- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Recently Accepted Articles ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 365792, 18 pages
Homotopy Analysis Method for Second-Order Boundary Value Problems of Integrodifferential Equations
1Department of Mathematics, Faculty of Science, Al Balqa Applied University, Salt 19117, Jordan
2Department of Mathematics, Faculty of Science, University of Jordan, Amman 11942, Jordan
Received 1 April 2012; Accepted 19 June 2012
Academic Editor: Gabriele Bonanno
Copyright © 2012 Ahmad El-Ajou 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.
- R. P. Kanwal, Linear Integral Differential Equations: Theory and Technique, Birkhauser, Boston, Ga, USA, 2nd edition, 1996.
- F. Bloom, “Asymptotic bounds for solutions to a system of damped integro-differential equations of electromagnetic theory,” Journal of Mathematical Analysis and Applications, vol. 73, no. 2, pp. 524–542, 1980.
- K. Holmåker, “Global asymptotic stability for a stationary solution of a system of integro-differential equations describing the formation of liver zones,” SIAM Journal on Mathematical Analysis, vol. 24, no. 1, pp. 116–128, 1993.
- L. K. Forbes, S. Crozier, and D. M. Doddrell, “Calculating current densities and fields produced by shielded magnetic resonance imaging probes,” SIAM Journal on Applied Mathematics, vol. 57, no. 2, pp. 401–425, 1997.
- A. J. Jerri, Introduction to Integral Equations with Applications, John Wiley & Sons, New York, NY, USA, 2nd edition, 1999.
- R. P. Agarwal, “Boundary value problems for higher order integro-differential equations,” Nonlinear Analysis, vol. 7, no. 3, pp. 259–270, 1983.
- R. P. Agarwal, Boundary Value Problems for High Ordinary Differential Equations, World Scientific, Singapore, 1986.
- Z. Wang, L. Liu, and Y. Wu, “The unique solution of boundary value problems for nonlinear second-order integral-differential equations of mixed type in Banach spaces,” Computers & Mathematics with Applications, vol. 54, no. 9-10, pp. 1293–1301, 2007.
- A. Saadatmandi and M. Dehghan, “Numerical solution of the higher-order linear Fredholm integro-differential-difference equation with variable coefficients,” Computers & Mathematics with Applications, vol. 59, no. 8, pp. 2996–3004, 2010.
- J. Zhao and R. M. Corless, “Compact finite difference method for integro-differential equations,” Applied Mathematics & Computation, vol. 177, no. 1, pp. 271–288, 2006.
- Q. M. Al-Mdallal, “Monotone iterative sequences for nonlinear integro-differential equations of second order,” Nonlinear Analysis, vol. 12, no. 6, pp. 3665–3673, 2011.
- S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Methods, Chapman & Hall, Boca Raton, Fla, USA, 2003.
- S. Liao, “On the homotopy analysis method for nonlinear problems,” Applied Mathematics & Computation, vol. 147, no. 2, pp. 499–513, 2004.
- S. J. Liao, “Homotopy analysis method: a new analytic method for nonlinear problems,” Applied Mathematics & Mechanics, vol. 19, no. 10, pp. 957–962, 1998.
- S. J. Liao and K. F. Cheung, “Homotopy analysis of nonlinear progressive waves in deep water,” Journal of Engineering Mathematics, vol. 45, no. 2, pp. 105–116, 2003.
- S. Liao, “Series solutions of unsteady boundary-layer flows over a stretching flat plate,” Studies in Applied Mathematics, vol. 117, no. 3, pp. 239–263, 2006.
- W. Wu and S. J. Liao, “Solving solitary waves with discontinuity by means of the homotopy analysis method,” Chaos, Solitons and Fractals, vol. 26, no. 1, pp. 177–185, 2005.
- Q. Sun, “Solving the Klein-Gordon equation by means of the homotopy analysis method,” Applied Mathematics & Computation, vol. 169, no. 1, pp. 355–365, 2005.
- 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.
- M. Zurigat, S. Momani, Z. Odibat, and A. Alawneh, “The homotopy analysis method for handling systems of fractional differential equations,” Applied Mathematical Modelling, vol. 34, no. 1, pp. 24–35, 2010.
- Z. Odibat, S. Momani, and H. Xu, “A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations,” Applied Mathematical Modelling, vol. 34, no. 3, pp. 593–600, 2010.
- 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.
- M. Zurigat, S. Momani, and A. Alawneh, “Analytical approximate solutions of systems of fractional algebraic-differential equations by homotopy analysis method,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1227–1235, 2010.
- O. Abu Arqub and A. El-Ajou, “Solution of the fractional epidemic model by homotopy analysis method,” Journal of King Saud University. In press.
- H. Vosughi, E. Shivanian, and S. Abbasbandy, “A new analytical technique to solve Volterra's integral equations,” Mathematical Methods in the Applied Sciences, vol. 34, no. 10, pp. 1243–1253, 2011.
- H. S. Nik, S. Effati, and R. Buzhabadi, “Analytic-approximate solution for an integro-differential equation arising in oscillating magnetic fields using homotopy analysis method,” Iranian Journal of Optimization, vol. 2, no. 3, pp. 518–535, 2010.
- N. A. Khan, A. Ara, and M. Jamil, “Approximations of the nonlinear Volterra's population model by an efficient numerical method,” Mathematical Methods in the Applied Sciences, vol. 34, no. 14, pp. 1733–1738, 2011.
- M. Zurigat, S. Momani, and A. Alawneh, “Homotopy analysis method for systems of fractional integro-differential equations,” Neural, Parallel & Scientific Computations, vol. 17, no. 2, pp. 169–186, 2009.
- H. M. Jaradat, F. Awawdeh, and O. Alsayyed, “Series solution to the high-order integro-differential equations,” Analele Universităţii din Oradea—Fascicola Matematică, vol. 16, pp. 247–257, 2009, Tom XVI.
- J. Cang, Y. Tan, H. Xu, and S. J. Liao, “Series solutions of non-linear Riccati differential equations with fractional order,” Chaos, Solitons and Fractals, vol. 40, no. 1, pp. 1–9, 2009.
- F. M. Allan, “Derivation of the Adomian decomposition method using the homotopy analysis method,” Applied Mathematics & Computation, vol. 190, no. 1, pp. 6–14, 2007.