fferential 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.
Cite this paper
Zubair, T. , Sajjad, M. , Madni, R. and Shabir, A. (2017) Hermite Solution of Bagley-Torvik Equation of Fractional Order. International Journal of Modern Nonlinear Theory and Application, 6, 104-118. doi: 10.4236/ijmnta.2017.63010.
. This equation is called Hermite equation, although the term is also used for the closely related equation
.
and 
the two branches of Hermite polynomial of degree
, which are orthogonal with respect to weigh function.
.
is given by
can be express in term of Hermite polynomials
coefficients is given by
.
(1*)
is given by
(2)
is given by
can be express in term of Hermite polynomials
(3)
are Hermites polynomials. Using (1*)-(3) and definition of fractional derivative, we get the following
(4)
and 
is given by 
and 
.
, we have following
(5)
is the unknown function, to be determined. The proposed technique for solving Equation (5) proceeds in the following three steps:
(6)
and
.
are Hermite polynomials of degree 
defined in Equation (6) and 
are unknown parameters, to be determined.
(7)
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).
and
. The 
-orthogonal projection 
be the mapping and we have
are the constants in the following form
, 
, 
, 
, 
, 
,
. The close form solution is
.
as
, 
, 
, 
, 
, 
,
. The close form solution is
.
as
, 
, 
, 
,
. The close form solution is
.
as
, 
, 
, 
, 
, 
, 
,
. The close form solution is
.
, 
, 
, 
, 
, 
, 
,
. The close form solution is
.
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.
