SOLVING FRACTIONAL DIFFUSION AND FRACTIONAL DIFFUSION-WAVE EQUATIONS BY PETROV-GALERKIN FINITE ELEMENT METHOD

Year 2014, Volume: 4 Issue: 2, 155 - 168, 01.12.2014

A. Esen Y. Uucar M. Yagmurlu O. Tasbozan

Abstract

In the last few years, it has become highly evident that fractional calculus has been widely used in several areas of science. Because of this fact, their numerical solutions also have become urgently important. In this manuscript, numerical solutions of both the fractional diffusion and fractional diffusion-wave equations have been obtained by a Petrov-Galerkin finite element method using quadratic B-spline base functions as trial functions and linear B-spline base functions as the test functions. In those equations, fractional derivatives are used in terms of the Caputo sense. While the L1 discretizaton formula has been applied to fractional diffusion equation, the L2 discretizaton formula has been applied to the fractional diffusion-wave equation. Finally, the error norms L2 and L∞ have been calculated for testing the accuracy of the proposed scheme.

Keywords

Finite element method, Petrov-Galerkin method, Fractional diffusion equation, Fractional diffusion-wave equation, Quadratic B-Spline, Linear B-Spline

References

• A. A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equa- tions, Elsevier, Amsterdam, 2006.
• C. elik and M. Duman, Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative, Journal of Computational Physics, 231(2012) 1743-1750.
• D.L. Logan, A First Course in the Finite Element Method (Fourth Edition), Thomson, 2007.
• H.G. Sun, W. Chen, K.Y. Sze, A semi-analytical Şnite element method for a class of time-fractional diffusion equations, arXiv: 1109.0641v1, [math-ph], 3 Sep 2011.
• I. Karatay, S.R. Bayramoglu, A. Sahin, Implicit difference approximation for the time fractional heat equation with the nonlocal condition, Appl. Numer. Math., 61 (2011) 1281–1288.
• I. Podlubny, Fractional Differential Equations, Acaademic Press, San Diego, 1999.
• J. Quintana-Murillo, S.B. Yuste, An Explicit Difference Method for solving Fractional Diffusion and Diffusion-Wave Equations in the Caputo Form, J. Comput. Nonlinear Dynam. 6, 021014 (2011).
• K. B. Oldham, J. Spanier, The Fractional Calculus, Academic, New York, 1974.
• Mainardi F., 1995, “Fractional Diffusive Waves in Viscoeslactic Solids.”, Non-linear Waves in Solids
• J.L. Wegner and F.R. Norwood, eds., ASME/AMR, FairŞeld, NJ, pp. 93-97. Mainardi, F. and Paradisi, P., 1997, “A Model of Diffusive Waves in Viscoelasticity Based on Fractional
• Calculus,” Proceedings of the 36th Conference on Decision and Control, O.R. Gonzales, ed., San Diego, CA, pp. 4961-4966.
• M.M. Meerschaert, C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J.Comput. Appl. Math. 172 (2004) 65-77.
• N.H. Sweilam, M.M. Khader and A.M.S. Mahdy, Crank-Nicolson Finite Difference Method For Solving Time-Fractional Diffusion Equation, 2(2012)1-9.
• O.P. Agrawal, Solution for a fractional diffusion-wave equation deŞned in a bounded domain, Nonlin. Dynam., 29 (2002) 145-155.
• P. M. Prenter, Splines and Variational Methods, New York, John Wileyi, 1975.
• R. Gorenflo, F. Mainardi, D. Moretti, G. Pagnini, P.Paradisi, Discrete random walk models for space- time fractional diffusion, Chem. Phys., 284 (2002) 521-541.
• S.B. Yuste, Weighted average Şnite difference methods for fractional diffusion equations, J. Comput. Phys., 216 (2006) 264-274.
• S.B. Yuste, L. Acedo, An explicit Şnite difference method and a new von Neumann-type stability analysis for fractional diffusion equations, SIAM J. Numer. Anal. 42 (2005) 1862-1874.
• S. Monami and Z. Odibat, Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Physics Letters A, 355(2006) 271-279.
• S. S. Ray, Exact Solutions for Time-Fractional Diffusion-Wave Equations by Decomposition Method, Phys. Scr., 75, 53-61, 2007.
• T.A.M. Langlands, B.I. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comput. Phys. 205 (2005) 719-736.
• V.E. Lynch, B.A. Carreras, D. del-Castillo-Negrete, K.M. Ferreira-Mejias, H.R. Hicks, Numerical methods for the solution of partial differential equations of fractional order, J. Comput. Phys. 192 (2003) 406-421.