All Title Author
Keywords Abstract


Tau-Path Following Method for Solving the Riccati Equation with Fractional Order

DOI: 10.1155/2014/207916

Full-Text   Cite this paper   Add to My Lib

Abstract:

A formulation for the fractional Legendre functions is constructed to find the solution of the fractional Riccati equation. The fractional derivative is described in the Caputo sense. The method is based on the Tau Legendre and path following methods. Theoretical and numerical results are presented. Analysis for the presented method is given. 1. Introduction Recently, many papers on fractional boundary value problems have been studied extensively. Several forms of them have been proposed in standard models, and there has been significant interest in developing numerical schemes for their solutions. Several numerical techniques are used to solve such problems such as Laplace and Fourier transforms [1, 2], Adomian decomposition and variational iteration methods [3, 4], eigenvector expansion [5], differential transform and finite differences methods [6, 7], power series method [8], collocation method [9], and wavelet method [10, 11]. Many applications of fractional calculus on various branches of science such as engineering, physics, and economics can be found in [12, 13]. Considerable attention has been given to the theory of fractional ordinary differential equations and integral equations [14, 15]. Additionally, the existence of solutions of ordinary and fractional boundary value problems using monotone iterative sequences has been investigated by several authors [16–20]. We consider the Riccati equation with fractional orders of the form where , and are continuous functions on and?? is a constant. Riccati equation with fractional order has been discussed by many researchers using different techniques such as collocation method based on Muntz polynomials [21], homotopy perturbation method [22], and series solution method [23]. In this paper we study the Tau-path following method for solving the Riccati equation with fractional order. We organize this paper as follows. In Section 2, we present basic definitions and results of fractional derivatives. We extend basic results to path following method for the fractional case. In Section 3, we introduce the fractional-order Legendre Tau method with path following method for solving the Riccati equation with fractional order. In Section 4, we present some numerical results to illustrate the efficiency of the presented method. Finally, we conclude with some comments in the last section. 2. Preliminaries In this section, we review the definition and some preliminary results of the Caputo fractional derivatives, as well as, the definition of the fractional-order Legendre functions and their properties. Definition

References

[1]  L. Gaul, P. Klein, and S. Kemple, “Damping description involving fractional operators,” Mechanical Systems and Signal Processing, vol. 5, no. 2, pp. 81–88, 1991.
[2]  I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, New York, NY, USA, 1999.
[3]  K. Al-Khaled and S. Momani, “An approximate solution for a fractional diffusion-wave equation using the decomposition method,” Applied Mathematics and Computation, vol. 165, no. 2, pp. 473–483, 2005.
[4]  S. Das, “Analytical solution of a fractional diffusion equation by variational iteration method,” Computers and Mathematics with Applications, vol. 57, no. 3, pp. 483–487, 2009.
[5]  L. E. Suarez and A. Shokooh, “An eigenvector expansion method for the solution of motion containing fractional derivatives,” Journal of Applied Mechanics, Transactions ASME, vol. 64, no. 3, pp. 629–634, 1997.
[6]  A. Arikoglu and I. Ozkol, “Solution of fractional integro-differential equations by using fractional differential transform method,” Chaos, Solitons and Fractals, vol. 40, no. 2, pp. 521–529, 2009.
[7]  M. M. Meerschaert and C. Tadjeran, “Finite difference approximations for two-sided space-fractional partial differential equations,” Applied Numerical Mathematics, vol. 56, no. 1, pp. 80–90, 2006.
[8]  Z. M. Odibat and N. T. Shawagfeh, “Generalized Taylor's formula,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 286–293, 2007.
[9]  Q. M. Al-Mdallal, M. I. Syam, and M. N. Anwar, “A collocation-shooting method for solving fractional boundary value problems,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 12, pp. 3814–3822, 2010.
[10]  Y. LI, “Solving a nonlinear fractional differential equation using Chebyshev wavelets,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 9, pp. 2284–2292, 2010.
[11]  J. L. Wu, “A wavelet operational method for solving fractional partial differential equations numerically,” Applied Mathematics and Computation, vol. 214, no. 1, pp. 31–40, 2009.
[12]  A. Kilbas, H. Srivastave, and J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Hollan Mathematics Studies, vol. 204, Elsevier Science, Amsterdam, The Netherlands, 2006.
[13]  S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivative, Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993.
[14]  V. Lakshmikantham and A. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis, Theory, Methods & Applications, vol. 69, no. 8, pp. 2677–2682, 2008.
[15]  S. Zhang, “Existence of solution for a boundary value problem of fractional order,” Acta Mathematica Scientia, vol. 26, no. 2, pp. 220–228, 2006.
[16]  R. P. Agarwal, M. Benchohra, and S. Hamani, “A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions,” Acta Applicandae Mathematicae, vol. 109, no. 3, pp. 973–1033, 2010.
[17]  M. Al-Refai and M. Ali Hajji, “Monotone iterative sequences for nonlinear boundary value problems of fractional order,” Nonlinear Analysis, Theory, Methods and Applications, vol. 74, no. 11, pp. 3531–3539, 2011.
[18]  V. Lakshmikantham and A. S. Vatsala, “General uniqueness and monotone iterative technique for fractional differential equations,” Applied Mathematics Letters, vol. 21, no. 8, pp. 828–834, 2008.
[19]  C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, NY, USA, 1992.
[20]  K. Oldham and J. Spanier, The Fractional Calculus, Academic, New York, NY, USA, 1974.
[21]  S. Esmaeili, M. Shamsi, and Y. Luchko, “Numerical solution of fractional differential equations with a collocation method based on Mntz polynomials,” Computers and Mathematics with Applications, vol. 62, no. 3, pp. 918–929, 2011.
[22]  N. A. Khan, A. Ara, and M. Jamil, “An efficient approach for solving the Riccati equation with fractional orders,” Computers and Mathematics with Applications, vol. 61, no. 9, pp. 2683–2689, 2011.
[23]  J. Cang, Y. Tan, H. Xu, and S.-J. Liao, “Series solutions of non-linear Riccati differential equations with fractional order,” Chaos, Solitons and Fractals, vol. 40, no. 1, pp. 1–9, 2009.
[24]  M. Caputo, “Linear models of dissipation whose Q is almost frequency independent, Part II Geophys,” Mathematics & Physical Sciences, vol. 13, no. 5, pp. 529–539, 1967.
[25]  W. A. Al-Salam, “On the product of two Legendre polynomials,” Mathematica Scandinavica, vol. 4, pp. 239–242, 1956.
[26]  H. I. Siyyam and M. I. Syam, “An accurate solution of the Poisson equation by the Chebyshev-Tau method,” Journal of Computational and Applied Mathematics, vol. 85, no. 1, pp. 1–10, 1997.
[27]  C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer, New York, NY, USA, 1988.
[28]  C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer, New York, NY, USA, 2006.
[29]  E. L. Allgower and K. Georg, Numerical Continuation Methods, Springer, New York, NY, USA, 1990.
[30]  M. I. Syam and S. M. Al-Sharo', “Collocation-continuation technique for solving nonlinear ordinary boundary value problems,” Computers and Mathematics with Applications, vol. 37, no. 10, pp. 11–17, 1999.
[31]  H. I. Siyyam and M. I. Syam, “The modified trapezoidal rule for line integrals,” Journal of Computational and Applied Mathematics, vol. 84, no. 1, pp. 1–14, 1997.
[32]  M. Merdan, “On the solutions fractional Riccati di¤erential equation with modifed Riemann-Liouville derivative,” International Journal of Differential Equations, vol. 2012, pp. 1–17, 2012.

Full-Text

comments powered by Disqus