We present two methods for solving a nonlinear system of fractional differential equations within Caputo derivative. Firstly, we derive operational matrices for Caputo fractional derivative and for Riemann-Liouville fractional integral by using the Bernstein polynomials (BPs). In the first method, we use the operational matrix of Caputo fractional derivative (OMCFD), and in the second one, we apply the operational matrix of Riemann-Liouville fractional integral (OMRLFI). The obtained results are in good agreement with each other as well as with the analytical solutions. We show that the solutions approach to classical solutions as the order of the fractional derivatives approaches 1. 1. Introduction Differential equations of fractional order have been subjected to many studies due to their frequent appearance in various applications in fluid mechanics, viscoelasticity, biology, physics, engineering, and so on. Recently, a large amount of literature was developed regarding the application of fractional differential equations in nonlinear dynamics (see, e.g., [1–11] and the references therein). Thus, a huge attention has been given to the solution of fractional ordinary differential equations, integral equations, and fractional partial differential equations of physical interest. As it is known, there exists no method that yields an exact solution for fractional differential equations. Various methods have been proposed in order to solve the fractional differential equations. These methods include the homotopy perturbation method [12–15], Adomian’s decomposition method [16–20], variation iteration method [12–14, 21–23], homotopy analysis method [24], differential transform method [25], operational matrices [26–28], and nonstandard finite difference scheme [29]. In this paper, we investigate the nonlinear system of fractional differential equations as and the initial condition where and . Also, are multivariable polynomial functions. The structure of the paper is given later. In Section 2, we present some preliminaries and properties in fractional calculus and Bernstein polynomials. In Section 3, we make operational matrices for product, power, Caputo fractional derivative, and Riemann-Liouville fractional integral by BPs. In Section 4, we apply two methods for solving nonlinear system of fractional differential equations by BPs. In Section 5, numerical examples are simulated to demonstrate the high performance of the proposed method. Conclusions are presented in Section 6. 2. Basic Tools In this section, we recall some basic definitions and properties of
References
[1]
J.-H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porous media,” Computer Methods in Applied Mechanics and Engineering, vol. 167, no. 1-2, pp. 57–68, 1998.
[2]
I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999.
[3]
R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
[4]
X. Gao and J. Yu, “Synchronization of two coupled fractional-order chaotic oscillators,” Chaos, Solitons and Fractals, vol. 26, no. 1, pp. 141–145, 2005.
[5]
J. G. Lu, “Chaotic dynamics and synchronization of fractional-order Arneodo's systems,” Chaos, Solitons & Fractals, vol. 26, no. 4, pp. 1125–1133, 2005.
[6]
J. G. Lu and G. Chen, “A note on the fractional-order Chen system,” Chaos, Solitons & Fractals, vol. 27, no. 3, pp. 685–688, 2006.
[7]
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, San Diego, Calif, USA, 2006.
[8]
D. Baleanu, O. G. Mustafa, and R. P. Agarwal, “An existence result for a superlinear fractional differential equation,” Applied Mathematics Letters, vol. 23, no. 9, pp. 1129–1132, 2010.
[9]
D. Baleanu, O. G. Mustafa, and R. P. Agarwal, “On the solution set for a class of sequential fractional differential equations,” Journal of Physics A, vol. 43, no. 38, Article ID 385209, 2010.
[10]
D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, World Scientific, Hackensack, NJ, USA, 2012.
[11]
S. Bhalekar, V. Daftardar-Gejji, D. Baleanu, and R. L. Magin, “Transient chaos in fractional Bloch equations,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3367–3376, 2012.
[12]
S. Momani and Z. Odibat, “Numerical approach to differential equations of fractional order,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 96–110, 2007.
[13]
S. Momani and Z. Odibat, “Homotopy perturbation method for nonlinear partial differential equations of fractional order,” Physics Letters A, vol. 365, no. 5-6, pp. 345–350, 2007.
[14]
S. Momani and Z. Odibat, “Numerical comparison of methods for solving linear differential equations of fractional order,” Chaos, Solitons & Fractals, vol. 31, no. 5, pp. 1248–1255, 2007.
[15]
Z. Odibat and S. Momani, “Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order,” Chaos, Solitons & Fractals, vol. 36, no. 1, pp. 167–174, 2008.
[16]
S. Momani and K. Al-Khaled, “Numerical solutions for systems of fractional differential equations by the decomposition method,” Applied Mathematics and Computation, vol. 162, no. 3, pp. 1351–1365, 2005.
[17]
H. Jafari and V. Daftardar-Gejji, “Solving a system of nonlinear fractional differential equations using Adomian decomposition,” Journal of Computational and Applied Mathematics, vol. 196, no. 2, pp. 644–651, 2006.
[18]
D. Lesnic, “The decomposition method for Cauchy advection-diffusion problems,” Computers & Mathematics with Applications, vol. 49, no. 4, pp. 525–537, 2005.
[19]
D. Lesnic, “The decomposition method for initial value problems,” Applied Mathematics and Computation, vol. 181, no. 1, pp. 206–213, 2006.
[20]
V. Daftardar-Gejji and H. Jafari, “Adomian decomposition: a tool for solving a system of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 301, no. 2, pp. 508–518, 2005.
[21]
Z. M. Odibat and S. Momani, “Application of variational iteration method to nonlinear differential equations of fractional order,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 7, no. 1, pp. 27–34, 2006.
[22]
S. Momani and Z. Odibat, “Analytical approach to linear fractional partial differential equations arising in fluid mechanics,” Physics Letters A, vol. 355, no. 4-5, pp. 271–279, 2006.
[23]
V. Daftardar-Gejji and H. Jafari, “An iterative method for solving nonlinear functional equations,” Journal of Mathematical Analysis and Applications, vol. 316, no. 2, pp. 753–763, 2006.
[24]
M. Zurigat, S. Momani, Z. Odibat, and A. Alawneh, “The homotopy analysis method for handling systems of fractional differential equations,” Applied Mathematical Modelling, vol. 34, no. 1, pp. 24–35, 2010.
[25]
V. S. Ertürk and S. Momani, “Solving systems of fractional differential equations using differential transform method,” Journal of Computational and Applied Mathematics, vol. 215, no. 1, pp. 142–151, 2008.
[26]
M. Alipour, D. Rostamy, and D. Baleanu, “Solving multi-dimensional FOCPs with inequality constraint by BPs operational matrices,” Journal of Vibration and Control, 2012.
[27]
D. Rostamy and K. Karimi, “Bernstein polynomials for solving fractional heat- and wave-like equations,” Fractional Calculus and Applied Analysis, vol. 15, no. 4, pp. 556–571, 2012.
[28]
D. Rostamy, M. Alipour, H. Jafari, and D. Baleanu, “Solving multi-term orders fractional differential equations by operational matrices of BPs with convergence analysis,” Romanian Reports in Physics, vol. 65, no. 2, 2013.
[29]
S. Momani, A. Abu Rqayiq, and D. Baleanu, “A nonstandard finite difference scheme for two-sided space-fractional partial differential equations,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 22, no. 4, Article ID 1250079, 2012.
[30]
E. W. Cheney, Introduction to Approximation Theory, AMS Chelsea Publishing, Providence, RI, USA, 2nd edition, 1982.
[31]
E. Kreyszig, Introduction Functional Analysis with Applications, John Wiley & Sons, New York, NY, USA, 1978.
[32]
M. Alipour and D. Rostamy, “Bernstein polynomials for solving Abel's integral equation,” The Journal of Mathematics and Computer Science, vol. 3, no. 4, pp. 403–412, 2011.
[33]
C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamic, Prentice-Hall, Englewood Cliffs, NJ, USA, 1988.
