Direct Solution of nth-Order IVPs by Homotopy Analysis Method
This article is part of Special Issue:
A. Sami Bataineh,
M. S. M. Noorani,
I. Hashim,
A. Sami Bataineh
Department of Mathematics, Irbid National University, Irbid 2600, Jordan inu.edu.jo
Search for more papers by this authorM. S. M. Noorani
Centre for Modelling and Data Analysis, School of Mathematical Sciences, Universiti Kebangsaan Malaysia (National University of Malaysia), 43600 Bangi Selangor, Malaysia ukm.my
Search for more papers by this authorCorresponding Author
I. Hashim
Centre for Modelling and Data Analysis, School of Mathematical Sciences, Universiti Kebangsaan Malaysia (National University of Malaysia), 43600 Bangi Selangor, Malaysia ukm.my
Search for more papers by this authorA. Sami Bataineh,
M. S. M. Noorani,
I. Hashim,
A. Sami Bataineh
Department of Mathematics, Irbid National University, Irbid 2600, Jordan inu.edu.jo
Search for more papers by this authorM. S. M. Noorani
Centre for Modelling and Data Analysis, School of Mathematical Sciences, Universiti Kebangsaan Malaysia (National University of Malaysia), 43600 Bangi Selangor, Malaysia ukm.my
Search for more papers by this authorCorresponding Author
I. Hashim
Centre for Modelling and Data Analysis, School of Mathematical Sciences, Universiti Kebangsaan Malaysia (National University of Malaysia), 43600 Bangi Selangor, Malaysia ukm.my
Search for more papers by this authorAcademic Editor: Tasawar Hayat
Abstract
Direct solution of a class of nth-order initial value problems (IVPs) is considered based on the homotopy analysis method (HAM). The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of the series solutions. The HAM gives approximate analytical solutions which are of comparable accuracy to the seven- and eight-order Runge-Kutta method (RK78).
References
- 1 Liao S.-J., The proposed homotopy analysis techniques for the solution of nonlinear problems, Ph.D. dissertation, 1992, Shanghai Jiao Tong University, Shanghai, China.
- 2 Liao S.-J., Beyond Perturbation: Introduction to the Homotopy Analysis Method, 2004, 2, Chapman & Hall/CRC, Boca Raton, Fla, USA, CRC Series: Modern Mechanics and Mathematics, MR2058313.
- 3 Liao S.-J., An approximate solution technique not depending on small parameters: a special example, International Journal of Non-Linear Mechanics. (1995) 30, no. 3, 371–380, https://doi.org/10.1016/0020-7462(94)00054-E, MR1336915, ZBL0837.76073.
- 4 Liao S.-J., 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. (1997) 32, no. 5, 815–822, https://doi.org/10.1016/S0020-7462(96)00101-1, MR1459007.
- 5 Liao S.-J., An explicit, totally analytic approximate solution for Blasius′ viscous flow problems, International Journal of Non-Linear Mechanics. (1999) 34, no. 4, 759–778, https://doi.org/10.1016/S0020-7462(98)00056-0, MR1688603.
- 6 Liao S.-J., On the homotopy analysis method for nonlinear problems, Applied Mathematics and Computation. (2004) 147, no. 2, 499–513, https://doi.org/10.1016/S0096-3003(02)00790-7, MR2012589, ZBL1086.35005.
- 7 Liao S.-J. and [email protected], Pop I., [email protected], Explicit analytic solution for similarity boundary layer equations, International Journal of Heat and Mass Transfer. (2004) 47, no. 1, 75–85, EID2-s2.0-0141907853, https://doi.org/10.1016/S0017-9310(03)00405-8, ZBL1045.76008.
- 8 Liao S.-J., Comparison between the homotopy analysis method and homotopy perturbation method, Applied Mathematics and Computation. (2005) 169, no. 2, 1186–1194, https://doi.org/10.1016/j.amc.2004.10.058, MR2174713, ZBL1082.65534.
- 9 Liao S.-J., A new branch of solutions of boundary-layer flows over an impermeable stretched plate, International Journal of Heat and Mass Transfer. (2005) 48, no. 12, 2529–2539, https://doi.org/10.1016/j.ijheatmasstransfer.2005.01.005, EID2-s2.0-17944369480.
- 10 Ayub M., Rasheed A., and Hayat T., Exact flow of a third grade fluid past a porous plate using homotopy analysis method, International Journal of Engineering Science. (2003) 41, no. 18, 2091–2103, https://doi.org/10.1016/S0020-7225(03)00207-6, MR1994304.
- 11 Hayat T., Khan M., and Asghar S., Homotopy analysis of MHD flows of an Oldroyd 8-constant fluid, Acta Mechanica. (2004) 168, no. 3-4, 213–232, https://doi.org/10.1007/s00707-004-0085-2, EID2-s2.0-2442526691, ZBL1063.76108.
- 12 Hayat T. and Khan M., Homotopy solutions for a generalized second-grade fluid past a porous plate, Nonlinear Dynamics. (2005) 42, no. 4, 395–405, https://doi.org/10.1007/s11071-005-7346-z, MR2190665, ZBL1094.76005.
- 13 Tan Y. and Abbasbandy S., Homotopy analysis method for quadratic Riccati differential equation, Communications in Nonlinear Science and Numerical Simulation. (2008) 13, no. 3, 539–546, https://doi.org/10.1016/j.cnsns.2006.06.006, EID2-s2.0-34848880798, ZBL1132.34305.
- 14 Abbasbandy S., The application of homotopy analysis method to nonlinear equations arising in heat transfer, Physics Letters A. (2006) 360, no. 1, 109–113, https://doi.org/10.1016/j.physleta.2006.07.065, MR2288118.
- 15 Abbasbandy S., The application of homotopy analysis method to solve a generalized Hirota-Satsuma coupled KdV equation, Physics Letters A. (2006) 15, 1–6.
- 16 Abbasbandy S., Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by means of the homotopy analysis method, Chemical Engineering Journal. (2008) 136, no. 2-3, 144–150, https://doi.org/10.1016/j.cej.2007.03.022, EID2-s2.0-38649139314.
- 17 Abbasbandy S. and Liao S.-J., A new modification of false position method based on homotopy analysis method, Applied Mathematics and Mechanics. (2008) 29, no. 2, 223–228, https://doi.org/10.1007/s10483-008-0209-z, MR2391551.
- 18 Bataineh A. S., Noorani M. S. M., and Hashim I., Modified homotopy analysis method for solving systems of second-order BVPs, Communications in Nonlinear Science and Numerical Simulation. (2009) 14, no. 2, 430–442, https://doi.org/10.1016/j.cnsns.2007.09.012, MR2458820.
- 19 Bataineh A. S., Noorani M. S. M., and Hashim I., Solving systems of ODEs by homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation. (2008) 13, no. 10, 2060–2070, https://doi.org/10.1016/j.cnsns.2007.05.026, MR2417577.
- 20 Bataineh A. S., Noorani M. S. M., and Hashim I., Solutions of time-dependent Emden-Fowler type equations by homotopy analysis method, Physics Letters A. (2007) 371, no. 1-2, 72–82, https://doi.org/10.1016/j.physleta.2007.05.094, EID2-s2.0-35348973440.
- 21 Bataineh A. S., Noorani M. S. M., and Hashim I., The homotopy analysis method for Cauchy reaction-diffusion problems, Physics Letters A. (2008) 372, no. 5, 613–618, https://doi.org/10.1016/j.physleta.2007.07.069, MR2378731.
- 22
Bataineh A. S.,
Noorani M. S. M., and
Hashim I., Series solution of the multispecies Lotka-Volterra equations by means of the homotopy analysis method, Differential Equations & Nonlinear Mechanics. (2008) 2008, 14, 816787, MR2425093, https://doi.org/10.1155/2008/816787, ZBL1160.34302.
10.1155/2008/816787 Google Scholar
- 23 Bataineh A. S., Noorani M. S. M., and Hashim I., Approximate analytical solutions of systems of PDEs by homotopy analysis method, Computers & Mathematics with Applications. (2008) 55, no. 12, 2913–2923, MR2401440, ZBL1142.65423.
- 24 Bataineh A. S., Noorani M. S. M., and Hashim I., Homotopy analysis method for singular IVPs of Emden-Fowler type, Communications in Nonlinear Science and Numerical Simulation. (2009) 14, no. 4, 1121–1131, https://doi.org/10.1016/j.cnsns.2008.02.004, MR2468944.
- 25 Hashim I., Abdulaziz O., and Momani S., Homotopy analysis method for fractional IVPs, Communications in Nonlinear Science and Numerical Simulation. (2009) 14, no. 3, 674–684, https://doi.org/10.1016/j.cnsns.2007.09.014, MR2449879.
- 26
Yabushita K.,
Yamashita M., and
Tsuboi K., An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method, Journal of Physics A. (2007) 40, no. 29, 8403–8416, https://doi.org/10.1088/1751-8113/40/29/015, MR2371242.
10.1088/1751-8113/40/29/015 Google Scholar
- 27 Cash J. R., A variable step Runge-Kutta-Nyström integrator for reversible systems of second order initial value problems, SIAM Journal on Scientific Computing. (2005) 26, no. 3, 963–978, https://doi.org/10.1137/S030601727, MR2126121, ZBL1121.65335.
- 28 Ramos H. and Vigo-Aguiar J., Variable-stepsize Chebyshev-type methods for the integration of second-order I.V.P.′s, Journal of Computational and Applied Mathematics. (2007) 204, no. 1, 102–113, https://doi.org/10.1016/j.cam.2006.04.032, MR2320339, ZBL1117.65106.
- 29
Ramos H. and
Vigo-Aguiar J., Variable stepsize Störmer-Cowell methods, Mathematical and Computer Modelling. (2005) 42, no. 7-8, 837–846, https://doi.org/10.1016/j.mcm.2005.09.011, MR2178513, ZBL1092.65060.
10.1016/j.mcm.2005.09.011 Google Scholar
- 30 Yahaya F., Hashim I., Ismail E. S., and Zulkifle A. K., Direct solutions of nth order initial value problems in decomposition series, International Journal of Nonlinear Sciences and Numerical Simulation. (2007) 8, no. 3, 385–392, EID2-s2.0-34548575151.
- 31
Chowdhury M. S. H. and
Hashim I., Direct solutions of nth-order initial value problems by homotopy-perturbation method, International Journal of Computer Mathematics. In presshttps://doi.org/10.1080/00207160802172224.
10.1080/00207160802172224 Google Scholar
- 32 He J.-H., Homotopy perturbation method: a new nonlinear analytical technique, Applied Mathematics and Computation. (2003) 135, no. 1, 73–79, https://doi.org/10.1016/S0096-3003(01)00312-5, MR1934316, ZBL1030.34013.
- 33 Genesio R. and Tesi A., Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems, Automatica. (1992) 28, no. 3, 531–548, https://doi.org/10.1016/0005-1098(92)90177-H, EID2-s2.0-0026866475, ZBL0765.93030.
