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  - 

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 

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.
References
   Deng, W. (2008) Finite Element Method for the Space and Time Fractional Fokker-Planck Equation. SIAM Journal on Numerical Analysis, 47, 204-226.
http://epubs.siam.org/doi/abs/10.1137/080714130
https://doi.org/10.1137/080714130

   Gao, G.H., Sun, Z.Z. and Zhang, Y.N. (2012) A Finite Difference Scheme for Fractional Sub-Diffusion Equations on an Unbounded Domain Using Artificial Boundary Conditions. Journal of Computational Physics, 231, 2865-2879.
https://www.hindawi.com/journals/mpe/2012/924956/
https://doi.org/10.1016/j.jcp.2011.12.028

   Momani, S., Odibat, Z. and Erturk, V.S. (2007) Generalized Differential Transform Method for Solving a Space- and Time-Fractional Diffusion-Wave Equation. Physics Letters A, 370, 379-387.
https://www.mutah.edu.jo/userhomepages/shmomani/public.htm
https://doi.org/10.1016/j.physleta.2007.05.083

   Odibat, Z. and Momani, S. (2008) A Generalized Differential Transform Method for Linear Partial Differential Equations of Fractional Order. Applied Mathematics Letters, 21, 194-199.
https://pdfs.semanticscholar.org/7250/38695674846cdbb7c5f5d2d9cbcb5f4e0fc9.pdf
https://doi.org/10.1016/j.aml.2007.02.022

   Hu, Y., Luo, Y. and Lu, Z. (2008) Analytical Solution of the Linear Fractional Differential Equation by Adomian Decomposition Method. Journal of Computational and Applied Mathematics, 215, 220-229.
https://pdfs.semanticscholar.org/58a6/966afe30ccc2f7bbc6b
17319d6a3d1a663ba.pdf
https://doi.org/10.1016/j.cam.2007.04.005

   El-Sayed, A.M.A. and Gaber, M. (2006) The Adomian Decomposition Method for Solving Partial Differential Equations of Fractal Order in Finite Domains. Physics Letters A, 359, 175-182.
https://www.researchgate.net/publication/223114291_The_A
domian_decomposition_method_for_solving_partial_differential
_equations_of_fractal_order_in_finite_domains
https://doi.org/10.1016/j.physleta.2006.06.024

   El-Sayed, A.M.A., Behiry, S.H. and Raslan, W.E. (2010) Adomian’s Decomposition Method for Solving an Intermediate Fractional Advection-Dispersion Equation. Computers & Mathematics with Applications, 59, 1759-1765.
http://www.sciencedirect.com/science/article/pii/S0898122109005537
https://doi.org/10.1016/j.camwa.2009.08.065

   Inc, M. (2008) The Approximate and Exact Solutions of the Space- and Time-Fractional Burgers Equations with Initial Conditions by Variational Iteration Method. Journal of Mathematical Analysis and Applications, 345, 476-484.
http://www.sciencedirect.com/science/article/pii/S0022247X08003739
https://doi.org/10.1016/j.jmaa.2008.04.007

   Odibat, Z. and Momani, S. (2009) The Variational Iteration Method: An Efficient Scheme for Handling Fractional Partial Differential Equations in Fluid Mechanics. Computers & Mathematics with Applications, 58, 2199-2208.
http://www.sciencedirect.com/science/article/pii/S0898122109001436
https://doi.org/10.1016/j.camwa.2009.03.009

   Wu, G.C. and Lee, E.W.M. (2010) Fractional Variational Iteration Method and Its Application. Physics Letters A, 374, 2506-2509.
https://zulfahmed.files.wordpress.com/2015/06/2010-physic
s-letters-section-a-general-atomic-and-solid-state-physics-37425.pdf
https://doi.org/10.1016/j.physleta.2010.04.034

   He, J.H. (2003) Homotopy Perturbation Method: A New Nonlinear Analytical Technique. Applied Mathematics and Computation, 135, 73-79.
https://www.researchgate.net/publication/242791050_Homo
topy_perturbation_method_A_new_nonlinear_analytical_technique
https://doi.org/10.1016/S0096-3003(01)00312-5

   Zubair, T., Hamid, M., Saleem, M. and Mohyud-Din, S.T. (2015) Numerical Solution of Infinite Boundary Integral Equations. International Journal of Modern Applied Physics, 5, 18-25.
https://www.researchgate.net/publication/311202189_Nume
rical_Solution_of_Infinite_Boundary_Integral_Equations

   Bin, Z. (2012) (G’/G)-Expansion Method for Solving Fractional Partial Differential Equations in the Theory of Mathematical Physics. Communications in Theoretical Physics, 58, 623-630.
https://doi.org/10.1088/0253-6102/58/5/02

   Usman, M. and Mohyud-Din, S.T. (2013) Traveling Wave Solutions of 7th Order Kaup Kuperschmidt and Lax Equations of Fractional-Order. International Journal of Advances in Applied Mathematics and Mechanics, 1, 17-34.

   Usman, M. and Mohyud-Din, S.T. (2014) U-Expansion Method for 5th Order Kaup Kuperschmidt and Lax Equation of Fractional Order. International Journal of Modern Mathematical Sciences, 9, 63-81.
http://www.modernscientificpress.com/journals/ViewArticle.
aspx?XBq7Uu+HD/8eRjFUGMqlRUFMGJSojFCd8JeYyjsMmViT
RlEyTDVgYMCmavqTokhF

   Alzaidy, J.F. (2013) Fractional Sub-Equation Method and Its Applications to the Space-Time Fractional Differential Equations in Mathematical Physics. British Journal of Mathematics & Computer Science, 3, 153-163.
https://www.researchgate.net/publication/271263179_Fracti
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

   Alzaidy, J.F. (2013) The Fractional Sub-Equation Method and Exact Analytical Solutions for Some Nonlinear Fractional PDEs. American Journal of Mathematical Analysis, 1, 14-19.
http://pubs.sciepub.com/ajma/1/1/3/

   Jafari, H., Yousefi, S.A., Firoozjaee, M.A., Momani, S. and Khalique, C.M. (2011) Application of Legendre Wavelets for Solving Fractional Differential Equations. Computers & Mathematics with Applications, 62, 1038-1045.
http://www.sciencedirect.com/science/article/pii/S0898122111003257
https://doi.org/10.1016/j.camwa.2011.04.024

   Khader, M.M., El Danaf, T.S. and Hendy, A.S. (2013) A Computational Matrix Method for Solving Systems of High Order Fractional Differential Equations. Applied Mathematical Modelling, 37, 4035-4050.
https://www.researchgate.net/publication/235916821_A_com
putational_matrix_method_for_solving_systems_of_high_order_f
ractional_differential_equations
https://doi.org/10.1016/j.apm.2012.08.009

   Zhu, L. and Fan, Q.B. (2012) Solving Fractional Nonlinear Fredholm Integro-Differential Equations by the Second Kind Chebyshev Wavelet. Communications in Nonlinear Science and Numerical Simulation, 17, 2333-2341.
live.oscarjournals.springer.com/track/pdf/10.1186/s13662-017-
viewlive.oscarjournals.springer.com
https://doi.org/10.1016/j.cnsns.2011.10.014

   Agarwal, N. (1953) A Propos d’unc Note de H4. Pierre Humbert, C. R. Se’ances Acad. Sci., 236, 2031-2032.
http://jnus.org/pdf/1/2014/1/1038.pdf

   Li, Y. and Zhao, W. (2010) Haar Wavelets Operational Matrix of Fractional Order Integration and Its Applications in Solving the Fractional Order Differential Equations. Applied Mathematics and Computation, 216, 2276-2285.
http://dl.acm.org/citation.cfm?id=2641414
https://doi.org/10.1016/j.amc.2010.03.063

   Khader, M.M. and Hendy, A.S. (2012) The Approximate and Exact Solutions of the Fractional-Order Delay Differential Equations Using Legendre Pseudo Spectral Method. International Journal of Pure and Applied Mathematics, 74, 287-297.
http://www.ijpam.eu/contents/2012-74-3/1/1.pdf

   Sweilam, N.H. and Khader, M.M. (2010) A Chebyshev Pseudo-Spectral Method for Solving Fractional Integro-Differential Equations. The ANZIAM Journal, 51, 464-475.
https://www.cambridge.org/core/journals/anziam-journal/arti
cle/div-classtitlea-chebyshev-pseudo-spectral-method-for-solvin
g-fractional-order-integro-differential-equationsdiv/D54540C52
DE837C74F79EDB24A32FE71
https://doi.org/10.1017/S1446181110000830

   Beheshti, S., Khosravian-Arab, H. and Zare, I. (2012) Numerical Solution of Fractional Differential Equations by Using the Jacobi Polynomials. Bulletin of the Iran Mathematical Society, 39, 6461-6470.
http://bims.iranjournals.ir/article_947_b3ed3a5b2e22cf9386624a6699f3ff0d.pdf

   Xu, C.-L. and Guo, B.-Y. (2002) Laguerre Pseudospectral Method for Non-Linear Partial Differential Equations. Journal of Computational Mathematics-International Edition, 20, 413-428.
http://dergi.cumhuriyet.edu.tr/cumuscij/article/view/5000118841

   Razzaghi, M. and Yousefi, S. (2001) Legendre Wavelets Method for the Solution of Nonlinear Problems in the Calculus of Variations. Mathematical and Computing Modelling, 34, 45-54.
http://www.sciencedirect.com/science/article/pii/S0895717701000486

   Khader, M.M. (2012) Introducing an Efficient Modification of the Homotopy Perturbation Method by Using Chebyshev Polynomials. Arab Journal of Mathematical Sciences, 18, 61-71.
http://www.sciencedirect.com/science/article/pii/S131951661100051X
https://doi.org/10.1016/j.ajmsc.2011.09.001

   Khader, M.M., Sweilam, N.H. and Mahdy, A.M.S. (2011) An Efficient Numerical Method for Solving the Fractional Diffusion Equation. Journal of Applied Mathematics and Bioinformatics, 1, 1-12.

   Hussaini, M.Y. and Zang, T.A. (1987) Spectral Methods in Fluid Dynamics. Annual Review of Fluid Mechanics, 19, 339-367.
http://www.annualreviews.org/doi/abs/10.1146/annurev.fl.19.010187.002011
https://doi.org/10.1146/annurev.fl.19.010187.002011

   Funaro, D. (1992) Polynomial Approximation of Differential Equations. Springer Verlag, New York, 8.
http://www.springer.com/gb/book/9783662138786

   Khader, M.M., El Danaf, T.S. and Hendy, A.S. (2012) Efficient Spectral Collocation Method for Solving Multi-Term Fractional Differential Equations Based on the Generalized Laguerre Polynomials. Fractional Calculus Application, 3, 1-14.
http://naturalspublishing.net/files/published/vk2130l11wp11o.pdf

   Khader, M.M. (2011) On the Numerical Solutions for the Fractional Diffusion Equation. Communications in Nonlinear Science and Numerical Simulations, 16, 2535-2542.
http://www.ijpam.eu/contents/2013-84-4/1/1.pdf

   Doha, E.H., Bahrawy, A.H. and Ezz-Eldien, S.S. (2011) Efficient Chebyshev Spectral Methods for Solving Multi-Term Fractional Orders Differential Equations. Applied Mathematical Modelling, 35, 5662-5672.
http://naturalspublishing.net/files/published/vk2130l11wp11o.pdf
https://doi.org/10.1016/j.apm.2011.05.011

   Dalir, M. and Bashour, M. (2010) Applications of Fractional Calculus. Applied Mathematical Sciences, 4, 1021-1032.
https://pdfs.semanticscholar.org/b9f3/cebf62c66c7bc06eab0
09aa1d60d70a19312.pdf

   Schneider, K., Kevlahan, N.K.R. and Farge, M. (1997) Comparison of an Adaptive Wavelet Method and Nonlinearly Filtered Pseudospectral Methods for Two-Dimensional Turbulence. Theoretical and Computational Fluid Dynamics, 9, 919-206.

   Erdelyi, A. (1955) Higher Transcendental Functions, McGrawHill, New York.
http://apps.nrbook.com/bateman/Vol3.pdf

   Bagley, R.L. and Torvik, P.J. (1983) Fractional Calculus—A Different Approach to the Analysis of Viscoelastically Damped Structures. AIAA Journal, 21, 741-748.
https://arc.aiaa.org/doi/abs/10.2514/3.8142
https://doi.org/10.2514/3.8142

   Podlubny, I. (1999) Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego.
http://www.sciepub.com/reference/3051

   Bagley, R.L. and Torvik, P.J. (1984) On the Appearance of the Fractional Derivative in the Behavior of Real Materials. Journal of Applied Mechanics, 51, 294-298.
http://appliedmechanics.asmedigitalcollection.asme.org/article.aspx?articleid=1407517
https://doi.org/10.1115/1.3167615

   Zahoor, M.A., Khan Raja, J.A. and Qureshi, I.M. (2010) Heuristic Computational Approach Using Swarm Intelligence in Solving Fractional Differential Equations. Proceedings of the 12th Annual Genetic and Evolutionary Computation Conference, Portland, 2023-2026.
https://www.hindawi.com/journals/cin/2012/721867/ref/

   Ray, S.S. and Bera, R.K. (2005) Analytical Solution of the Bagley Torvik Equation by Adomian Decomposition Method. Applied Mathematics and Computation, 168, 398-410.
http://www.ijpam.eu/contents/2016-110-2/3/

   Zahoor, R.M.A., Khan, J.A. and Qureshi, I.M. (2009) Evolutionary Computation Technique for Solving Riccati Differential Equation of Arbitrary Order. World Academy of Science, Engineering and Technology, 58, 531-536.