Numerical Solutions for Linear Fractional Differential Equations of Order \(1 < \alpha< 2\) Using Finite Difference Method (ffdm)

3601
Downloads

3445
Views
Authors
Ramzi B. Albadarneh
 Department of Mathematics, Hashemite University, Zarqa Jordan.
Iqbal M. Batiha
 Department of Mathematics, Al alBayt University, MafraqJordan.
Mohammad Zurigat
 Department of Mathematics, Al alBayt University, MafraqJordan.
Abstract
The major goal of this paper is to find accurate solutions for linear fractional differential equations
of order \(1 < \alpha < 2\) . Hence, it is necessary to carry out this goal by preparing a new method called
Fractional Finite Difference Method (FFDM). However, this method depends on several important
topics and definitions such as Caputo's definition as a definition of fractional derivative, Finite
Difference Formulas in three types (Forward, Central and Backward) for approximating the second
and third derivatives and Composite Trapezoidal Rule for approximating the integral term in the
Caputo's definition. In this paper, the numerical solutions of linear fractional differential equations
using FFDM will be discussed and illustrated. The purposed problem is to construct a method
to find accurate approximate solutions for linear fractional differential equations. The efficiency of
FFDM will be illustrated by solving some problems of linear fractional differential equations of order
\(1 < \alpha< 2\).
Keywords
 Finite difference formulas
 composite trapezoidal rule
 numerical solutions
 linear fractional differential equation.
MSC
References

[1]
O. Abdulaziz, I. Hashim, S. Momani, Solving systems of fractional differential equations by homotopy perturbation method , Phys. Lett., 372 (2008), 451459.

[2]
O. P. Agrawal , A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dynam., 38 (2004), 323337.

[3]
R. B. Albadarneh, N. T. Shawagfeh, Z. S. AboHammour, General (n+1)explicit finite difference formulas with proof, Appl. Math. Sci., 6 (2012), 9951009.

[4]
M. Alipour, D. Rostamy, Solving nonlinear fractional differential equations by bernstein polynomials operational matrices , J. Math. Comput. Sci., 5 (2012), 185196.

[5]
R. L. Burden, J. D. Faires, Numerical Analysis , 3rd ed., PWS Publishers, USA (1986)

[6]
M. Caputo, Linear models of dissipation whose Q is almost frequency independent II , Geophys. J. R. Astr. Soc., 13 (1967), 529539.

[7]
V. DaftardarGejji, H. Jafari, Solving a multiorder fractional differential equation using Adomian decomposition, , Appl. Math. Comput., 189 (2007), 541548.

[8]
K. Diethelm, G. Walz, Numerical solution of fractional order differential equations by extrapolation, Numer. Algorithms., 16 (1997), 231253.

[9]
F. B. M. Duarte, J. A. T. Machado, Chaotic phenomena and fractional order dynamics in the trajectory control of redundant manipulators , Nonlinear Dynam., 29 (2002), 315–342.

[10]
N. Engheta, On fractional calculus and fractional multipoles in electromagnetism, IEEE Trans. Antennas Propagation , 44 (1996), 554566.

[11]
I. Hashim, O. Abdulaziz, S. Momani, Homotopy analysis method for fractional IVPs, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 674684.

[12]
S. Irandoustpakchin, Exact solutions for some of the fractional differential equations by using modiffication of He's variational iteration method , Math. Sci., 5 (2011), 5160.

[13]
Y. Keskin, O. Karaoglu, S. Servi, G. Oturanc, The approximate solution of highorder linear fractional differential equations with variable coefficients in terms of generalized Taylor polynomials, Math. Comput. Appl., 16 (2011), 617629.

[14]
C. Lederman, J.M. Roquejoffre, N. Wolanski, Mathematical justification of a nonlinear integrodifferential equation for the propagation of spherical , Ann. Mat. Pura Appl., 183 (2004), 173239

[15]
R. L. Magin , Fractional calculus models of complex dynamics in biological tissues, Comput. Math. Appl., 59 (2010), 15861593.

[16]
F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, Springer, Vienna (1997)

[17]
J. H. Mathews, K. K. Fink, Numerical Methods Using Matlab , PrenticeHall Inc., New Jersey (2004)

[18]
F. C. Meral, T. J. Royston, R. Magin, Fractional calculus in viscoelasticity: an experimental study, Commun. Nonlinear Sci. Num. Simul., 15 (2010), 939945.

[19]
Z. Odibat, S. Momani, V. S. Erturk, Generalized differential transform method: application to differential equations of fractional order, Appl. Math. Comput., 197 (2008), 467477.

[20]
K. B. Oldham, Fractional differential equations in electrochemistry, Adv. Eng. Soft., 41 (2010), 912.

[21]
M. Rehman, R. A. Khan, A numerical method for solving boundary value problems for fractional differential equations, Appl. Math. Model., 36 (2012), 894907.

[22]
D. Rostamy, M. Alipour, H. Jafari, D. Baleanu, Solving multiTerm orders fractional differential equations by operational matrices of BPs with convergence analysis , Rom. Rep. Phys., 65 (2013), 334349.

[23]
W. Rudin, Principles of mathematical analysis , McGrawHill Book Company, New YorkToronto London (1953)

[24]
S. L. Salas, G. j. Etgen, H. Einar, Calculus: one and several variables, John Wiley and Sons, New York (1995)

[25]
M. Weilbeer , effcient numerical methods for fractional differential equations and their analytical background, Papier ieger, USA (2005)

[26]
G. Wu, E. W. M. Lee, Fractional variational iteration method and its application, Phys. Lett., 374 (2010), 25062509.

[27]
C. Yang, J. Hou, An approximate solution of nonlinear fractional differential equation by Laplace transform and Adomian polynomials, J. Inform. Comput. Sci., 10 (2013), 213222.

[28]
V. G. Ychuk, B. Datsko, V. Meleshko, Mathematical modeling of time fractional reaction diffusion systems, J. Comput. Appl. Math., 220 (2008), 215225.