NUMERICAL SOLUTION FOR A CLASS OF FRACTIONAL VARIATIONAL PROBLEM VIA SECOND ORDER B-SPLINE FUNCTION

In this paper, we solve a class of fractional variational problems (FVPs) by using operational matrix of fractional integration which derived from second order spline (B-spline) basis function. The fractional derivative is defined in the Caputo and Riemann-Liouville fractional integral operator. The B-spline function with unknown coefficients and B-spline operational matrix of integration are used to replace the fractional derivative which is in the performance index. Next, we applied the method of constrained extremum which involved a set of Lagrange multipliers. As a result, we get a system of algebraic equations which can be solve easily. Hence, the value for unknown coefficients of B-spline function is obtained as well as the solution for the FVPs. Finally, the illustrative examples shown the validity and applicability of this method to solve FVPs.


INTRODUCTION
Fractional calculus is a branch of mathematical analysis that studies the noninteger order of differentiation and integration [14]. It is an old mathematical topic with history as long as ordinary calculus because it has been developed since the year of 1695 due to some theories discovered by G.W. Leibniz and L Hopital [13].
In the past few decades, fractional calculus has attracted the attention of many mathematicians and physicists as they realized that the fractional calculus has various fascinating applications in solving real life problems which include numerous fields of science, physics, as well as biology. For instance, electrical circuit [3], complex dynamic modeling in biological tissues [16], economics and finance ( [15], [25]), the model of viscoelastic ( [12], [17], [18]), chemistry [19], and thermodynamics [24]. Hence, fractional calculus problem has become a significant topic to be studied by researchers, as example in [20,21]. In most cases, there is no exact solution for fractional calculus problem. Therefore, the numerical methods of fractional calculus problem are needed to find the approximate solution. Among the methods are including spectral tau method [5], finite different method [28], and others.
On the other hand, the variational problems are the problems that involved finding the maximum or minimum value of a certain function. Once it contains fractional order derivative in the performance index, it will become fractional variatonal problems (FVPs). In this paper, we consider FVPs as follow: subject to the boundary conditions where 0 ≤ t ≤ T , n − 1 < α ≤ n, D α is a fractional operator and α is a positive fractional number. We assume that L(t, x(t), D α x(t)) is continuous functions.
This FVPs was introduced by Riewe during 1996 where the non-integer order derivatives were used in non-conservative systems of mechanics [23]. In recent years, many mathematicians show their interest toward FVPs and solve it using various type of methods such as Euler Lagrange equation ( [1], [2]), Jacobi polynomials [4], Legendre polynomials [6], Rayleigh-Ritz method [11] and many more. However, all these methods are based on smooth function (i.e. polynomials are smooth).
Hence, the main aim for this paper is to propose a new numerical scheme based on operational matrix via piecewise function, which is second order of Bspline function to solve the FVPs. The numerical scheme is through second order B-spline operational matrix of integration where the fractional derivative is in the sense of Riemann-Liouville. This B-spline operational matrix of integration was successfully used in [9] to solve the fractional integro-differential equations and fractional partial differential equations. Here, we extend it to solve the FVPs. The approach used in [9] for obtaining fractional integration of B-spline functions is via Laplace transform. This method is used as approximation of function due to it properties that have explicit formula, symmetric and also compact support of B-spline.
This present paper is arranged as follow. In section 2, some necessary preliminaries of fractional integration and derivatives are investigated as well as brief explanation for B-spline function. In section 3, the derivations for B-spline operational matrix of integration are discussed here. In section 4, the numerical scheme is presented to show how the operational matrix of fractional integration derive from B-spline function and be used to solve FVPs. In section 5, the proposed method is applied to solve numerical examples of FVPs. Finally, the conclusion for this paper is given in section 6.

PRELIMINARIES
In this section, we briefly describe some important definitions that will be using in this paper.
2.1. Fractional integration and derivative. The Riemann-Liouville fractional integration of order α > 0 of f (t) is described as : where Γ( . ) is gamma function. Below shows several important properties of Riemann Liouville integral: Meanwhile the Caputo fractional derivative of order α > 0 of f (t) is described as : where n − 1 < α ≤ n, n N and n = α .
2.2. B-spline function. B-spline is a piecewise polynomial function and possesses a high degree of smoothness at the knots which is known as the places where the pieces of polynomial are join together.
Definition 2.1. The cardinal B-splines, N n (t) of order n are described via this convolution product: Then, the two-scale relationship of order n for the B-spline function is : In this paper, we will using second order B-spline function, hence, n = 2, so from (9), we have We use the second order B-spline function to solve FVP as describe in problem (1). We only focused on domain [0, 1]. Therefore, we define second order B-spline scaling function, ϕ 2,j,k (t) as where N 2,j,k = N 2 (2 j t − k) and j is involving in discretization step, which j must be 2 j ≥ 3. Hence, we have the B-spline function of order 2 as in [9].
2.3. Function Approximation. For a fixed j = J, a function of f (t) which is defined on interval [0, 1] can be approximated by the second order B-spline function as follow, where the vectors of coefficient, C and Φ J (t) are shown below with r = −1: Then, c k can be find using Practically, we can also find the dual function,φ 2,j,k (t) by following equation.

B-SPLINE OPERATIONAL MATRIX OF FRACTIONAL INTEGRATION
In this section, we will briefly discuss the derivation of B-spline operational matrix of fractional integration. For this part, we follow the work in [9]. We use Laplace transform to calculate the fractional integration for second order B-spline function with k = 0, 1, . . . , 2 j − 2. In this process, we first change ϕ n,j,k (t) to unit step function, we get From (18), we convert it into Laplace transform: By using definition from Riemann-Liouville fractional integration (3), we have the Laplace transform of I α ϕ 2,j,k (t) as Then, take the inverse Laplace for both sides of equation (19), we will get The unit step function in (20) can be easily rewrite in piecewise function. Hence, by letting a = 1 2 jα Γ(α + 2) , the fractional integration of second order B-spline function is achieved as follow: 3.1. Operational Matrices. Here we will explain more on operational matrix of fractional integration for second order B-spline function. First, letΦ J be the vector of dual function of Φ J where order 2 and r = −1.
Theorem 3.1. If the matrix P 2 = [p i,r ] is describe as : then we have : (a) P 2 which is a (2 j + 1)(2 j + 1) symmetric matrix : The vector of dual function forΦ J (t) can be obtained by using linear combination of Φ J (t) [27] which interpret as : Proof. The following formula is simple to be applied for proving the theorem [9] : where j ≥ n, i, k = r, . . . , 2 j − 1 and r = 1 − n.
Theorem 3.2. Consider vector of second-order B-spline, Φ J (t) and let α > 0, then where F α is the B-spline operational matrix of fractional integration and is defined as follow: where element of E 2 is given as e i,k , e i,k = 1 0 (I α ϕ 2,j,k (t))ϕ 2,j,i (t)dt (26) and Proof. Multiply both sides of equation (24) with Φ T J (t) and integrate from 0 to 1, we have By using equation (23), we have where each of element E 2 is e i,k = 1 0 (I α ϕ 2,j,k (t))ϕ 2,j,i (t)dt.

THE NUMERICAL SCHEME
In this section, we describe in details on how to solve the FVPs in (1) and (2) by applying the operational matrix of fractional integration which is derived from second order B-spline function.
Firstly, we approximate D α x(t) with B-spline function, Φ J (t), where C is the unknown coefficients of B-spline function and both of C and Φ J (t) are given in (14) and (15) respectively. After that, by doing integration for (27), we get where F α is the second order B-spline operational matrix of integration. From equation (27) and (28), we can approximate the performance index of FVPs in (1) as follow: The integral in (29) may be solved by any numerical integration such as Gauss quadrature. Alternative, we can solve the integral by using Maple as it is more convenient to be use and computer oriented. Then, let assume where λ is refer as unknown Lagrange multiplier and b j is the initial boundary condition given in (2).
In this way, the J * we get in (30) is expanded by unknown coefficient of Bspline function, c 1 , c 2 , . . . , c n and also unknown Lagrange multiplier as in (31). So, by employing the necessary conditions of optimality (32) for the performance index and the boundary condition (2), we can reduce the problem of FVPs to a system of algebraic equations which can easily solved using any numerical methods. δJ * δc 1 = 0, δJ * δc 2 = 0, . . . , δJ * δc n = 0, δJ * δλ n = 0 (32) Once we get the value for coefficients, we can simply calculate the value for J using equation (29).

NUMERICAL EXAMPLE
In this section, we will provide two problems for FVPs and solve it by using our proposed scheme which is via second order B-spline function.
Example 1: We consider the problem based on example given in paper by Mohammed, O. H. [22].
subject to the boundary condition The exact solution for α = 1 is In paper [22], a scheme via hat basis function is applied to obtain the numerical solution for FVPs. Here, we compare our present result and result in [22] with the exact solution as shown in TABLE (1). From this comparison, we can see that our new numerical scheme gives more accurate result. Figure (1) shows the comparison of our numerical solution of equation (33) for α = 1, 0.5 and 0.6 with the exact solution. The numerical solution is comparable with those in [22]. Example 2: We consider the following problem as in paper by Ezz-Eldien, S. S. [7].
In TABLE (2), the absolute error of x(t) is listed with different value of N using our present method. Meanwhile, TABLE (3) shows the absolute error for the performance index by using the proposed method which is B-spline function.

CONCLUSION
In this paper, we have constructed a new numerical scheme based on second order B-spline function for a type of Fractional Variational Problems, FVPs. In our