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In this paper, a new methodology of fractional derivatives based upon Hermite polynomial is projected. The fractional derivatives are demonstrated according to Caputo sense. Hermite collocation technique is introduced to express the definite results of Bagley-Torvik Equations. The appropriateness and straightforwardness of numerical plan is presented by graphs and error tables.

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 [

We give some basic definitions and properties of the fractional calculus theory which are used further in this paper.

Definition 1. A real function

Definition 2. The Riemann-Liouville fractional integral operator of order

Properties of the operator

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

Definition 3. The fractional derivative of

for

We use the ceiling function

Bagley-Torvik equation assumes an extremely vital part to study the performance of different material by application of fractional calculus [

with initial condition

with boundary condition at

where

where

has the fraction derivative

Further, we will discuss mathematical modeling of BT equation with feed-for- ward artificial neural network. The solution

tion along with its arbitrary order derivative

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

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

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

whose solutions are the Physicists Hermites Polynomials, which is the second kind of Hermite polynomials.

The Hermite polynomials is given by

where

and also

Here we have

Further we have orthogonality

A function

where

where

The explicit formula of Hermites polynomials is

where

Further we have

where

A function

where

where

Note that only for

a) Methodology

Consider the multi order fractional differential equation (1) as

where

Step 1: According to the proposed algorithm we assume the following trial solution

where

where

Step 2: Substituting Equation (6) into Equation (5), we get

Using (4) we have

Step 3: Further we Assume suitable collocation point for Equation (7). There- fore, we obtained system has

b) Approximation by Hermite Polynomials [

Let us define

Due to the orthogonality property, we can write it as

where

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

Consider the trial solutions for

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

Consider the trial solutions for

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

This equation can be simplify by using

Consider the trial solutions for

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

Finally, we get the approximate solution

which is exact solution.

Case 5. In Equation (1), we take

The numerical solution is represented in

0 | 0.00000E+00 | 0.00000E+00 |

0.1 | 9.00000E−05 | 6.42250E−45 |

0.2 | 1.28000E−03 | 9.13422E−44 |

0.3 | 5.67000E−03 | 4.04617E−43 |

0.4 | 1.53600E−02 | 1.09611E−42 |

0.5 | 3.12500E−02 | 2.23003E−42 |

0.6 | 5.18400E−02 | 3.69936E−42 |

0.7 | 7.20300E−02 | 5.14014E−42 |

0.8 | 8.19200E−02 | 5.84590E−42 |

0.9 | 6.56100E−02 | 4.68200E−42 |

1.0 | 0.00000E+00 | 0.00000E+00 |

0 | 0.00000E+00 | 4.00000E−100 |

0.1 | 9.08224E−32 | 7.57685E−45 |

0.2 | 1.80074E−31 | 1.07760E−43 |

0.3 | 2.65365E−31 | 4.77341E−43 |

0.4 | 3.42665E−31 | 1.29312E−42 |

0.5 | 4.05488E−31 | 2.63085E−42 |

0.6 | 4.44066E−31 | 4.36426E−42 |

0.7 | 4.44538E−31 | 6.06400E−42 |

0.8 | 3.88121E−31 | 6.89662E−42 |

0.9 | 2.50300E−31 | 5.52352E−42 |

1.0 | 8.00000E−100 | 2.00000E−99 |

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.

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. https://doi.org/10.4236/ijmnta.2017.63010