Modified methods for solving two classes of distributed order linear fractional differential equations

This paper introduces two methods for the numerical solution of distributed order linear fractional differential equations. The first method focuses on initial value problems (IVPs) and based on the αth Caputo fractional definition with the shifted Chebyshev operational matrix of fractional integration. By applying this method, the IVPs are converted into simple linear differential equations which can be easily handled. The other method focuses on boundary value problems (BVPs) based on Picard's method frame. This method is based on iterative formula contains an auxiliary parameter which provides a simple way to control the convergence region of solution series. Several numerical examples are used to illustrate the accuracy of the proposed methods compared to the existing methods. Also, the response of mechanical system described by such equations is studied.

[1]  X. Y. Li,et al.  A numerical method for solving distributed order diffusion equations , 2016, Appl. Math. Lett..

[2]  Teodor M. Atanackovic,et al.  Distributed-order fractional wave equation on a finite domain. Stress relaxation in a rod , 2010, 1005.3379.

[3]  N. Shimizu,et al.  Nonlinear Fractional Derivative Models of Viscoelastic Impact Dynamics Based on Entropy Elasticity and Generalized Maxwell Law , 2011 .

[4]  John T. Katsikadelis,et al.  Numerical solution of distributed order fractional differential equations , 2014, J. Comput. Phys..

[5]  Ali H. Bhrawy,et al.  A review of operational matrices and spectral techniques for fractional calculus , 2015 .

[6]  A. Bhrawy,et al.  A new formula for fractional integrals of Chebyshev polynomials: Application for solving multi-term fractional differential equations , 2013 .

[7]  Yangquan Chen,et al.  Application of numerical inverse Laplace transform algorithms in fractional calculus , 2011, J. Frankl. Inst..

[8]  Neville J. Ford,et al.  Distributed order equations as boundary value problems , 2012, Comput. Math. Appl..

[9]  H. N. Hassan,et al.  Single and dual solutions of fractional order differential equations based on controlled Picard’s method with Simpson rule , 2017 .

[10]  Zhi-Zhong Sun,et al.  Some high-order difference schemes for the distributed-order differential equations , 2015, J. Comput. Phys..

[11]  H. Vázquez-Leal,et al.  A comparison of HPM, NDHPM, Picard and Picard–Padé methods for solving Michaelis–Menten equation , 2015 .

[12]  Ahmed G. Radwan,et al.  Controlled Picard Method for Solving Nonlinear Fractional Reaction–Diffusion Models in Porous Catalysts , 2017 .

[13]  Miklós Rontó,et al.  A new approach to non-local boundary value problems for ordinary differential systems , 2015, Appl. Math. Comput..

[14]  Kai Diethelm,et al.  Numerical analysis for distributed-order differential equations , 2009 .

[15]  T. Atanacković.  A generalized model for the uniaxial isothermal deformation of a viscoelastic body , 2002 .

[16]  Michele Caputo,et al.  Mean fractional-order-derivatives differential equations and filters , 1995, ANNALI DELL UNIVERSITA DI FERRARA.

[17]  Y. Chen,et al.  Design, implementation and application of distributed order PI control. , 2013, ISA transactions.

[18]  Stevan Pilipović,et al.  On a fractional distributed-order oscillator , 2005 .

[19]  Zhi-Zhong Sun,et al.  Two alternating direction implicit difference schemes with the extrapolation method for the two-dimensional distributed-order differential equations , 2015, Comput. Math. Appl..

[20]  Moonyong Lee,et al.  Deterministic analysis of distributed order systems using operational matrix , 2016 .

[21]  Tom T. Hartley,et al.  Fractional-order system identification based on continuous order-distributions , 2003, Signal Process..

[22]  Kendall E. Atkinson.  An introduction to numerical analysis , 1978 .

[23]  M. L. Morgado,et al.  Numerical approximation of distributed order reaction-diffusion equations , 2015, J. Comput. Appl. Math..

[24]  Stevan Pilipović,et al.  On a nonlinear distributed order fractional differential equation , 2007 .

[25]  R. Gorenflo,et al.  FRACTIONAL RELAXATION AND TIME-FRACTIONAL DIFFUSION OF DISTRIBUTED ORDER , 2006 .

[26]  Fawang Liu,et al.  Compact difference scheme for distributed-order time-fractional diffusion-wave equation on bounded domains , 2015, J. Comput. Phys..

[27]  Carl F. Lorenzo,et al.  Fractional System Identification: An Approach Using Continuous Order-Distributions , 1999 .

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

[29]  H. H. G. Hashem,et al.  Picard and Adomian decomposition methods for a quadratic integral equation of fractional order , 2014 .

[30]  Teodor M. Atanackovic,et al.  Semilinear ordinary differential equation coupled with distributed order fractional differential equation , 2008, 0811.2871.

[31]  H. N. Hassan,et al.  A new approach for a class of nonlinear boundary value problems with multiple solutions , 2015 .

[32]  J. Katsikadelis.  The fractional distributed order oscillator. a numerical solution , 2012 .