Skip to main content
SpringerLink
Log in

Numerical solutions of the initial value problem for fractional differential equations by modification of the Adomian decomposition method

  • Research Paper
  • Published:
Fractional Calculus and Applied Analysis Aims and scope Submit manuscript

Abstract

In this paper, we extend a reliable modification of the Adomian decomposition method presented in [34] for solving initial value problem for fractional differential equations.

In order to confirm the applicability and the advantages of our approach, we consider some illustrative examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. Adomian, R. Rach, Inversion of nonlinear stochastic operators. J. Math. Anal. Appl. 91 (1983), 39–46.

    Article  MATH  MathSciNet  Google Scholar 

  2. G. Adomian, Stochastic Systems. Academic Publ., New York, 1983.

    MATH  Google Scholar 

  3. G. Adomian, Nonlinear Stochastic Operator Equations. Academic Publ., Orlando, 1986.

    MATH  Google Scholar 

  4. G. Adomian, Nonlinear Stochastic Systems Theory and Applications to Physics. Kluwer, Academic Publ., Dordrecht, 1989.

    Book  MATH  Google Scholar 

  5. G. Adomian, R. Rach, R. Meyers, An efficient methodology for the physical sciences. Kybernetes. 20 (1991), 24–34.

    Article  MATH  MathSciNet  Google Scholar 

  6. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method. Kluwer, Academic Publ., Dordrecht, 1994.

    Book  MATH  Google Scholar 

  7. N. Bellomo, R. Riganti, Nonlinear Stochastic System Analysis in Physics and Mechanics. World Scientific, Singapore and River Edge, NJ, 1987.

    Book  Google Scholar 

  8. V. Daftardar-Gejji, H. Jafari, Adomian decomposition: a tool for solving a system of fractional differential equations. J. Math. Anal. Appl. 301 (2005), 508–518.

    Article  MATH  MathSciNet  Google Scholar 

  9. V. Daftardar-Gejji, H. Jafari, An iterative method for solving nonlinear functional equations. J. Math. Anal. Appl. 316 (2006), 753–763.

    Article  MATH  MathSciNet  Google Scholar 

  10. M. Dehghan, J. Manafian, A. Saadatmandi, Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numer. Methods Partial Differential Equations. 26 (2010), 448–479.

    MATH  MathSciNet  Google Scholar 

  11. K. Diethelm, N.J. Ford, A.D. Freed, Detailed error analysis for a fractional Adams method. Numer. Algorithms. 36 (2004), 31–52.

    Article  MATH  MathSciNet  Google Scholar 

  12. E.H. Doha, A.H. Bhrawy, S.S. Ezz-Eldien, Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations. Appl. Math. Model. 35 (2011), 5662–5672.

    Article  MATH  MathSciNet  Google Scholar 

  13. E.H. Doha, A.H. Bhrawy, S.S. Ezz-Eldien, A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order. Comput. Math. Appl. 62 (2011), 2364–2373.

    Article  MATH  MathSciNet  Google Scholar 

  14. J.S. Duan, Z.H. Liu, F.K. Zhang, T. Chaolu, Analytic solution and numerical solution to endolymph equation using fractional derivative. Ann. Differential Equations 24 (2008), 9–12.

    MATH  MathSciNet  Google Scholar 

  15. S. Esmaeili, M. Shamsi, Y. Luchko, Numerical solution of fractional differential equations with a collocation method based on Muntz polynomials. Comput. Math. Appl. 62 (2011), 918–929.

    Article  MATH  MathSciNet  Google Scholar 

  16. I. Hashim, O. Abdulaziz, S. Momani, Homotopy analysis method for fractional IVPs. Commun. Nonlinear Sci. Numer. Simul. 14 (2009), 674–684.

    Article  MATH  MathSciNet  Google Scholar 

  17. A.A. Hemeda, Variational iteration method for solving wave equation. Comput. Math. Appl. 56 (2008), 1948–1953.

    Article  MATH  MathSciNet  Google Scholar 

  18. A.A. Hemeda, Variational iteration method for solving nonlinear partial differential equations. Chaos Solitons Fractals 39 (2009), 1297–1303.

    Article  MATH  MathSciNet  Google Scholar 

  19. A.A. Hemeda, Variational iteration method for solving nonlinear coupled equations in 2-dimensional space in fluid mechanics. Int. J. Contemp. Math. Sci. 7 (2012), 1839–1852.

    MATH  MathSciNet  Google Scholar 

  20. A.A. Hemeda, Homotopy perturbation method for solving partial differential equations of fractional order. Int. J. Math. Anal. 6 (2012), 2431–2448.

    MATH  MathSciNet  Google Scholar 

  21. A.A. Hemeda, Homotopy perturbation method for solving systems of nonlinear coupled equations. Appl. Math. Sci. 6 (2012), 4787–4800.

    MATH  MathSciNet  Google Scholar 

  22. A.A. Hemeda, New iterative method: application to nthorder integro differential equations. Int. Math. Forum 7 (2012), 2317–2332.

    MATH  MathSciNet  Google Scholar 

  23. A.A. Hemeda, Formulation and solution of nth-order derivative fuzzy integro differential equation using new iterative method with a reliable algorithm. J. Appl. Math. 2012 (2012), 1–17.

    Article  MathSciNet  Google Scholar 

  24. A.A. Hemeda, New iterative method: An application for solving fractional physical differential equations. Abstr. Appl. Anal. 2013 (2013), 1–9.

    MathSciNet  Google Scholar 

  25. H. Jafari, V. Daftardar-Gejji, Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method. Appl. Math. Comput. 180 (2006), 700–706.

    Article  MATH  MathSciNet  Google Scholar 

  26. H. Jafari, V. Daftardar-Gejji, Solving a system of nonlinear fractional differential equations using Adomian decomposition. J. Comput. Appl. Math. 196 (2006), 644–651.

    Article  MATH  MathSciNet  Google Scholar 

  27. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, 2006.

    MATH  Google Scholar 

  28. C. Li, Y. Wang, Numerical algorithm based on Adomian decomposition for fractional differential equations. Comput. Math. Appl. 57 (2009), 1672–1681.

    Article  MATH  MathSciNet  Google Scholar 

  29. K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York, 1993.

    MATH  Google Scholar 

  30. S. Momani, A numerical scheme for the solution of multi-order fractional differential equations. Appl. Math. Comput. 182 (2006), 761–770.

    Article  MATH  MathSciNet  Google Scholar 

  31. Z. Odibat, S. Momani, H. Xu, A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations. Appl. Math. Model. 34 (2010), 593–600.

    Article  MATH  MathSciNet  Google Scholar 

  32. K.B. Oldham, J. Spanier, The Fractional Calculus. Academic Publ., New York, 1974.

    MATH  Google Scholar 

  33. I. Podlubny, Fractional Differential Equations. Academic Puvl., San Diego, 1999.

    MATH  Google Scholar 

  34. R. Rach, A.M. Wazwaz, J.S. Duan, A reliable modification of the Adomian decomposition method for higher-order nonlinear differential equations. Kybernetes. 42 (2013), 282–308.

    Article  MathSciNet  Google Scholar 

  35. S.S. Ray, R.K. Bera, Solution of an extraordinary differential equation by Adomian decomposition method. J. Appl. Math. 2004 (2004), 331–338.

    Article  MATH  MathSciNet  Google Scholar 

  36. A. Saadatmandi, M. Dehghan, A new operational matrix for solving fractional-order differential equations. Comput. Math. Appl. 59 (2010), 1326–1336.

    Article  MATH  MathSciNet  Google Scholar 

  37. M.G. Sakar, F. Erdogan, A. Yildirim, Variational iteration method for the time-fractional Fornberg-Whitham equation. Comput. Math. Appl. 63 (2012), 1382–1388.

    Article  MATH  MathSciNet  Google Scholar 

  38. K. Scott, Y.P. Sun, Approximate analytical solutions for models of three-dimensional electrodes by Adomian’s decomposition method. Modern Aspects of Electrochemistry 41 (2007), 221–304.

    Article  Google Scholar 

  39. S.E. Serrano, Hydrology for Engineers, Geologists, and Environmental Professionals: An Integrated Treatment of Surface, Subsurface, and Contaminant Hydrology. Second revised ed., HydroScience, Ambler, PA, 2010.

    Google Scholar 

  40. S.E. Serrano, Engineering Uncertainty and Risk Analysis: A Balanced Approach to Probability, Statistics, Stochastic Modeling, and Stochastic Differential Equations. Second revised ed., HydroScience, Ambler, PA, 2011.

    Google Scholar 

  41. N.T. Shawagfeh, Analytical approximate solutions for nonlinear fractional differential equations. Appl. Math. Comput. 131 (2002), 517–529.

    Article  MATH  MathSciNet  Google Scholar 

  42. A.M. Wazwaz, A First Course in Integral Equations. World Scientific, Singapore and River Edge, NJ, 1997.

    Book  MATH  Google Scholar 

  43. A.M. Wazwaz, Partial Differential Equations: Methods and Applications. Balkema, Lisse, 2002.

    Google Scholar 

  44. A.M. Wazwaz, Partial Differential Equations and Solitary Waves Theory. Higher Education, Beijing, Springer, Berlin, 2009.

    Book  MATH  Google Scholar 

  45. A.M. Wazwaz, Linear and Nonlinear Integral Equations: Methods and Applications. Higher Education, Beijing, Springer, Berlin, 2011.

    Book  Google Scholar 

  46. G.C. Wu, A fractional characteristic method for solving fractional partial differential equations. Appl. Math. Lett. 24 (2011), 1046–1050.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Ave, Tehran, Iran

    Neda Khodabakhshi & S. Mansour Vaezpour

  2. Department of Chemical and Materials Engineering Faculty of Engineering, King Abdulaziz University, P.O. Box: 80204, Jeddah, 21589, Saudi Arabia

    Dumitru Baleanu

  3. Department of Mathematics and Computer Sciences, Çankaya University, Faculty of Art and Sciences, Ankara, Turkey

    Dumitru Baleanu

  4. Institute of Space Sciences, P.O.BOX MG-23, R 76900, Magurele-Bucharest, Romania

    Dumitru Baleanu

Authors
  1. Neda Khodabakhshi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Mansour Vaezpour
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dumitru Baleanu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Mansour Vaezpour.

About this article

Cite this article

Khodabakhshi, N., Mansour Vaezpour, S. & Baleanu, D. Numerical solutions of the initial value problem for fractional differential equations by modification of the Adomian decomposition method. fcaa 17, 382–400 (2014). https://doi.org/10.2478/s13540-014-0176-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.2478/s13540-014-0176-2

MSC 2010

Key Words and Phrases

Search

Navigation