1. Introduction
Numerical analysis is the study of set of rules that use numerical estimation for the problems of mathematical analysis as distinguished from discrete mathematics. Fractional differential equations are operational and most effective tool to describe different physical phenomena such as rheology, diffusion processes, damping laws, and so on. Many technics have been delegated to solve differential equation of fractional order. Different structures are used to resolve the issues of nonlinear physical models of fractional orders like Finite element method [1] , Finite difference method [2] , differential transformation method [3] [4] , Adomian’s decomposition method [5] [6] [7] , variational iteration method [8] [9] [10] , Homotopy perturbation technique [11] , Zubair decomposition method (ZDM) [12] , (G’/G)-expansion method [13] , (U’/U)-expansion method [14] , U- expansion method [15] , Fractional sub numerical announcement method [16] [17] , Legendre wavelets technique [18] , Chebyshev wavelets framework [19] [20] [21] , Haar wavelets schema [22] , Legendre Method [23] , Chebyshev strategy [24] , Jacobi polynomial scheme [25] and collocation scheme [26] [27] [28] [29] . All the mentioned approaches have certain limitations like excessive computational work, less efficiency to tackle nonlinearity and divergent solution due to which many issues arise. All these disputes can be fixed with the help of orthogonal polynomials, which is a vital thought in close estimation and structures. These orthogonal polynomials are the reason of powerful strategies of spectral methods [30] [31] [32] . Starting late, Khader [33] displayed a capable numerical procedure for enlightening the fractional order physical problems using the Chebyshev polynomials. In the [34] two Chebyshev spectral frameworks for measuring multi-term fractional problems are displayed. The author (Tamour Zubair) devolve a new wavelets algorithm to construct the numerical solution of nonlinear Bagley-Torvik equation of fractional order which will have less computational works, straight forward and better accuracy as compare to the existing technique. It is to be emphasized that proposed algorithm is tremendously simple but highly effective Moreover, this new pattern is proficient for reducing the computational work to a tangible level while still retaining a very high level of accuracy.
2. Basic Definitions
Fractional Calculus [35] - [40]
We give some basic definitions and properties of the fractional calculus theory which are used further in this paper.
Definition 1. A real function 
is said to be in the space 
if there exists a real number
, such that
, where 
, and it is said to be in the space 
iff
.
Definition 2. The Riemann-Liouville fractional integral operator of order
, of a function 
is defined as
Properties of the operator 
can be found in literature, we mention only the following: For 
and
:
1) 
2) 
3) 
The Riemann-Liouville derivative has certain drawbacks when trying to model real-world processes with fractional differential equations. Therefore, we shall introduce a improved fractional differential operator 
proposed by
.
Definition 3. The fractional derivative of 
in the Caputo sense is defined as
for
. For the Caputo’s derivative we have 
is a constant,
We use the ceiling function 
to denote the smallest integer greater than or equal to
, and
. Recall that for
, the Caputo differential operator coincides with the usual differential operator of integer order.
3. Bagley-Torvik Equations
Bagley-Torvik equation assumes an extremely vital part to study the performance of different material by application of fractional calculus [40] [41] . It has increased its significance in many fields of industrial and applied sciences. Precisely, the equation with 1/2 order derivative or 3/2 order derivative can be model the frequency dependent damping materials. The summed up form of Bagley-Torvik equation is given
(1)
with initial condition
with boundary condition at
, for
, is given by
where 
is the nonlinear operator of the equation, 
is unknown function. 
and 
are the constant coefficients, T is the constant representing the span of input in close interval [0,T], and 
are contents. When we have
where 
is mass of the rigid plate, 
is stiffness of the spring, S is the area of plate immersed in Newtonian fluid, 
is the velocity, and 
is the fluid density then equation (1) represents the motion of large thin plate in a Newtonian fluid [39] . Similarly, linearly damped fractional oscillator with the damping term
has the fraction derivative
.
Further, we will discuss mathematical modeling of BT equation with feed-for- ward artificial neural network. The solution 
of the fractional differential equa-
tion along with its
can be approximated by the following continuous mapping as a neural network methodology [41] [42] [43] [44] :
where 
and 
are bounded real valued adaptive parameters, h is the number of neurons and 
is the active function taken as exponential function.Fractional differential equation neural networks (FDN-NNs) can be approximate as
for
, we get
Using Definition 4, for
, we get
Figure 1. FDE-NN architecture of Bagley-Torvik equation.
The mathematical model can be the linear combinations of the networks represented above. The FDE-NN architecture formulated for Bagley-Torvik equation can be seen in Figure 1. It is clear that the solution 
can be approximated with 
subject to finding appropriate unknown weights.
4. Hermite Polynomials [45]
It is classical orthogonal polynomials play very important role in probability. It has wide applications in numerical analysis as finite element methods as shape functions for beams. They are also applicable in physical quantum theory. Hermite polynomials are categorized into two kinds
The Probabilists Hermite polynomials are the solutions of
where 
and 
is a constant, with the boundary conditions that 
should be polynomially bounded at infinity. The above equation can be written in the form of eigen value problem
solutions are the Eigen functions of the differential operator
. This equation is called Hermite equation, although the term is also used for the closely related equation
whose solutions are the Physicists Hermites Polynomials, which is the second kind of Hermite polynomials.
The Hermite polynomials is given by
where 
and also
.
and 
the two branches of Hermite polynomial of degree
, which are orthogonal with respect to weigh function.
Here we have
.
Further we have orthogonality 
is given by
A function 
can be express in term of Hermite polynomials
where 
coefficients is given by
where
.
5. Fractional Form of Hermite Polynomials [35] - [40]
The explicit formula of Hermites polynomials is
(1*)
where 
is given by
Further we have
(2)
where 
is given by
A function 
can be express in term of Hermite polynomials
(3)
where 
are Hermites polynomials. Using (1*)-(3) and definition of fractional derivative, we get the following
(4)
where 
and 
is given by 
and 
.
Note that only for
, we have following
a) Methodology
Consider the multi order fractional differential equation (1) as
(5)
where 
is the unknown function, to be determined. The proposed technique for solving Equation (5) proceeds in the following three steps:
Step 1: According to the proposed algorithm we assume the following trial solution
(6)
where 
and
.
where 
are Hermite polynomials of degree 
defined in Equation (6) and 
are unknown parameters, to be determined.
Step 2: Substituting Equation (6) into Equation (5), we get
Using (4) we have
(7)
Step 3: Further we Assume suitable collocation point for Equation (7). There- fore, we obtained system has 
equations and 
unknowns. Solving this system gives the unknown coefficients using Conjugate Gradient Method. Putting these constant into trial solution, we can obtained the approximate/exact solutions of linear/nonlinear fractional differential Equation (5).
b) Approximation by Hermite Polynomials [45]
Let us define 
and
. The 
-orthogonal projection 
be the mapping and we have
Due to the orthogonality property, we can write it as
where 
are the constants in the following form
6. Numerical Simulation
In this section, we apply new algorithm to construct approximate/exact solutions fractional differential equation. Numerical results are very encouraging.
Case 1 In Equation (1), we take
, 
, 
, 
, 
, 
,
. The close form solution is
.
Consider the trial solutions for 
as
Using the trail solution into Equation (1) and proceed it according to Step 1 and Step 2, then we collocate it further to generate the system of equations. Solve the system of equations along with initial conditions, we get the values of constants
Finally, we get the approximate solution
which is exact solution.
Case 2 In Equation (1), we take
, 
, 
, 
, 
, 
,
. The close form solution is
.
Consider the trial solutions for 
as
Using the trail solution into Equation (1) and proceed it according to Step 1 and Step 2, then we collocate it further to generate the system of equations. Solve the system of equations along with initial conditions, we get the values of constants
Finally, we get the approximate solution
which is exact solution.
Case 3 In Equation (1), we take
, 
, 
, 
,
. The close form solution is
.
This equation can be simplify by using
Consider the trial solutions for 
as
Using the trail solution into Equation (1) and proceed it according to Step 1 and Step 2, then we collocate it further to generate the system of equations. Solve the system of equations along with initial conditions, we get the values of constants
Finally, we get the approximate solution
which is exact solution.
Case 4 In Equation (1), we take
, 
, 
, 
, 
, 
, 
,
. The close form solution is
.
Using the trail solution into Equation (1) and proceed it according to Step 1 and Step 2, then we collocate it further to generate the system of equations. Solve the system of equations along with initial conditions, we get the values of constants
Finally, we get the approximate solution
which is exact solution.
Case 5. In Equation (1), we take
, 
, 
, 
, 
, 
, 
,
. The close form solution is
.
The numerical solution is represented in Table 1 in case of 
and
, while the error for various values of 
and 
are repre- sented in Table 2. There is a graphical comparison between exact and approximate solution represented in Figure 2.
Table 1. Numerical comparison between exact and approximate solution for deferent values of 
Table 2. Numerical comparison between exact approximate solutions for different values of 
Figure 2. Graphysical comparision between exact and approximted solution.
7. Conclusions
All the facts and findings of the paper are summarized as follow:
・ This paper provides novel study of Bagley-Torvik equations of fractional order in different situations by using newly suggested Hermite Polynomial scheme.
・ Implementation of this methodology is moderately relaxed and with the help of this suggested algorithm, complicated problems can be tackled.
・ It is to be highlighted that the suggested comparison gives attentive respond regarding some particular issues for values of M, which demonstrates viability of the proposed framework. Likewise, the reliability of the application provided this technique a more comprehensive suitability.
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