Numerical analysis for distributed-order differential equations

In this paper we present and analyse a numerical method for the solution of a distributed-order differential equation of the general form @!"0^mA(r,D"*^ru(t))dr=f(t) where m is a positive real number and where the derivative D"*^r is taken to be a fractional derivative of Caputo type of order r. We give a convergence theory for our method and conclude with some numerical examples.

[1]  I. M. Sokolov,et al.  Distributed-Order Fractional Kinetics , 2004 .

[2]  R. Gorenflo,et al.  AN OPERATIONAL METHOD FOR SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS WITH THE CAPUTO DERIVATIVES , 1999 .

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

[4]  C. Lubich,et al.  Fractional linear multistep methods for Abel-Volterra integral equations of the second kind , 1985 .

[5]  Kai Diethelm,et al.  Multi-order fractional differential equations and their numerical solution , 2004, Appl. Math. Comput..

[6]  N. Ford,et al.  Numerical Solution of the Bagley-Torvik Equation , 2002, BIT Numerical Mathematics.

[7]  Michele Caputo,et al.  Diffusion with space memory modelled with distributed order space fractional differential equations , 2003 .

[8]  Kai Diethelm,et al.  Multi-term fractional differential equations, multi-order fractional differential systems and their numerical solution , 2008 .

[9]  N. Ford,et al.  Pitfalls in fast numerical solvers for fractional differential equations , 2006 .

[10]  Kai Diethelm,et al.  Numerical solution methods for distributed order differential equations , 2001 .

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

[12]  M. Caputo.  Linear Models of Dissipation whose Q is almost Frequency Independent-II , 1967 .

[13]  Alain Roger Nkamnang.  Diskretisierung von mehrgliedrigen Abelschen Integralgleichungen und gewöhnlichen Differentialgleichungen gebrochener Ordnung , 1999 .

[14]  Rudolf Gorenflo,et al.  Cauchy and Nonlocal Multi-Point Problems for Distributed Order Pseudo-Differential Equations, Part One , 2005 .

[15]  C. Lubich,et al.  A Stability Analysis of Convolution Quadraturea for Abel-Volterra Integral Equations , 1986 .

[16]  Kai Diethelm,et al.  Efficient Solution of Multi-Term Fractional Differential Equations Using P(EC)mE Methods , 2003, Computing.

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

[18]  K. Diethelm.  AN ALGORITHM FOR THE NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER , 1997 .

[19]  C. Lubich.  Convolution Quadrature Revisited , 2004 .

[20]  N. Ford,et al.  The numerical solution of linear multi-term fractional differential equations: systems of equations , 2002 .

[21]  Alan D. Freed,et al.  Detailed Error Analysis for a Fractional Adams Method , 2004, Numerical Algorithms.

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