Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-05T08:41:26.206Z Has data issue: false hasContentIssue false
  • English
  • Français

Treatment for third-order nonlinear differential equations based on the Adomian decomposition method

Part of: Functional-differential and differential-difference equations

Published online by Cambridge University Press:  01 April 2017

Xueqin Lv  and
Jianfang Gao
Xueqin Lv
Affiliation:
School of Mathematics and Sciences, Harbin Normal University, Harbin, Heilongjiang, 150025, China email [email protected]
Jianfang Gao
Affiliation:
School of Mathematics and Sciences, Harbin Normal University, Harbin, Heilongjiang, 150025, China email [email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The Adomian decomposition method (ADM) is an efficient method for solving linear and nonlinear ordinary differential equations, differential algebraic equations, partial differential equations, stochastic differential equations, and integral equations. Based on the ADM, a new analytical and numerical treatment is introduced in this research for third-order boundary-value problems. The effectiveness of the proposed approach is verified by numerical examples.

MSC classification

Primary: 34K10: Boundary value problems
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 20 , Issue 1 , 2017 , pp. 1 - 10
Copyright
© The Author(s) 2017 

References

Adomian, G., Solving frontier problems of physics: the decomposition method (Kluwer Academic, Boston, MA, 1994).Google Scholar
Adomian, G., ‘Explicit solutions of nonlinear partial differential equations’, Appl. Math. Comput. 88 (1997) 117–126.Google Scholar
Adomian, G., ‘Solutions of nonlinear P.D.E.’, Appl. Math. Lett. 11 (1998) no. 3, 121–123.Google Scholar
Al-Hayani, W. and Casasus, L., ‘The Adomian decomposition method in turning point problems’, Appl. Math. Comput. 177 (2005) 187–203.CrossRefGoogle Scholar
Bougoffa, L. and Rach, R. C., ‘Solving nonlocal boundary value problems for first- and second-order differential equations by the Adomian decomposition method’, Kybernetes 42 (2013) no. 4, 641–664.Google Scholar
Dehghan, M., ‘Application of the Adomian decomposition method for two-dimensional parabolic equation subject to nonstandard boundary specifications’, Appl. Math. Comput. 157 (2004) 549–560.Google Scholar
Dib, A., Haiahem, A. and Bou-said, B., ‘An analytical solution of the MHD Jeffery–Hamel flow by the modified Adomian decomposition method’, Comput. Fluids 102 (2014) 111–115.Google Scholar
Duan, J. S. and Rach, R. C., ‘A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations’, Appl. Math. Comput. 218 (2011) no. 8, 4090–4118.Google Scholar
Ebaid, A., ‘A new numerical solution for the MHD peristaltic flow of a bio-fluid with variable viscosity in a circular cylindrical tube via Adomian decomposition method’, Phys. Lett. A 372 (2008) 5321–5328.Google Scholar
Ebaid, A., ‘Exact solutions for the generalized Klein–Gordon equation via a transformation and exp-function method and comparison with Adomian’s method’, J. Comput. Appl. Math. 223 (2009) 278–290.CrossRefGoogle Scholar
Ebaid, A., ‘A new analytical and numerical treatment for singular two-point boundary value problems via the Adomian decomposition method’, J. Comput. Appl. Math. 235 (2011) no. 8, 1914–1924.Google Scholar
Eldabe, N. T., Elghazy, E. M. and Ebaid, A., ‘Closed form solution to a second order boundary value problem and its application in fluid mechanics’, Phys. Lett. A 363 (2007) 257–259.Google Scholar
El-Kalla, I. L., ‘Convergence of Adomian’s method applied to a class of Volterra type integro-differential equations’, Int. J. Differ. Equ. Appl. 10 (2005) no. 2, 225–234.Google Scholar
Ewing, R. E. and Lin, T., ‘A class of parameter estimation techniques for fluid flow in porous media’, Adv. Water Resour. 14 (1991) 89–97.Google Scholar
Formaggia, L., Nobile, F., Quarteroni, A. and Veneziani, A., ‘Multiscale modelling of the circulatory system: a preliminary analysis’, Comput. Vis. Sci. 2 (1999) 75–83.Google Scholar
Geng, F., ‘A novel method for nonlinear two-point boundary value problems: combination of ADM and RKM’, Appl. Math. Comput. 217 (2011) 4676–4681.Google Scholar
Guo, L.-J., ‘Existence of positive solutions for nonlinear third-order three-point boundary value problems’, Appl. Math. Comput. 68 (2008) 3151–3158.Google Scholar
Hashim, I., ‘Adomian decomposition method for solving BVPs for fourth-order integro-differential equations’, Appl. Math. Comput. 193 (2006) 658–664.Google Scholar
Hopkins, B., ‘Third-order boundary value problems with sign-changing solutions’, Appl. Math. Comput. 67 (2007) 126–137.Google Scholar
Lesnic, D., ‘A computational algebraic investigation of the decomposition method for time-dependent problems’, Appl. Math. Comput. 119 (2001) 197–206.Google Scholar
Lesnic, D. and Elliott, L., ‘The decomposition approach to inverse heat conduction’, J. Math. Anal. Appl. 232 (1999) no. 82, 82–98.CrossRefGoogle Scholar
Li, Y., ‘Positive periodic solutions for fully third-order ordinary differential equations’, Appl. Math. Comput. 59 (2010) 3464–3471.Google Scholar
Lv, X. and Cui, M., ‘Solving a singular system of two nonlinear ODEs’, Appl. Math. Comput. 198 (2008) 534–543.Google Scholar
Lv, X. and Cui, M., ‘Existence and numerical method for nonlinear third-order boundary-value problem in the reproducing kernel space’, Bound. Value Probl. 2010 (2010) 1–13.Google Scholar
Mehrkanoon, S., ‘A direct variable step block multistep method for solving general third-order ODEs’, Numer. Algorithms 57 (2011) 53–66.Google Scholar
Shawagfeh, N. T., ‘Analytic approximate solution for a nonlinear oscillator equation’, Comput. Math. Appl. 31 (1996) 135–141.Google Scholar
Shi, P., ‘Weak solution to evolution problem with a nonlocal constraint’, SIAM J. Anal. 24 (1993) 46–58.Google Scholar
Wazwaz, A. M., ‘A reliable modification of Adomian’s decomposition method’, Appl. Math. Comput. 102 (1999) 77–96.Google Scholar
Wazwaz, A. M., ‘A reliable algorithm for obtaining positive solutions for nonlinear boundary value problems’, Math. Comput. Simulation 41 (2001) 1237–1244.Google Scholar
Wazwaz, A. M., ‘The modified decomposition method applied to unsteady flow of gas through a porous medium’, Appl. Math. Comput. 118 (2001) 123–132.Google Scholar
Wazwaz, A. M., ‘A reliable algorithm for solving boundary value problems for higher-order integro-differential equations’, Appl. Math. Comput. 118 (2001) 327–342.Google Scholar
Wazwaz, A. M., ‘The numerical solution of fifth-order boundary value problems by the decomposition method’, Appl. Math. Comput. 136 (2001) 259–270.Google Scholar