Hybrid approximations for fractional calculus

Abstract. In this paper, a numerical method for solving the fractional differential equations is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Taylor polynomials are presented. The Riemann-Liouville fractional integral operator for hybrid functions is given. This operator is then utilized to reduce the solution of the initial value problems for fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.


Introduction
Fractional differential equations (FDEs) are generalized from integer order ones, which are obtained by replacing integer order derivatives by fractional ones. Many physical systems are modeled using fractional differential equations (FDEs). (see for example [1][2][3][4] and references therein).
The available sets of orthogonal functions can be divided into three classes. The first class includes sets of piecewise constant basis functions (e.g., block-pulse, Haar, Walsh, etc.). The second class consists of sets of orthogonal polynomials (e.g., Chebyshev, Laguerre, Legendre, etc.). The third class is the set of sine-cosine functions in the Fourier series. Orthogonal functions have been used when dealing with various problems of the dynamical systems. The main advantage of using orthogonal functions is that they reduce the dynamical system problems to those of solving a system of algebraic equations by using the operational matrices of differentiation or integration. These matrices can be uniquely determined based on the particular orthogonal functions. Special attention has been given to applications of the Walsh functions, rationalized Haar functions and Legendre wavelets [14][15][16]. The Bernoulli polynomials and Taylor series are not based on orthogonal functions. Nevertheless, they possess the operational matrix of integration. In recent years, the hybrid functions consisting of the combination of block-pulse functions with Legendre polynomials, Chebyshev polynomials, Taylor series, or Bernoulli polynomials [17][18][19][20] have been successfully used for solving selected smooth and non-smooth problems arising in diverse areas of science and engineering. Among the hybrid functions, the structures of the hybrid of block-pulse functions and Taylor polynomials are much simpler than the other hybrid functions [21]. The novel contributions for using the hybrid of block-pulse functions and Taylor polynomials together with their error analysis are given in [21]. In 2015, to the best of our knowledge, the Riemann-Liouville fractional integral operator for hybrid of block-pulse functions and Bernoulli polynomials was derived directly for the first time [22,23].
In the present paper, a new numerical method for solving the initial value problem for fractional order differential equations is presented. The method is based upon hybrid functions approximation. These hybrid functions, which consist of block-pulse functions and Taylor polynomials, are included. An exact Riemann-Liouville fractional integral operator for the hybrid of block-pulse functions and Taylor polynomials is given. This operator is then utilized to reduce the solution of the fractional order differential equations to the solution of algebraic equations.

Definition 1 The Riemann-Liouville fractional integral operator of order α is defined as [2]
where t α−1 * f (t) is the convolution product of t α−1 and f (t).
For the Riemann-Liouville fractional integral, we have Definition 2 Caputo's fractional derivative of order α is defined as [2] ( where α > 0 is the order of the derivative, and n is the smallest integer greater than α.

Function approximation
A function f (t) defined over the interval [0, t f ) may be expanded as where If the infinite series in Eq. (5) is truncated, then Eq. (5) can be written as

Riemann-Liouville fractional integral operator for hybrid of block-pulse functions and Taylor series
We now derive the operator I α for B(t) in Eq. (6) given by To obtain I α b nm (t), we use the Laplace transform. By using Eq.
where µ c (t) is the unit step function defined as By taking the Laplace transform from Eq. (8) and using From Eq. (1) we get Taking the inverse Laplace transform of Eq. (9) yields By using Eq. (10), we have

Problem statement
We focus on the following problem: Caputo fractional differential equation with the initial conditions

Numerical method
In this section, we use the hybrid of block-pulse functions and Taylor polynomials for solving Eq. (11) with the initial conditions in Eq. (12).
In this case, we expand D α f (t) by the hybrid functions as using Eq. (2) and (13), we obtain from Eq. (14), we get Substituting Eqs. (13)- (15) in Eq. (11), we get a system of algebraic equations. Next, we collocate these equations at the Newton-cotes nodes t i given by These equations give N(M + 1) algebraic equations, which can be solved for the unknown vector A T using Newton's iterative method.

Illustrative Example
In this section two examples are given to demonstrate the applicability and accuracy of our method.

Consider the equation [24]
with where q 1 = 0.0159 and q 2 = 0.1379. The exact solution of this problem is [24] Here, we solve this problem by using the Hybrid functions with N = 1 and M = 2. Let Then, by using Eqs. (7) and (19), we have By substituting Eqs. (19) and (21) in Eq. (17), we get . (22) By collocating Eq. (22) at the Newton-cotes nodes given in Eq. (16), we get which is the exact solution.

Example 2
Consider fractional Riccati equation [25] subject to the initial state f (0) = 0. To solve by using the present method, we let from Eqs. (7) and (24), we get By substituting Eqs. (24) and (25) in Eq. (23) and collocating at the Newton-cotes nodes given in Eq. (16), we get N(M + 1) nonlinear algebraic equations which can be solved for the unknown vector A using Newton's iterative method. The exact solution of this problem, when q = 1, is The numerical results for f (t) with N = 1, M = 5, and q = 0.7, 0.8, 0.9, 1 are plotted in Figure 1. The approximate solutions, using the present method, are in agreement with the exact solutions for q = 1 and, as indicated in Figure 2

Conclusion
A general formulation for the Riemann-Liouville fractional integral operator for hybrid of block-pulse functions and Taylor polynomials has been derived. This operator is used to approximate numerical solution of FDEs. In the limit, as q approaches an integer value, the scheme provides solution for the integer-order differential equations. The solution obtained using the present method shows that this approach can solve the problem effectively.