Fractional differential equations (FDEs) are modelled in different fields of science and engineering such as control engineering, electromagnetism, image processing, fluid flow, statistical mechanics, In general, most of FDEs do not have exact solutions, therefore some investigators studied various methods for finding approximate solutions. These techniques include, Adomian decomposition method (ADM) , homotopy perturbation method (HPM) , Bezier curve method (BCM) , finite difference method, etc. For example, He  proposed HPM when this method is an approach which searches for an analytical approximate solution of linear and nonlinear problems, also, the differential transform method (DTM) is applied to use for solving FBIVP .
BCM is used for solving dynamical systems, (see ). Also BCM is used for solving delay differential equations and switched systems (see ). Authors in  proposed the utilization of BCM on some linear optimal control systems with pantograph delays. Also, to solve the quadratic Riccati differential equation and the Riccati differential-difference equation, BCM is utilized (see ). In this study, BCM is extended for solving FBIVP as follows:
where is derivative operator of fractional order α, x(t) is unknown function on [0,1], x′(0) is the derivative of x with respect to t at t = 0, and x0, and λ are given constant.
The outline of this sequel is as follows: In Section 2, background materials are stated. In Section 3, problem statement is presented. Section 4 is devoted to numerical examples for the precision of the proposed technique. Finally, the conclusion is presented in Section 5.
2 Background materials
Several definitions of a fractional derivative of order α > 0 existed.
The caputo’s fractional derivative of order α is stated in 
where α0 > 0 and n1 is the smallest integer greater than α.
2.1 Function approximation
Utilizing Bezier curves, this technique is to approximate the solutions x(t) where x(t) is given in Eq. (2). Define the Bezier polynomials of degree n that approximate over the interval t ∊ [t0, tf] as follows:
where h = tf – t0, t0 = 0, tf = 1, CT = [c0,c1,…,cn]T, and
is the Bernstein polynomial with degree n for t ∊ [t0,tf], and cr is the control point .
3 Problem statement
One may consider the following FBIVP:
By BCM, one may have
where w1(x) = 1,
then, one may have
by Eq. (7), one can obtain a system of n+1 linear equations with n+1 unknown coefficients ci. Also by utilizing many subroutine algorithm for solving this linear equations, one can find the unknown coefficients ci, i = 0, 1, …, n.
4 Numerical application
In this section, some numerical examples are presented to illustrate the proposed method.
The following FBIVP is considered (see ):
Using the described technique with n = 5, one may have the following xapprox(t), Figures 1, 2, Table 1 and Table 2. The computation takes 6 s of CPU time when it is performed by Maple 16. Also the value of approximate solution with the stated technique is more accurate than that with the stated technique in  (see Table 2).
The following FBIVP is considered (see ):
Using the described technique with n = 5, one may have the following xapprox(t), Figures 3, 4, Table 4 and Table 3. Also the value of approximate solution with the stated technique is more accurate than that with the stated technique in  (see Table 3).
In this study, BCM is used to solve a class of FBIVP. The achieved results by the BCM are in good agreement with the given exact solutions. The study shows that the method is effective and is a simple technique to solve FBIVP.
One may extend this method for optimal control problems governed by fuzzy ordinary differential equations (FODEs), but it has a complicated manipulation. But the method is simple, by solving various numerical examples, accuracy can be found in comparison of other methods.
The author wishes to acknowledge financial support from the University of Venda.
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About the article
Published Online: 2017-12-29
Conflict of Interest: The author declares that there is no conflict of interest in the use of the above mentioned software.
Citation Information: Open Physics, Volume 15, Issue 1, Pages 1045–1048, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2017-0131.
© 2017 F. Ghomanjani and S. Shateyi. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0