Chelyshkov-Tau Approach for Solving Bagley-Torvik Equation

There are few numerical techniques available to solve the Bagley-Torvik equation which occurs considerably frequently in various offshoots of applied mathematics and mechanics. In this paper, we show that Chelyshkov-tau method is a very effective tool in numerically solving this equation. To show the accuracy and the efficiency of the method, several problems are implemented and the comparisons are given with other methods existing in the recent literature. The results of numerical tests confirm that Chelyshkov-tau method is superior to other existing ones and is highly accurate.


Introduction
Recently, the fractional derivative has attracted a lot of attentions due to widely applied in various fields of physics and engineering. Fractional derivative is an excellent tool to describe memory and genetic characteristics of various materials and processes. Many phenomena in various fields of science and engineering such as control, porous media, electrochemistry, viscoelasticity, electromagnetic, etc. can be described by Fractional differential Equation (FDE). The evolution of effective and perfectly appropriate method for numerically solving FDEs has received great attention over the past years.
In this paper, we develop a new approach called Chelyshkov-  ( ) u x represents motion of the rigid plate. The questions of existence and uniqueness of the solution to this initial value problem have been discussed in [1] [2], so there is no need to go into these matters here.
Orthogonal polynomials play an important role in mathematics as well as in applications in mathematical physics, engineering and computer science.
Chelyshkov polynomials are the most recent set of orthogonal polynomials.

Basic Definitions of Fractional
In this section, we introduce the basic necessary definitions and primary facts of the fractional calculus theory which will be more used in this work [26] [27].
The Riemann-Liouville fractional derivative operator D α is given by However, its derivative has Some drawbacks when we try to make a model for a real phenomenon using fractional differential equations. Therefore, we will provide a modified fractional differential operator D α proposed by by Caputo [28].

Definition 2. The Caputo derivative definition is defined as
Hence, 0 α > and m is the smallest integer greater than or equal to α . For the Caputo fractional derivative we have It can be said when α ∈  , the Caputo differential operator matches with the integer-order differential operator.
For more details on fractional derivative definitions, theorems and its properties, you can see [26] [27].

Some Properties of Chelyshkov Polynomials
We first review some important concepts and basics of the Chelyshkov and the orthogonality condition of Chelyshkov polynomials [32] is Also it follows from this relation that ( ) By using the Cauchy integral formula for derivative and the Rodrigues type representation, we can get the integral relation x provide a natural way to solve, expand, and interpret solutions. Actually, these polynomials can be expressed in terms of the Jacobi polynomials ( ) , k P α β by the following relation, where the coefficients j a are the unknown Chelyshkov coefficients and , 0,1, , Then we can convert the solution expressed by (7) and its derivative (8) to  (5) then the finite series can be converted Proof. The proof is straightforward using Equation (4).

The Description of Chelyshkov Scheme
Let us seek the solution of (1) expressed in terms of Chelyshkov polynomials as Replacing each terms of (1) with the corresponding approximations defined in (7), (8) and (10) and we obtain the following theorem.
Equation (12), which can be written in the matrix form ( ) (13) can be written as (14) As in a typical tau method [34] we generate N-1 linear equations by applying The boundary condition is derived from Equation (2) (16) Equations (15) and (16)  Consequently, ( ) u x given in Equation (7) can be calculated..

Numerical Results
In this section, we apply the Chelyshkov- By applying Equation (15) By applying Equation (16), we have which is the exact solution.
is the kth derivative of the Mittag-Leffler function with parameters λ and µ given by       [19] for underlying the variational iteration method (VIM), the fractional iteration method (FIM) and with analogous results of El-Gamel and Abd El-Hady [13] for underlying Legendre-collocation method.    [38] for underlying Haar wavelets method.

Conclusion
In this paper, Chelyshkov operational matrix of fractional derivative has been derived. Our approach was based on the tau method. The proposed technique is easy to implement efficiently and yield accurate results. Moreover, only a small number of Chelyshkov polynomials is needed to obtain a satisfactory result. In addition, an interesting feature of this method is to find the analytical solution if the equation has an exact solution that is polynomial functions. Numerical examples are included and a comparison is made with an existing result.