Numerical solution of Fractional differential equation by Wavelets and Hybrid functions

abstract: In this paper, we introduce methods based on operational matrix of fractional order integration for solving a typical n-term non-homogeneous fractional differential equation (FDE). We use Block pulse wavelets matrix of fractional order integration where a fractional derivative is defined in the Caputo sense. Also we consider Hybrid of Block-pulse functions and shifted Legendre polynomials to approximate functions. By the use of these methods we translate an FDE to an algebraic linear equations which can be solved. Methods have been tested by some numerical examples.


Introduction
In recent decades, the fractional calculus and fractional differential equations have bean attracted much attention and increasing interest. Fractional differential equations are generalized from integer order ones, which are achieved by replacing integer order derivatives by fractional ones. In recent years, studies on application of the FDE in science has attracted increasing attention [1,2,4,7,17]. For instance, Bagley and Torvik formulated the motion of a rigid plate immersing in a Newtonian fluid [9]. It shows that the use of fractional derivatives for the mathematical modeling of viscoelastic materials is quite natural [9]. It should be mentioned that the main reasons for the theoretical development are mainly the wide use of polymers in various fields of engineering [9]. Also in 1991, S. Westerlund suggested 232 A.H. Refahi Sheikhani and M. Mashoof using fractional derivatives for the description of propagation of plane electromagnetic waves in an isotropic and homogeneous, lossy dielectric and in the paper on electrochemically polarizable media, published in 1993, [9]. Caputo suggested the fractional-order version of the relationship between electric field and electric flux density [9]. An FDE in time domain can be described as the following for a n ( a D αn t y(t)) + · · · + a 1 ( a D α1 t y(t)) + a 0 ( a D α0 t y(t)) = u(t), (1.1) subject to the initial conditions where a i ∈ R, 0 < α 1 < α 2 < ... < α n , and a D αi t y(t) denotes the caputo fractional derivative of order α i . We can see the conditions of existence and uniqueness of solutions to the FDE in [9]. Moreover several numerical methods have been used to approximate the solution of fractional differential equations, such as finite difference method [11], collocation [12] method and other methods [13,5,18]. Any time function can be synthesized completely to a tolerable degree of accuracy by using set of orthogonal functions. For such accurate representation of a time function, the orthogonal set should be "complete" [10]. In this paper we will apply Block-pulse and Hybrid functions based on Block-pulse wavelet and Shifted Legendre polynomials to approximate the solution of (1.1) under conditions (1.2). We begin by introducing some necessary definitions and theorems of the fractional calculus theory and wavelets. In Section 3 the operational matrices of fractional order integration for Block-pulse wavelets and Hybrid functions are obtained. Also in Section 4 the proposed methods are applied to several examples. The conclusion is given at the end.

Preliminaries
In this Section, we present some basic definitions and properties of fractional calculus and wavelets [9,10,6,3].
Definition 2.1. The shifted Legendre polynomials are defined on the interval [0, 1] and can be determined with the aid of the following recurrence formulae [13]: where P 0 (x) = 1 and P 1 (x) = 2x − 1.
The functions b i are disjoint and orthogonal [10].
T )) may be expanded by the Block-pulse functions as: where The Block-pulse coefficients f i are obtained as Proof: In [10]. ✷ Now we define the Hybrid functions of Block-Pulse and shifted Legendre polynomials.

Operational Matrix of Fractional Order Integration
In this Section we will introduce operational matrix methods based on Blockpulse and Hybrid functions of Block-pulse and shifted Legendre polynomials to numerical solution of fractional order differential equations.
Fractional integration of the block-pulse function vector has given as where F (α) is the Block-pulse operational matrix of the fractional order integration and [14] Now let Hy n,m ≃ ΦB mn (x) and I α Hy n,m (x) = Q (α) Hy n,m (x) then we can construct operational matrix of fractional order integration for Hybrid functions as: . In this section, we analyze the errors of a function is expanded in terms of Bluck-pulse or Hybrid functions of Block-Pulse and shifted Legendre polynomials. (4.1) Proof: In [19].
where f 2 =< f, f >. Therefore there exist the unique coefficients such that where c i,j are defined in Definition 2.3, for more details refer to [20].
Proof: Let f i be the Taylor polynomial of order n−1 for f on [ iT m , (i+1)T m ), therefore Since C T Hy n,m (x), is the best approximation f out of X, f i ∈ X, from above equation we have ✷ Theorems 4.2 shows that the error of Hybrid functions of Block-Pulse and shifted Legendre polynomials reduces to zero very fast as n and m increase.

Numerical Examples
In order to show the efficiency of operational matrix of fractional order integration by Block-pulse and Hybrid functions of Block-pulse and shifted Legendre polynomials for solving initial value problems as (1.1), we apply it to solve different types of FDE which exact solutions are known.  Example 5.1. Consider the equation when q 1 = 0.0159 and q 2 = 0.1379, The exact solution of this problem is 1+ 1 2 t 2 . By integrating from both sides of order q 2 , we have y(x) + 3I(y(x)) + 2I 2−q2 (y(x)) + I 2−q1 (y(x)) + 5I 2 (y(x)) = I 2 (f (x)) + 1 + 3I(1) + 2I 2−q2 (1) + I 2−q1 (1). (5.2) In this example we use the operational matrix of fractional order integration with respect to Block-pulse wavelet and Hybrid functions of Block-Pulse and shifted Legendre polynomials. From Section 3 , we can approximate solution y(x), f (x) and 1 as follows If we use Block-pulse wavelet then W (x) = B m (x), if we apply Hybrid functions of Block-Pulse and shifted Legendre polynomials then we have W (x) = Hy n,m (x). By substituting above equations in (16) we have or Hy n,m (x), respectively. By solving above algebraic equations by existing methods [8,21], we can fond C y . Results are shown in fig .1 for W (x) = B 32 (x). From Fig. 1.a we can see that the numerical solution (block-pulse simulation) is coincide with the exact solution in much points. Also from Fig. 1.b we found that the error ranges between −0.15 and 0.15. The exact solution is y(x) = 1 + x, [13]. By taking I 2 from boot side of (19) we have x y(x) + y(x)) = I 2 (1 + x), (5.6) now from (8), (9) and initial conditions we can see that above equation becomes to y(x) − x − 1 + I 1 2 (y(x)) + I 1 2 (−x − 1) + I 2 (y(x)) = I 2 (1 + x).
which is the integral representation of (5.5).
In this example we use the operational matrix of fractional order integration with respect to Block-pulse wavelet. By applying theorem 2.1, we can approximate solution y(x) and 1 + x as follows by substituting (22) in (21) and using operational matrices we have From (3.2) we can see that the entries of principal diagonal of upper triangular matrix I m + F ( 1 2 ) + F (2) are positive and thus the matrix I m + F ( 1 2 ) + F (2) is nonsingular. This shows that the algebraic equations (23) have a unique solution as Therefore the numerical solution can be regarded as 1 + x, which is the exact solution. Table 1: Absolute error for α = 1.5, a = 12 for example 4.2, case 1.
x In this example we consider an FDE appearing in applied problems [9]. 0 D α x y(x) + ay(x) = f (x) , t > 0, y k (0) = a k , k = 0, 1, · · · , n − 1, (5.10) where n − 1 < α ≤ n. For 0 < α ≤ 2 and a k = 0 this equation is called the Relaxation-oscillation equation. Case 1. Consider a k = 0 and f (x) ≡ H(x), where H(x) is the Heaviside function. In this case The analytical solution of [9] is The integral representation of (5.10) for 1 < α < 2 is we solve the problem, by applying the Hybrid functions of Block-Pulse and shifted Legendre polynomials described in the previous Sections on [0, 3) for α = 1.5. The algebraic equations corresponding to (5.12) are of the form C(I + aG (α) ) = C f G (α) where G (α) is defined in example 5.1. Fig. 2 shows the numerical results and absolute error generated by Hybrid functions (Hy 8,3 (x)) for a = 2 with α = 1.5. Also the absolute error for a = 12 are shown in Table 1 generated by Block pulse and Hybrid functions. From Table 1, we can see that the operational matrix methods achieve a good approximation with the exact solution. Case 2. In this case, we consider f (x) = 0, 0 ≤ α ≤ 1 and a 0 = 1 , a 1 = 0, the analytical solution is [13] Table 2. Table 2 shows that the Hybrid operational matrix method gives an efficient numerical solution for α = 1.

Conclusion
The fractional differential equations play an important role in physics, chemical mixing, chaos theory, and biological system as well. The fundamental goal of this work has been to apply an efficient method for the solution of FDE with initial values .In this paper we presented the operational matrix of fractional order integration method to solve FDE with initial values. This method transforms FDE into algebraic equations. Examples show that the method has been successfully applied to find the approximate solutions of the FDE with initial values. Figures and Tables show that this method is extremely effective and practical for this sort of approximate solutions.