Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-13T04:45:07.376Z Has data issue: false hasContentIssue false
  • English
  • Français

A NEW FORMULA FOR ADOMIAN POLYNOMIALS AND THE ANALYSIS OF ITS TRUNCATED SERIES SOLUTION FOR FRACTIONAL NON-DIFFERENTIABLE INITIAL VALUE PROBLEMS

Part of: Error analysis and interval analysis Real functions Equations and systems with constant coefficients

Published online by Cambridge University Press:  20 November 2013

M. M. KHADER
M. M. KHADER*
Affiliation:
Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt
*
mohamedmbd@yahoo.com
Rights & Permissions [Opens in a new window]

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.

A new formula for Adomian polynomials is introduced and applied to obtain truncated series solutions for fractional initial value problems with nondifferentiable functions. These kinds of equations contain a fractional single term which is examined using Jumarie fractional derivatives and fractional Taylor series for nondifferentiable functions. The property of nonlocality of these equations is examined, and the existence and uniqueness of solutions are discussed. Convergence and error analysis for the Adomian series solution are also studied. Numerical examples show the accuracy and efficiency of this formula for solving initial value problems for high-order fractional differential equations.

Keywords

Adomian polynomialsJumarie fractional derivativesnondifferentiable initial value problemsnonlocality propertyerror and convergence analysis

MSC classification

Secondary: 65K10: Optimization and variational techniques 65G99: None of the above, but in this section 26A33: Fractional derivatives and integrals 35E99: None of the above, but in this section 68U20: Simulation
Type
Research Article
Information
The ANZIAM Journal , Volume 55 , Issue 1 , July 2013 , pp. 69 - 92
Copyright
Copyright ©2013 Australian Mathematical Society 

References

Abassy, T. A., “Improved Adomian decomposition method”, Comput. Math. Appl. 59 (2010) 42–54; doi:10.1016/j.camwa.2009.06.009.CrossRefGoogle Scholar
Adomian, G., Stochastic systems (Academic Press, New York, 1983).Google Scholar
Adomian, G., Solving frontier problems of physics: the decomposition method (Kluwer Academic, Boston, MA, 1994).CrossRefGoogle Scholar
Almeida, R. and Torres, D. F. M., “Fractional variational calculus for nondifferentiable functions”, Comput. Math. Appl. 61 (2011) 3097–3104; doi:10.1016/j.camwa.2011.03.098.CrossRefGoogle Scholar
Behiry, S. H., Hashish, H., El-Kalla, I. L. and Elsaid, A., “A new algorithm for the decomposition solution of nonlinear differential equations”, Comput. Math. Appl. 54 (2007) 459–466; doi:10.1016/j.camwa.2006.12.027.CrossRefGoogle Scholar
El-Kalla, I. L., “Error estimate of the series solution to a class of nonlinear fractional differential equations”, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 1408–1413; doi:10.1016/j.cnsns.2010.05.030.CrossRefGoogle Scholar
Hosseini, M. M. and Nasabzadeh, H., “On the convergence of Adomian decomposition method”, Appl. Math. Comput. 182 (2006) 536–543; doi:10.1016/j.amc.2006.04.015.Google Scholar
Jumarie, G., “Modified Riemann–Liouville derivative and fractional Taylor series of nondifferentiable functions. Further results”, Comput. Math. Appl. 51 (2006) 1367–1376; doi:10.1016/j.camwa.2006.02.001.CrossRefGoogle Scholar
Jumarie, G., “New stochastic fractional models for Malthusian growth, the Poissonian birth process and optimal management of populations”, Math. Comput. Modelling 44 (2006) 231–254; doi:10.1016/j.mcm.2005.10.003.CrossRefGoogle Scholar
Jumarie, G., “Laplace’s transform of fractional order via the Mittag–Leffler function and modified Riemann–Liouville derivative”, Appl. Math. Lett. 22 (2009) 1659–1664; doi:10.1016/j.aml.2009.05.011.CrossRefGoogle Scholar
Jumarie, G., “Table of some basic fractional calculus formulae derived from a modified Riemann–Liouville derivative for nondifferentiable functions”, Appl. Math. Lett. 22 (2009) 378–385; doi:10.1016/j.aml.2008.06.003.CrossRefGoogle Scholar
Khader, M. M., “On the numerical solutions for the fractional diffusion equation”, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 2535–2542; doi:10.1016/j.cnsns.2010.09.007.CrossRefGoogle Scholar
Khader, M. M., “Introducing an efficient modification of the homotopy perturbation method by using Chebyshev polynomials”, Arab J. Math. Sci. 18 (2012) 61–71; doi:10.1016/j.ajmsc.2011.09.001.CrossRefGoogle Scholar
Khader, M. M., “Introducing an efficient modification of the variational iteration method by using Chebyshev polynomials”, Appl. Appl. Math. 7 (2012) 283–299; http://www.pvamu.edu/pages/7659.asp.Google Scholar
Khader, M. M., El Danaf, T. S. and Hendy, A. S., “Efficient spectral collocation method for solving multi-term fractional differential equations based on the generalized Laguerre polynomials”, J. Fract. Calc. Appl. 3 (13) (2012) 1–14.Google Scholar
Khader, M. M. and Hendy, A. S., “The approximate and exact solutions of the fractional-order delay differential equations using Legendre pseudospectral method”, Int. J. Pure Appl. Math. 74 (2012) 287–297; http://www.ijpam.eu/contents/2012-74-3/1/1.pdf.Google Scholar
Khader, M. M. and Hendy, A. S., “A numerical technique for solving fractional variational problems”, Math. Meth. Appl. Sci. 36 (2013) 1281–1289; doi:10.1002/mma.2681.CrossRefGoogle Scholar
Manuela, M., “Analysis of Adomian series solution to a class of nonlinear ordinary systems of Raman type”, Appl. Math. E-Notes 11 (2011) 50–60.Google Scholar
Miller, K. S. and Ross, B., An introduction to the fractional calculus and fractional differential equations (Wiley, New York, 1993).Google Scholar
Momani, S., “An explicit and numerical solutions of the fractional KdV equation”, Math. Comput. Simulation 70 (2005) 1110–1118; doi:10.1016/j.matcom.2005.05.001.CrossRefGoogle Scholar
Momani, S. and Odibat, Z., “Numerical approach to differential equations of fractional order”, J. Comput. Appl. Math. 207 (2007) 96–110; doi:10.1016/j.cam.2006.07.015.CrossRefGoogle Scholar
Odibat, Z. and Momani, S., “Application of variational iteration method to nonlinear differential equations of fractional order”, Int. J. Nonlinear Sci. Numer. Simul. 1 (2006) 15–27; doi:10.1515/IJNSNS.2006.7.1.27.Google Scholar
Podlubny, I., Fractional differential equations (Academic Press, New York, 1999).Google Scholar
Rach, R. and Duan, J. S., “Near field and far field approximations by the Adomian and asymptotic decomposition methods”, Appl. Math. Comput. 131 (2011) 5910–5922; doi:10.1016/j.amc.2010.12.093.Google Scholar
Saha, S. and Bera, R. K., “An approximate solution of nonlinear fractional differential equations by Adomian decomposition method”, J. Appl. Math. Comput. 167 (2005) 561–571; doi:10.1016/j.amc.2004.07.020.CrossRefGoogle Scholar
Shawagfeh, N. T., “Analytical approximate solutions for nonlinear fractional differential equations”, Appl. Math. Comput. 131 (2002) 517–529; doi:10.1016/S0096-3003(01)00167-9.Google Scholar
Sweilam, N. H. and Khader, M. M., “A Chebyshev pseudo-spectral method for solving fractional order integro-differential equations”, ANZIAM 51 (2010) 464–475; doi:10.1017/S1446181110000830.CrossRefGoogle Scholar
Sweilam, N. H., Khader, M. M. and Adel, M., “On the stability analysis of weighted average finite difference methods for fractional wave equation”, Fract. Differ. Calc. 2 (2012) 17–29; doi:10.7153/fdc-02-02.Google Scholar
Sweilam, N. H., Khader, M. M. and Al-Bar, R. F., “Numerical studies for a multi-order fractional differential equation”, Phys. Lett. A 371 (2007) 26–33; doi:10.1016/j.physleta.2007.06.016.CrossRefGoogle Scholar
Sweilam, N. H., Khader, M. M. and Mahdy, A. M. S., “Crank–Nicolson finite difference method for solving time-fractional diffusion equation”, J. Fract. Calc. Appl. 2 (2012) 1–9.Google Scholar
Sweilam, N. H., Khader, M. M. and Mahdy, A. M. S., “Numerical studies for fractional-order logistic differential equation with two different delays”, J. Appl. Math. 2012 (2012) 1–14; doi:10.1155/2012/764894.CrossRefGoogle Scholar
Sweilam, N. H., Khader, M. M. and Mahdy, A. M. S., “Numerical studies for solving fractional-order logistic equation”, Int. J. Pure Appl. Math. 78 (2012) 1199–1210; available at http://www.ijpam.eu/contents/2012-78-8/12/12.pdf.Google Scholar
Khader, M. M., “Numerical treatment for solving fractional Riccati differential equation”, J. Egyptian Math. Soc. 21 (2013) 32–37; doi:10.1016/j.joems.2012.09.005.CrossRefGoogle Scholar
Sweilam, N. H., Khader, M. M. and Nagy, A. M., “Numerical solution of two-sided space-fractional wave equation using finite difference method”, J. Comput. Appl. Math. 235 (2011) 2832–2841; doi:10.1016/j.cam.2010.12.002.CrossRefGoogle Scholar