Issues‎ > ‎vol17n3‎ > ‎

Numerical Solution of Fractional Differential Equations by using Fractional Spline Functions.


Faraidun K. Hamasalh1, Pshtiwan O. Muhammad1

1Faculty of Science and Science Education, School of Science, Sulaimani Univ., Sulaimani, Iraq



Abstract
In this paper, we consider a new suitable lacunary fractional interpolation with the idea
of the spline function of polynomial form, and the method applied to solve linear
fractional differential equations. The results obtained are in good agreement with the
exact analytical solutions and the numerical results presented by two examples, results
also show that the technique introduced here is robust and easy to apply.

Key Words: Fractional integral and derivative, Caputo Derivative, Taylor’s expansion, Error bound, Spline functions.



References

[1] Podlubny, I.“ FractionalDifferentional Equations,” Academic Press, San Diego, (1999).
[2] Kilbas,A. A., Srivastava,H. M. and Trujillo, J. J. “ Theory and Applications of Fractional Differential Equations,”
Minsk, Belarus, 1st edition, 2006.
[3] Barkari, E., Metzler, R. and Klafter,J. “ From continuous time random walks to the fractional Fokker-Planck
equation,” Phys. Rev. E 61 (1) (2000) 132-138.
[4] Yuste,S. B.Acedo, L. and Lindenberg,K. “ Reaction front in an A+BC reaction-subdiffusion process,” Phys. Rev.
E 69 (3) (2004) 036126.
[5] Wang,J. R. andZhou,Y. “ Existence and controllability results for fractional semilinear differential inclusions,”
Nonlinear Anal. RWA 12 (2011) 3642-3653.
[6] Haghighi,A. R.,Aghababa,M. P.and Roohi,M. “ Robust stabilization of a class of three- dimensional uncertain
fractional-order non-autonomous systems,” Int. J. Industrial Mathematics, 6 (2014) 133-139.
[7] Wang,J. R. andZhou,Y. “ A class of fractional evolution equations and optimal controls,” Nonlinear Anal. RWA
12 (2011) 262-272.
[8] Wang,J. R. and Zhou,Y. and Wei,W. “ Fractional Schrodinger equations with potential and optimal controls,”
Nonlinear Anal. RWA 13 (2012) 2755-2766.
[9] Hall,M. G. andBarrick,T. R. “ From diffusion-weighted MRI to anomalous diffusion imaging,” Magn. Reson.
Med, 59 (2008) 447-455.
[10] Oldham,K. B. and Spanier,J. “ The Fractional Calculus, Academic Press,” New York. NY, USA, 1974.
[11] Momani,S. “ Analytical approximate solution for fractional heat-like and wave-like equations with variable
coefficients using the decomposition method,” Applied Mathematics and Computation, vol. 165, no. 2, pp. 459–
472, 2005.

[12] Momani,S., Odibat,Z. and Alawneh,A. “ Variational iteration method for solving the space- and time-fractional
KdV equation,” Numerical Methods for Partial Differential Equations, vol. 24, no. 1, pp. 262–271, 2008.
[13] Odibat,Z.and Momani,S.“ Thevariational iteration method: an efficient scheme for handling fractional
partial differential equations in fluid mechanics,” Computers and Mathematics with Applications. An International
Journal, vol. 58, no. 11-12, pp. 2199–2208, 2009.
[14] Galeone L. and Garrappa R.“ Fractional Adams-Moulton methods,” Math. Comp. simulation 79 (2008) 1358-1367.
[15] Diethelm,K. “ An algorithm for the numerical solution of differential equations of fractional order,” Electronic
Transactions on Numerical Analysis, vol. 5, pp. 1–6, 1997.
[16] Hamasalh, F. K.“ Appliedlacunary interpolation for solving Boundary value problems,” International Journal of
Modern Engineering Research, Vol. 2 Issue. 1, pp-118-123. ISSN: 2249-6645.
[17] Saxena,A. “ (0, 1, 2, 4) Interpolation by G-splines,” Acta Math. Hung. 51 (1988) 261-271.
[18] Fawzy,T. “ (0, 1, 3) Lacunary Interpolation by G-splines,” Annales Univ. Sci. Budapest, Section Mathematics,
vol. XXIX (1989) pp. 63-67. Zbl. 654, 41006
[19] Richard Herrmann, “ Fractional calculus : an introduction for physicists ,” GigaHedron, Germany, 2nd edition,2014.
[20] Usero,D. “ Fractional Taylor Series for Caputo Fractional Derivatives. Construction of Numerical Schemes,”
Dpto. deMatemáticaAplicada, Universidad Complutense de Madrid, Spain, 2008.
[21] Ishteva, M. “ Properties and applications of the Caputo fractional operator,” Msc. Thesis, Dept. of Math.,
Universität Karlsruhe (TH), Sofia, Bulgaria, 2005.
[22] Micula,G.,Fawzy,T. and Ramadan,Z. “ A polynomial spline approximation method for solving system of ordinary
differential equations,” Babes-Bolyai Cluj-Napoca. Mathematica, vol. 32, no. 4, pp. 55–60, 1987.
[23] Ramadan,M. A. “ Spline solutions of first order delay differential equations,” Journal of the Egyptian
Mathematical Society, vol. 13, no. 1, pp. 7–18, 2005.
[24] Ramadan,M. A., El-Danaf,T. S. and Sherif,M. N. “ Numerical solution of fractional differential eqautions using
polynomial spline functions,” submitted.
[25] Zahra W. K. and Elkholy S. M.,“ Quadratic spline solution for boundary value problem of fractional order,”
Numer Algor,59:373-391,2012.