Solving Nonlinear Fractional Differential Equations by Bernstein Polynomials Operational Matrices

Volume 5, Issue 3, pp 185 - 196 http://dx.doi.org/10.22436/jmcs.05.03.06

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Authors

Mohsen Alipour - Department of Mathematics, Imam Khomeini International University, P.O. Box 34149-16818, Qazvin, Iran. Davood Rostamy - Department of Mathematics, Imam Khomeini International University, P.O. Box 34149-16818, Qazvin, Iran.

Abstract

In this paper, we solve nonlinear fractional differential equations by Bernstein polynomials. Firstly, we derive the Bernstein polynomials (BPs) operational matrix for the fractional derivative in the Caputo sense, which has not been undertaken before. This method reduces the problems to a system of algebraic equations. The results obtained are in good agreement with the analytical solutions and the numerical solutions in open literatures. Also, the solutions approach to classical solutions as the order of the fractional derivatives approach to 1.

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Keywords

• Nonlinear fractional differential equations
• Bernstein polynomials
• operational matrix
• Caputo derivative

MSC

•  34A08
•  34A12
•  34A25
•  34A45

References

• [1] X. Gao, J.Yu , Synchronization of two coupled fractional-order chaotic oscillators, Chaos Sol. Fract. , 26 (1) (2005), 141–145.


• [2] J. G. Lu, Chaotic dynamics and synchronization of fractional-order Arneodo’s systems, Chaos Sol. Fract., 26 (4) (2005), 1125–1133.


• [3] J. G. Lu, G. Chen, A note on the fractional-order Chen system, Chaos Sol. Fract., 27 (3) (2006), 685–688.


• [4] I. Podlubny, Fractional Differential Equations, Academic Press, NewYork (1999)


• [5] S. Momani, Z. Odibat, Homotopy perturbation method for nonlinear partial differential equations of fractional order, Phys. Lett. A , 365 (5-6) (2007), 345–350.


• [6] S. Momani, Z. Odibat, Numerical comparison of methods for solving linear differential equations of fractional order, Chaos Soliton. Fract. , 31 (5) (2007), 1248–1255.


• [7] Z. Odibat, S. Momani, Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos Soliton. Fract. , 36 (1) (2008), 167–174.


• [8] M. Zurigat, S. Momani, Z. Odibat, A. Alawneh, The homotopy analysis method for handling systems of fractional differential equations, Applied Mathematical Modelling , 34 (2010), 24–35.


• [9] S. Momani, Al-Khaled, Numerical solutions for systems of fractional differential equations by the decomposition method, Appl. Math. Comput., 162 (3) (2005), 1351–1365.


• [10] H. Jafari, V. D. Gejji, Solving a system of nonlinear fractional differential equations using Adomain decomposition, Appl. Math. Comput., 196 (2006), 644–651.


• [11] V. Daftardar-Gejji, H. Jafari, Adomian decomposition: a tool for solving a system of fractional differential equations, J. Math. Anal. Appl., 301 (2) (2005), 508–518.


• [12] Z. Odibat, S. Momani, Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlinear Sci. Numer. Simulat, 1 (7) (2006), 15–27.


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


• [14] V. S. Ertürk, S. Momani, Solving systems of fractional differential equations using differential transform method, Journal of Computational and Applied Mathematics , 215 (2008), 142 – 151.


• [15] M. Caputo, Linear models of dissipation whose Q is almost frequency independent II, Geophys. J. Roy. Astronom. Soc. , 13 (1967), 529-539.


• [16] Y. Luchko, R. Gorneflo, The initial value problem for some fractional differential equations with the Caputo derivative, Preprint series A08-98, Fachbereich Mathematik und Informatik, Freie Universitat Berlin (1998)


• [17] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, Inc., New York (1993)


• [18] K. B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York (1974)


• [19] E. Kreyszig, Introduction Functional Analysis with Applications , John Wiley and Sons Incorporated, (1978.)


• [20] C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods in Fluid Dynamic, Prentice-Hall, Englewood Cliffs, NJ (1988)


• [21] Z. Odibat, S. Momani, Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order, Chaos. Solitons and Fractals, 36 (2008), 167-174.