An efficient scheme for solving a system of time- fractional order differential-algebraic equations by using fractional Laplace iteration method

Authors

  • Sameer Hasan

    Al-Mustansiriay University
  • Eman Namah

    Baghdad University

How to Cite

Hasan, S., & Namah, E. (2016). An efficient scheme for solving a system of time- fractional order differential-algebraic equations by using fractional Laplace iteration method. International Journal of Advanced Mathematical Sciences, 5(1), 1-7. https://doi.org/10.14419/ijams.v5i1.6889

DOI:

https://doi.org/10.14419/ijams.v5i1.6889

Keywords:

Riemann-Liouville Derivative, Analytic Solution, Fractional Laplace Iteration Method, Mittag-Leffller Functions, System of Time -Fractional Order Differential-Algebraic Equations.

Abstract

In this article, we propose an efficient algorithm for solving system of time- fractional differential-algebraic equations by using a fractional Laplace iteration method. The scheme is tested for some examples and the results demonstrate reliability and accuracy of this method.

References

  1. [1] R. Hilfer, "Applications of fractional calculus in physics", World Scientific, Singapore, 2000. https://doi.org/10.1142/3779.

    [2] R. Gorenflo, F. Mainardi, "Fractional calculus: Integral and differential equations of fractional order, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics", Springer, New York, (1997) 223-276. https://doi.org/10.1007/978-3-7091-2664-6_5.

    [3] A. Carpinteri, F. Mainardi, "Fractals and Fractional Calculus in Continuum Mechanics", Springer Verlag, Wien, New York, (1997). https://doi.org/10.1007/978-3-7091-2664-6.

    [4] F. Mainardi, "Fractional caculus: Some basic problem in continuum and statistical mechanics" in A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer, New York, (1997) 291-348. https://doi.org/10.1007/978-3-7091-2664-6_7.

    [5] U. M. Ascher, L R. Petzold, "Projected implicit Runge-Kutta methods for differential-algebraic equations", SIAM Journal on Numerical Analysis, 28 (1991) 1097-1120. https://doi.org/10.1137/0728059.

    [6] N. Guzel, M. Bayram, "On the numerical solution of differential-algebraic equations with index-3", Appl. Math. Comput, 175 (2) (2006) 1320-1331. https://doi.org/10.1016/j.amc.2005.08.025.

    [7] N. Guzel, M. Bayram, "Numerical solution differential-algebraic equations with index-2", Appl. Math. Comput., 174 (2) (2006) 1279-1289. https://doi.org/10.1016/j.amc.2005.05.035.

    [8] W. Wang, "An algorithm for solving DAEs with mechanization", Appl. Math. Comput., 167 (2) (2005) 1350-1372. https://doi.org/10.1016/j.amc.2004.08.010.

    [9] E. Celik, M. Bayram, "The numerical solution of physical problems modeled as a systems of differential-algebraic equations (DAEs)", J. Franklin Inst., 342 (1) (2005) 1-6. https://doi.org/10.1016/j.jfranklin.2004.07.004.

    [10] S. Momani, Z. Odibat, "Numerical approach to differential equations of fractional order", J. Comput. Appl. Math., 207 (2007) 96-110. https://doi.org/10.1016/j.cam.2006.07.015.

    [11] S. Momani, Z. Odibat, "Homotopy perturbation method for nonlinear partial differential equation of fractional order", Phys. Lett. A 365 (2007) 345-350. https://doi.org/10.1016/j.physleta.2007.01.046.

    [12] Z. Odibat, S. Momani, "Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order", Chaos. Solitions Fractals, 36(1) (2006) 167-174. https://doi.org/10.1016/j.chaos.2006.06.041.

    [13] F. Soltanian, Mehdi .Dehghan, S. M. Karbasi, "Solution of the differential algebraic equations via homotopy perturbation method and their engineering applications", International Journal of Comput. Math., 9 (2010) 1950-1974. https://doi.org/10.1080/00207160802545908.

    [14] B. Barani, S.M. Hosseini, M. Saffarzadeh, S. Javanmardi, "Analytical approach of differential-algebraic equation of frational order via homotopy perturbation method", Communications on Advanced Compt. Scien. With Appli., 2014 (2014) 1-8.

    [15] M. M. Hosseini, "Adomian decomposition method for solution of differential algebraic equations", Journal of Comput. and Appl. Math., 197 (2006) 495-501. https://doi.org/10.1016/j.cam.2005.11.012.

    [16] M. Hosseini, "Adomian decomposition method for solution of nonlinear differential algebraic equations", Appl. Math. And Comput., 181 (2006) 1737-1744. https://doi.org/10.1016/j.amc.2006.03.027.

    [17] E. Celike, M. Bayramb, T. Yelogu, "Solution of differential-algebraic equations (ADEs) by Adomian decomposition method", Inter. J. Pure Appl. Math. Sci., 3(1) (2006) 93-100.

    [18] S. Momani, K. Al. Khaled, "Numerical solution for system of fractional differential equations by the decomposition method", Appl. Math. Comput., 162 (2005) 1351-1365. https://doi.org/10.1016/j.amc.2004.03.014.

    [19] H. Jafari, V. D. Gejji, "Solving a system of nonlinear fractional equations usingf Adomian decomposition", Appl. Math. Comput., 196 (2006) 644-651. https://doi.org/10.1016/j.cam.2005.10.017.

    [20] S. J. Liao, "The proposed homotopy analysis technique for the solution of nonlinear problems", Ph. D. Tesis, Shanghnai Jiao Tong University, 1992.

    [21] M. Zurigat, S. Momani, A. Alawneh, "Analytical approximate solution of systems of fractional algebraic- differential equations by homotopy analysis method", Comput. And Math. With Appl., 59 (2010) 1227-1235. https://doi.org/10.1016/j.camwa.2009.07.002.

    [22] F. Soltanian, S. M. Karbasi, M. M. Hosseini, "Application of He, s variational iteration method for solution of differential-algebraic equations", Chaos, Sloitons and Fractals, 41(1) (2004) 436-445. https://doi.org/10.1016/j.chaos.2008.02.004.

    [23] Z. Odibat, S.Momani, "Application of variation iteration method to nonlinear differential equations of fractional order", Int. Nonlin. Sci. Numer. Simulat., 1 (7) (2006) 15-27.

    [24] S. Momani, Z. Odibat, "Numerical comparsion of methods for solving linear differential equation of fractional order", Chaos Solitons Fractals, 31 (2007) 1248-1255. https://doi.org/10.1016/j.chaos.2005.10.068.

    [25] S. Gupta, D. Kumar, J. Singh, "Numerical study for systems of fractional differential equations via Laplace transform", Journal of the Egypian Math. Society, 23 (2015) 256-262.

    [26] H. Latifizadeh, E. Hesameddini,"Flexibility and efficiency of new analytical method for solving systems of linear and nonlinear differential equations", International Journal of Science and Engineering Investigations", 2 ( July 2013) 44-51.

    [27] I. Podlubny, "Fractional differential equations", New York, NY: A cademic Press, 1999.

    [28] S. Kazem, "Exact solution of some linear fractional differential equations by laplace transform", Tnternational Journal of Nonlinear Science, 16 (1) (2013) 3-11.

Downloads

How to Cite

Hasan, S., & Namah, E. (2016). An efficient scheme for solving a system of time- fractional order differential-algebraic equations by using fractional Laplace iteration method. International Journal of Advanced Mathematical Sciences, 5(1), 1-7. https://doi.org/10.14419/ijams.v5i1.6889