Local Polynomial Smoother for Solving Bagley-Torvik Fractional Differential Equations

Local polynomial smoother (LPS) is a weighted local least-squares nonparametric method. It provides a local Taylor series fit of the data at any location and can be directly used in a differential equation to provide a numerical scheme. In this article, we introduce this new nonparametric idea based on local polynomial smoother, for acquiring the numerical solution of the Bagley-Torvik fractional-order differential equations. Furthermore, this paper will present a numerical comparison with some methods, such as legendre operational matrix and pseudo-spectral method. The efficiency and accuracy of the LPS method are demonstrated by two numerical examples.


Introduction
The fractional differential equations appear more and more frequently in different research areas and engineering applications.Phenomena in electromagnetics, acoustics, viscoelasticity, and electrochemistry and material science are also described by differential equations of fractional order.The solution of the differential equation containing fractional derivative is much involved.For example, fractional-order derivatives have been successfully used to model damping forces with memory effect or to describe state feedback controllers [1][2][3][4].Consequently, an effective and easy to use method for solving such equations is needed.it was undoubtedly the book written by Oldham and Spanier [3] which played an outstanding role in the development of this subject.
Recently, Su and Yan [24], have used local polynomial smoother method for the numerical solution of integro-differential equations.Later, Su and Yan [25,26], manage to solve ordinary differential equations and partial differential equations respectively by using this method.Numerical results demonstrate that local polynomial fitting method is more accurate, simply and efficient.In this study, we consider the following initial value problem in the case of the inhomogeneous Bagley-Torvik equation, then compared with other methods: with the boundary conditions: where y (0) (t) = y(t) and y(t) is an unknown function, P (t), R(t), S(t), and f (t) are functions defined on the interval a ≤ t ≤ b, and b, γ 0 and γ 1 are real constants.The main destination of this paper is to explore a fast algorithm for numerical solution of Bagley-Torvik FDEs (1) using a new nonparametric method, local polynomial smoother (LPS).In Section 2, we present the definition of fractional-order integration and fractional-order differentiation.In Section 3, local polynomial smoother method are described and their main thinking are reviewed.In Section 4, three parameters selections, the kernel K and the bandwidth h, are discussed in LPS method.In Section 5, we employ local polynomial Smoother method for solving of the Bagley-Torvik FDEs (1).Finally, in Section 7, we present some numerical tests to validate the theoretical results and we make comparisons of the methods discussed in the article.

Basic definitions
We first give the definition of fractional-order integration and fractional-order differentiation [17].For the concept of fractional derivative, we will adopt Caputos definition, which is a modification of the Riemann-Liouville definition and has the advantage of dealing properly with initial value problems.Definition 2.1 A real function f (x), x > 0, is said to be in the space Cµ, µ ∈ R if there exists a real number p > µ such that f (x) = x p f 1 (x), where The Riemann-Liouville fractional integral operator of order α ≥ 0 of a function, f ∈ Cµ, µ ≥ −1, is defined as The properties of the operator J α can be found in [34,35]; we mention only the following: For f ∈ Cµ, µ ≥ −1, α, β ≥ 0 and γ > −1: x α+γ .
The Riemann-Liouville derivative has certain disadvantages when trying to model real-world phenomena using fractional differential equations.Therefore, we will introduce a modified fractional differential operator D α * proposed by Caputos work on the theory of viscoelasticity [39].

Local polynomial smoother
Local polynomial smoother technique was first proposed by Fan.J [27][28][29].And, this method was also mentioned and used by Su [30] .In order to describe the basic ideas of the LPS, firstly, we introduce the mathematical thoughts of local polynomial smoother.This idea were detailed mentioned in [27][28][29].The basic idea is to form a locally optimal Taylor series to the data by minimizing the L 2 norm of the residual between the polynomial model and the data.Since the form of Smoother function is not specified, so the data points with long distance from t 0 provides little information to y(t 0 ).Therefore, we can only use the local data points around t 0 .We suppose that y(t) has p+1 derivative at t 0 , by the Taylor expansion , for point t ,located in the neighborhood of this pointt 0 , we can use the p-order multivariate polynomials to locally approximate y(t), and the surrounding local point of t 0 , we model y(t) as Taylor series expansion around any pint t 0 is: where parameter β j depends on t 0 , so called local parameter.Obviously, the local parameter β j = y (j) (t0) j! fits the local model with local data and it can be minimized, where h controls the size of the bandwidth of local area.Using matrix notation to represent the local polynomial smoother is more convenient.Below is the design matrix corresponding to (3) with t and Y : The weighted least squares problem (3) can be written as here, Furthermore, we can get the estimation ŷ(t 0 ), where e 1 is a column vector (the same size of β) with the first element equal to 1, and the rest equal to zero, that is, . The selection of K does not influence the results much.We selected the quartic kernel as follows

LPS for Bagley-Torvik equation
In this section, the LPS method for solving (1) is outlined.As we know, datas Y i at t = t i in fractional differential equation are unknown, consequently, instead of estimating data at a specific point t i , we want to find a best match of some unknown data Y i , i=1,2,....We could vary Taylor series residual density (4) at t i with respect to these unknown data.Let equation( 8) be an approximate solution of (1): where, The conventional collocation method can be expressed We can adjust the value of parameter n flexibly, for example n=30 ,50, 70.It is required that the approximate solution (8) satisfies the (1-2) at the point t = t i .Putting expression (8) into equation (1), we can obtain expression as follows: In order to satisfy unique solution theorem of integral-differential equation, we should take initial conditions (2) into account.Consequently, putting (8) to initial conditions (2), then we can acquire the approximate expression (11) as follows: Consequently, according to the deduction process of equation ( 4), we can obtain easily the minimization function of problems (1-2), that is to say, the minimization function (Taylor series residual density) can be varied as (11): Combination equations ( 9), ( 10) with (11), we can acquire the system which can be expressed (12): then, the matrix form( 5) can be written as follows by using (13)(14).15) where Putting (14-15) to (6), where In this paper, we choose the optimal weight function Then estimated set of coefficients β i can be obtained by solving matrix system solution, therefore, approximate-value ŷ(t 0 ) at point t 0 can be obtained.Repeating the above steps, we can acquire ŷ(t i ) at point where n can be selected any positive integer.

Numerical examples
In this section, in order to illustrate the description above and to show the efficiency of the mentioned method for solving equation (1-2), we give the values of the exact solution y(t), In tables and figures, the approximate solution ŷ(t), the absolute error function e AE (t) = |y(t) − ŷ(t)| at the selected points of the given interval.We have performed all numerical computations in MATLAB.Example 1 ( [34]) First, we consider the Bagley-Torvik equation with the boundary conditions y(0) = 1 and y(1) = 2, and the exact solution y(t) = 1+t, certainly, To simplify the procedures, giving n=11, p=1, h=0.5, we calculate the estimator ŷ(t i ) i=1, 2, ...,11 at point t = t i , such as ŷ(t 0 )   with the boundary conditions y(0)=0 and y(1)=0 and the exact solution y(x) = x 2 −x.

Conclusions
In this paper, we propose a fast algorithm based on local polynomial smoother which is applied for the numerical solution of Bagley-Torvik fractional differential equations (FDEs).The error result of LPS solution is very small and calculation processes are simple and feasible.Compared with Legendre operational matrix and pseudo-spectral method, LPS methods converges to solutions with fewer number of nodes and possess more accuracy.Moreover, it is more flexible to resolve problems just only adjusting

Table 1
Comparison of the absolutes for the numerical results and absolute errors at point t i = 0.1: 0.1: 1.0 for Example 1.