modified homotopy analysis method for solving systems of second-order BVPs,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 2, pp. 430–442, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
- A. S. Bataineh, M. S. M. Noorani, and I. Hashim, “Solving systems of ODEs by homotopy analysis method,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 10, pp. 2060–2070, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- A. S. Bataineh, M. S. M. Noorani, and I. Hashim, “Solutions of time-dependent Emden-Fowler type equations by homotopy analysis method,” Physics Letters A, vol. 371, no. 1-2, pp. 72–82, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- A. S. Bataineh, M. S. M. Noorani, and I. Hashim, “The homotopy analysis method for Cauchy reaction-diffusion problems,” Physics Letters A, vol. 372, no. 5, pp. 613–618, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
- A. S. Bataineh, M. S. M. Noorani, and I. Hashim, “Series solution of the multispecies Lotka-Volterra equations by means of the homotopy analysis method,” Differential Equations & Nonlinear Mechanics, vol. 2008, Article ID 816787, 14 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- A. S. Bataineh, M. S. M. Noorani, and I. Hashim, “Approximate analytical solutions of systems of PDEs by homotopy analysis method,” Computers & Mathematics with Applications, vol. 55, no. 12, pp. 2913–2923, 2008. View at Zentralblatt MATH · View at MathSciNet
- A. S. Bataineh, M. S. M. Noorani, and I. Hashim, “Homotopy analysis method for singular IVPs of Emden-Fowler type,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 4, pp. 1121–1131, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
- I. Hashim, O. Abdulaziz, and S. Momani, “Homotopy analysis method for fractional IVPs,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 3, pp. 674–684, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- K. Yabushita, M. Yamashita, and K. Tsuboi, “An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method,” Journal of Physics A, vol. 40, no. 29, pp. 8403–8416, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
- J. R. Cash, “A variable step Runge-Kutta-Nyström integrator for reversible systems of second order initial value problems,” SIAM Journal on Scientific Computing, vol. 26, no. 3, pp. 963–978, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- H. Ramos and J. Vigo-Aguiar, “Variable-stepsize Chebyshev-type methods for the integration of second-order I.V.P.'s,” Journal of Computational and Applied Mathematics, vol. 204, no. 1, pp. 102–113, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- H. Ramos and J. Vigo-Aguiar, “Variable stepsize Störmer-Cowell methods,” Mathematical and Computer Modelling, vol. 42, no. 7-8, pp. 837–846, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- F. Yahaya, I. Hashim, E. S. Ismail, and A. K. Zulkifle, “Direct solutions of order initial value problems in decomposition series,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 3, pp. 385–392, 2007.
- M. S. H. Chowdhury and I. Hashim, “Direct solutions of -order initial value problems by homotopy-perturbation method,” International Journal of Computer Mathematics. In press. View at Publisher · View at Google Scholar
- 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 · View at MathSciNet
- R. Genesio and A. Tesi, “Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems,” Automatica, vol. 28, no. 3, pp. 531–548, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH