An Efficient Formulation of Chebyshev Tau Method for Constant Coefficients Systems of Multi-order FDEs

The objective of the present work is to introduce a computational approach employing Chebyshev Tau method for approximating the solutions of constant coefficients systems of multi-order fractional differential equations. For this purpose, a series representation for the exact solutions in a neighborhood of the origin is obtained to monitor their smoothness properties. We prove that some derivatives of the exact solutions of the underlying problem often suffer from discontinuity at the origin. To fix this drawback and design a high order approach a regularization procedure is developed. In addition to avoid high computational costs, a suitable strategy is implemented such that approximate solutions are obtained by solving some triangular algebraic systems. Complexity and convergence analysis of the proposed scheme are provided. Various practical test problems are presented to exhibit capability of the given approach.

[1]  Fawang Liu,et al.  Numerical simulation of the fractional Bloch equations , 2014, J. Comput. Appl. Math..

[2]  Nasser Hassan Sweilam,et al.  Two computational algorithms for the numerical solution for system of fractional differential equations , 2015 .

[3]  S. Dragomir.  Some Gronwall Type Inequalities and Applications , 2003 .

[4]  William A. Wakeham,et al.  A fractional differential equation for a MEMS viscometer used in the oil industry , 2009 .

[5]  Ivo Petráš,et al.  Chaos in the fractional-order Volta’s system: modeling and simulation , 2009 .

[6]  P. Mokhtary.  Numerical treatment of a well-posed Chebyshev Tau method for Bagley-Torvik equation with high-order of accuracy , 2015, Numerical Algorithms.

[7]  T. A. Zang,et al.  Spectral Methods: Fundamentals in Single Domains , 2010 .

[8]  Tadeusz Kaczorek,et al.  Positive Linear Systems Consisting of $n$ Subsystems With Different Fractional Orders , 2011, IEEE Transactions on Circuits and Systems I: Regular Papers.

[9]  Ali H. Bhrawy,et al.  Shifted fractional-order Jacobi orthogonal functions: Application to a system of fractional differential equations , 2016 .

[10]  Hossein Jafari,et al.  Adomian decomposition: a tool for solving a system of fractional differential equations , 2005 .

[11]  Hao Zhu,et al.  Chaos and synchronization of the fractional-order Chua’s system , 2009 .

[12]  Stefan Siegmund,et al.  Asymptotic behavior of solutions of linear multi-order fractional differential systems , 2017, 1708.08131.

[13]  Jafar Biazar,et al.  Modeling the pollution of a system of lakes , 2006, Appl. Math. Comput..

[14]  A. Hendy,et al.  A computational matrix method for solving systems of high order fractional differential equations , 2013 .

[15]  M. L. Morgado,et al.  A Hybrid Numerical Scheme for Fractional-Order Systems , 2018, Innovation, Engineering and Entrepreneurship.

[16]  P. Mokhtary,et al.  Numerical analysis of an operational Jacobi Tau method for fractional weakly singular integro-differential equations , 2017 .

[17]  E. Hille,et al.  Lectures on ordinary differential equations , 1968 .

[18]  Yan-Qiao Wei,et al.  Numerical solutions for systems of fractional order differential equations with Bernoulli wavelets , 2019, Int. J. Comput. Math..

[19]  R. Bagley,et al.  On the Fractional Calculus Model of Viscoelastic Behavior , 1986 .

[20]  N. Ford,et al.  Analysis of Fractional Differential Equations , 2002 .

[21]  Tao Tang,et al.  Convergence Analysis of Spectral Galerkin Methods for Volterra Type Integral Equations , 2012, J. Sci. Comput..

[22]  Jie Shen,et al.  Spectral Methods: Algorithms, Analysis and Applications , 2011 .

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

[24]  Richard L. Magin,et al.  Solving the fractional order Bloch equation , 2009 .

[25]  Zaid M. Odibat,et al.  Analytic study on linear systems of fractional differential equations , 2010, Comput. Math. Appl..

[26]  N. Sweilam,et al.  Numerical studies for a multi-order fractional differential equation , 2007 .

[27]  Shaher Momani,et al.  Solving systems of fractional differential equations by homotopy-perturbation method , 2008 .

[28]  Junzhi Yu,et al.  Dynamic analysis of a fractional-order Lorenz chaotic system , 2009 .

[29]  P. Mokhtary,et al.  On the numerical solution of a class of linear fractional integro-differential algebraic equations with weakly singular kernels , 2019, Applied Numerical Mathematics.

[30]  E. Demirci,et al.  A Fractional Order SEIR Model with Density Dependent Death Rate ABSTRACT | FULL TEXT , 2011 .

[31]  Elif Demirci,et al.  A method for solving differential equations of fractional order , 2012, J. Comput. Appl. Math..

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

[33]  Kewang Chen,et al.  Chaotic behavior in a new fractional-order love triangle system with competition , 2015 .

[34]  R. F. Camargo,et al.  Analysis of fractional-order models for hepatitis B , 2018, Computational and Applied Mathematics.

[35]  G. H. Erjaee,et al.  Chebyshev operational matrix method for solving multi-order fractional ordinary differential equations , 2013 .

[36]  Shaher Momani,et al.  Numerical approach to differential equations of fractional order , 2007 .

[37]  Wei-Ching Chen,et al.  Nonlinear dynamics and chaos in a fractional-order financial system , 2008 .

[38]  S. Momani,et al.  Solving systems of fractional differential equations using differential transform method , 2008 .

[39]  Yiming Chen,et al.  Numerical algorithm to solve system of nonlinear fractional differential equations based on wavelets method and the error analysis , 2015, Appl. Math. Comput..

[40]  Fanhai Zeng,et al.  Numerical Methods for Fractional Calculus , 2015 .

[41]  P. Mokhtary.  Discrete Galerkin Method for Fractional Integro-Differential Equations , 2015, 1501.01111.

[42]  P. Mokhtary,et al.  Reconstruction of exponentially rate of convergence to Legendre collocation solution of a class of fractional integro-differential equations , 2015, J. Comput. Appl. Math..

[43]  Dumitru Baleanu,et al.  New spectral techniques for systems of fractional differential equations using fractional-order generalized Laguerre orthogonal functions , 2014 .

[44]  Fawang Liu,et al.  Multi-term time-fractional Bloch equations and application in magnetic resonance imaging , 2017, J. Comput. Appl. Math..

[45]  P. Mokhtary,et al.  Generalized Jacobi–Galerkin method for nonlinear fractional differential algebraic equations , 2018 .

[46]  F. Ghanbari,et al.  Numerical solution of a class of fractional order integro-differential algebraic equations using Müntz-Jacobi Tau method , 2019, J. Comput. Appl. Math..

[47]  Mohd. Salmi Md. Noorani,et al.  Series Solutions of Systems of Nonlinear Fractional Differential Equations , 2009 .

[48]  Mokhtary Payam,et al.  Convergence analysis of spectral Tau method for fractional Riccati differential equations , 2014 .