### Hermite collocation method for fractional order differential equations

#### Abstract

This paper focuses on the approximate solutions of the higher order fractional differential equations with multi terms by the help of Hermite Collocation method (HCM). This new method is an adaptation of Taylor's collocation method in terms of truncated Hermite Series. With this method, the differential equation is transformed into an algebraic equation and the unknowns of the equation are the coefficients of the Hermite series solution of the problem. This method appears as a useful tool for solving fractional differential equations with variable coefficients. To show the pertinent feature of the proposed method, we test the accuracy of the method with some illustrative examples and check the error bounds for numerical calculations.

#### Full Text:

PDF#### References

Kilbas, A.A., Sirvastava, H.M., Trujillo, J.J. (2006). Theory and Application of Fractional Differential Equations. North-Holland Mathematics Studies. Vol 204, Amsterdam.

Mainardi, F., Luchko, Yu., Pagnini, G. (2001). The fundamental Solution of the Space-Time Fractional Diffusion Equation. Frac. Calc. Appl. Anal., 4(2), 153-152.

Ross, B. (1975). Fractional Calculus and its Applications. Lecture Notes in Mathematics. Vol. 457, Springer Verlag.

Stanislavsky, A.A. (2004) Fractional Oscillator. Phys. Rev. E., 70(5).

Trujillo, J.J. (1999). On a Riemann-Liouville Generalised Taylor’s Formula. J. Math. Anal. Appl., 231, 255-265.

Podlubny, I. (1999). Fractional Differential Equations. Academic Press, New York.

Ciesielski, M., Leszcynski, J. (2003). Numerical simulations of anomalous diffusion. Computer Methods Mech. Conference, Gliwice Wisla Poland.

Yuste, S.B. (2006). Weighted average finite difference methods for fractional diffusion equations. J. Comput. Phys., 1, 264-274.

Odibat, Z.M. (2006). Approximations of fractional integrals and Caputo derivatives. Appl. Math. Comput., 527-533.

Odibat, Z.M. (2009). Computational algorithms for computing the fractional derivatives of functions. Math. Comput. Simul., 79(7), 2013-2020.

Ford, N.J., Connolly, A.Joseph. (2009). Systemsbased decomposition schemes for the approximate solution of multi-term fractional differential equations. J. Comput. Appl. Math., 229, 382-391.

Ford, N.J. (2001). Simpson, A. Charles, The numerical solution of fractional differential equations: speed versus accuracy. Numer. Alg., 26, 333-346.

Sweilam, N.H., Khader, M.M., Al-Bar, R.F. (2007). Numerical studies for a multi order fractional differential equations. Phys. Lett. A, 371, 26-33.

Momani, S., Odibat, Z. (2006). Analytical solution of a time-fractional Navier-Stokes equation by adomian decomposition method. Appl. Math. Comput., 177, 488-494.

Odibat, Z., Momani, S. (2007). Numerical approach to differential equations of fractional order. J. Comput.Appl. Math., 207(1), 96-110.

Odibat, Z., Momani, S. (2008). Numerical methods for nonlinear partial differential equations of fractional order. Appl. Math. Model., 32, 28-39.

Odibat, Z., Momani, S. (2006). Application of variational iteration method to equation of fractional order. Int. J. Nonlinear Sci. Numer. Simul., 7, 271-279.

Wu, J.L. (2009). A wavelet operational method for solving fractional partial differential equations numerically. Appl. Math. Comput., 214, 31-40.

Erturk, V.S., Momani, S. (2008). Solving systems of fractional differential equations using differential transform method. J. Comput. Appl. Math., 215, 142-151.

Yalcınbas, S., Konuralp, A., Demir, D.D., Sorkun, H.H. (2010). The solution of the fractional differential equation with the generalized Taylor collocation method. IJRRAS, August 2010.

Abd-Elhameed, W.M., Youssri, Y.H. (2016). A Novel Operational Matrix of Caputo Fractional Derivatives of Fibonacci Polynomials: Spectral Solutions of Fractional Differential Equations. Entropy, 18(10),345.

Abd-Elhameed, W.M., Youssri, Y.H. (2016). Spectral Solutions for Fractional Differential Equations via a Novel Lucas Operational Matrix of Fractional Derivatives. Romanian Journal of Physics, 61(5-6), 795-813.

Abd-Elhameed, W.M., Youssri, Y.H., (2015). New Spectral Solutions of Multi-Term Fractional-Order Initial Value Problems With Error Analysis. CMES: Computer Modeling in Engineering and Sciences, 105(5), 375-398.

Youssri, H., Abd-Elhameed W.M. (2016). Spectral Solutions for Multi-Term Fractional Initial Value Problems Using a New Fibonacci Operational Matrix of Fractional Integration. Progress in Fractional Differentiation and Applications, 2(2), 141-151.

Kumar, P., Agrawal, O.P. (2006). An approximate method for numerical solution of fractional differential equations. Signal Process, 86, 2602-2610.

Kumar, P. (2006). New numerical schemes for the solution of fractional differential equations. Southern Illinois University at Carbondale. Vol 134.

Kumar, P., Agrawal, O.P. (2006). Numerical scheme for the solution of fractional differential equations of order greater than one. J. Comput. Nonlinear Dyn., 1, 178-185.

Yavuz M., Ozdemir, N. (2018). A Different Approach to the European Option Pricing Model with New Fractional Operator. Mathematical Modelling of Natural Phenomena, 13(1),1-12.

Avci, D., Eroglu Iskender, B.B, Ozdemir, N. (2017). Conformable heat equation on a radial symmetric plate. Thermal Science, 21(2), 819-826.

Yavuz, M., Ozdemir, N. (2018). Numerical inverse Laplace homotopy technique for fractional heat equations. Thermal Science, 22(1), 185-194.

Dattoli, G. (2004). Laguerra and generalized Hermite polynomials: the point of view of the operation method. Integral Transforms Special Func., 15, 93-99.

Akgonullu, N., Sahin, N., Sezer, M. (2010). A Hermite Collocation Method for the Approximate Solutions of Higher-Order Linear Fredholm Integro-Differential Equations. Numerical Methods for Partial Differential Eqautions,27(6), 1708-1721.

Caputo, M. (1967). Linear models of dissipation whose Q is almost frequency independent II. Geophys. J. Royal Astron. Soc., 13, 529-539.

Samko,S., Kilbas, A., Marichev, O. (1993). Fractional integral and derivatives theory and applications. Gordon and Breach, New York.

Diethelm, K., Ford, N.J. (2002). Numerical Solution of the Bagley-Torvik Equation. BIT, 42, 490-507.

Diethelm K. (1997). An algorithm for the numerical solution of differential equations of fractional order. Electronic Transactions on Numerical Analysis,5, 1-6.

DOI: http://dx.doi.org/10.11121/ijocta.01.2018.00610

### Refbacks

- There are currently no refbacks.

Copyright (c) 2018 Fatma AYAZ

This work is licensed under a Creative Commons Attribution 4.0 International License.