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.

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.

Zubair, T. , Sajjad, M. , Madni, R. and Shabir, A. (2017) Hermite Solution of Bagley-Torvik Equation of Fractional Order.

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https://doi.org/10.1016/j.physleta.2007.05.083

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https://doi.org/10.1016/j.jmaa.2008.04.007

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http://www.ijaamm.com/uploads/2/1/4/8/21481830/paper_2_sayed.pdf

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onal_Sub-Equation_Method_and_its_Applications_to_the_Space-

Time_Fractional_Differential_Equations_in_Mathematical_Physics

https://doi.org/10.9734/BJMCS/2013/2908

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http://www.sciencedirect.com/science/article/pii/S0898122111003257

https://doi.org/10.1016/j.camwa.2011.04.024

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https://www.researchgate.net/publication/235916821_A_com

putational_matrix_method_for_solving_systems_of_high_order_f

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