Nonstandard finite difference schemes for a fractional-order Brusselator system

In this paper, we discuss numerical methods for fractional order problems. Some nonstandard finite difference schemes are presented and investigated. The application in the simulation of a fractional-order Brusselator system is hence presented. By means of some numerical experiments, we show the effectiveness of the proposed approach.

[1]  C. Lubich Discretized fractional calculus , 1986 .

[2]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[3]  Ronald E. Mickens,et al.  Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition , 2007 .

[4]  Dynamically consistent discrete Lotka-Volterra competition models derived from nonstandard finite-difference schemes , 2007 .

[5]  Roberto Garrappa,et al.  On Multistep Methods for Differential Equations of Fractional Order , 2006 .

[6]  Ronald E. Mickens,et al.  A nonstandard finite-difference scheme for the LotkaVolterra system , 2003 .

[7]  Changpin Li,et al.  Synchronization in fractional-order differential systems , 2005 .

[8]  K. Diethelm The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type , 2010 .

[9]  Dumitru Baleanu,et al.  A Nonstandard Finite difference Scheme for Two-Sided Space-fractional Partial differential equations , 2012, Int. J. Bifurc. Chaos.

[10]  I. Podlubny Fractional differential equations , 1998 .

[11]  Dumitru Baleanu,et al.  Positive Solutions of an Initial Value Problem for Nonlinear Fractional Differential Equations , 2012 .

[12]  M. T. Cicero FRACTIONAL CALCULUS AND WAVES IN LINEAR VISCOELASTICITY , 2012 .

[13]  O. Agrawal,et al.  Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering , 2007 .

[14]  H. Kober ON FRACTIONAL INTEGRALS AND DERIVATIVES , 1940 .

[15]  V. V. Gafiychuk,et al.  Stability analysis and limit cycle in fractional system with Brusselator nonlinearities , 2008 .

[16]  Roberto Garrappa,et al.  On some explicit Adams multistep methods for fractional differential equations , 2009 .

[17]  Roberto Garrappa,et al.  On linear stability of predictor–corrector algorithms for fractional differential equations , 2010, Int. J. Comput. Math..

[18]  O. Agrawal,et al.  Advances in Fractional Calculus , 2007 .

[19]  R. Hilfer Applications Of Fractional Calculus In Physics , 2000 .

[20]  D. Baleanu,et al.  Stability analysis of Caputo fractional-order nonlinear systems revisited , 2011, Nonlinear Dynamics.

[21]  Yury F. Luchko,et al.  Algorithms for the fractional calculus: A selection of numerical methods , 2005 .

[22]  Ronald E. Mickens,et al.  Finite-difference models of ordinary differential equations: influence of denominator functions , 1990 .

[23]  R. Magin Fractional Calculus in Bioengineering , 2006 .

[24]  K. Diethelm,et al.  Fractional Calculus: Models and Numerical Methods , 2012 .

[25]  Shaher Momani,et al.  The non-standard finite difference scheme for linear fractional PDEs in fluid mechanics , 2011, Comput. Math. Appl..

[26]  Ahmed M. A. El-Sayed,et al.  On the fractional-order logistic equation , 2007, Appl. Math. Lett..

[27]  F. Mainardi Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models , 2010 .

[28]  Ingo Schäfer,et al.  Fractional Calculus via Functional Calculus: Theory and Applications , 2002 .

[29]  Dumitru Baleanu,et al.  Stability of q-fractional non-autonomous systems , 2013 .

[30]  Changpin Li,et al.  Does the fractional Brusselator with efficient dimension less than 1 have a limit cycle , 2007 .

[31]  Jinhu Lü,et al.  Stability analysis of linear fractional differential system with multiple time delays , 2007 .

[32]  P. Butzer,et al.  AN INTRODUCTION TO FRACTIONAL CALCULUS , 2000 .

[33]  Roberto Garrappa,et al.  On the use of matrix functions for fractional partial differential equations , 2011, Math. Comput. Simul..

[34]  Luigi Fortuna,et al.  Fractional Order Systems: Modeling and Control Applications , 2010 .

[35]  Mingxiang Chen,et al.  Analysis of and numerical schemes for a mouse population model in Hantavirus epidemics , 2006 .

[36]  N. Ford,et al.  A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations , 2013 .

[37]  R. Mickens Nonstandard Finite Difference Models of Differential Equations , 1993 .

[38]  D. Matignon Stability results for fractional differential equations with applications to control processing , 1996 .

[39]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[40]  W. Ames Mathematics in Science and Engineering , 1999 .

[41]  Igor Moret,et al.  On the Convergence of Krylov Subspace Methods for Matrix Mittag-Leffler Functions , 2011, SIAM J. Numer. Anal..

[42]  Ahmed Gomaa Radwan,et al.  Stability and non-standard finite difference method of the generalized Chua's circuit , 2011, Comput. Math. Appl..

[43]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[44]  Roberto Garrappa,et al.  On accurate product integration rules for linear fractional differential equations , 2011, J. Comput. Appl. Math..