About Site Map Submit Contact Us Log in | Create an account
Create an account Log In
Average Rating
User Rating:
Visitors Rating:
My rating:

Write review
  • Last update: 5 years ago
  • Total downloads: 125
  • Operating system: Linux, Mac OS X, Win All, BSD, Solaris
  • Publisher: Roberto Garrappa
See full specifications

scripts default iconPredictor-corrector PECE method for fractional differential equations (Scripts) Publisher's description

FDE12 solves an initial value problem for a non-linear differential equation of fractional order (FDE)

FDE12 solves an initial value problem for a non-linear differential equation of fractional order (FDE).

This is an implementation of the predictor-corrector method of Adams-Bashforth-Moulton described in [1]. Convergence and accuracy of the method are studied in [2]. The implementation with multiple corrector iterations has been proposed and discussed for multiterm FDEs in [3]. In this implementation the discrete convolutions are evaluated by means of the FFT algorithm described in [4] allowing to keep the computational cost proportional to N*log(N)^2 instead of N^2 as in the classical implementation; N is the number of time-point in which the solution is evaluated, i.e. N = (TFINAL-T)/H. The stability properties of the method implemented by FDE12 have been studied in [5].


[T,Y] = FDE12(ALPHA,FDEFUN,T0,TFINAL,Y0,h) integrates the initial value problem for the FDE, or the system of FDEs, of order ALPHA > 0
Y^(k)(T0) = Y0(:,k+1), k=0,...,m-1
where m is the smallest integer grater than ALPHA and D^ALPHA is the fractional derivative according to the Caputo's definition. FDEFUN is a function handle corresponding to the vector field of the FDE and for a scalar T and a vector Y, FDEFUN(T,Y) must return a column vector. The set of initial conditions Y0 is a matrix with a number of rows equal to the size of the problem (hence equal to the number of rows of the output of FDEFUN) and a number of columns depending on ALPHA and given by m. The step-size H>0 is assumed constant throughout the integration.

[T,Y] = FDE12(ALPHA,FDEFUN,T0,TFINAL,Y0,H,PARAM) solves as above with the additional set of parameters for the FDEFUN as FDEFUN(T,Y,PARAM).

[T,Y] = FDE12(ALPHA,FDEFUN,T0,TFINAL,Y0,H,PARAM,MU) solves the FDE with the selected number MU of multiple corrector iterations. The following values for MU are admissible:
MU = 0 : the corrector is not evaluated and the solution is provided just by the predictor method (the first order rectangular rule);
MU > 0 : the corrector is evaluated by the selected number MU of times; the classical PECE method is obtained for MU=1;
MU = Inf : the corrector is evaluated for a certain number of times until convergence of the iterations is reached (for convergence the difference between two consecutive iterates is tested).
The defalut value for MU is 1

[T,Y] = FDE12(ALPHA,FDEFUN,T0,TFINAL,Y0,H,PARAM,MU,MU_TOL) allows to specify the tolerance for testing convergence when MU = Inf. If not specified, the default value MU_TOL = 1.E-6 is used.


[1] K. Diethelm, A.D. Freed, The Frac PECE subroutine for the numerical solution of differential equations of fractional order, in: S. Heinzel, T. Plesser (Eds.), Forschung und Wissenschaftliches Rechnen 1998, Gessellschaft fur Wissenschaftliche Datenverarbeitung, Gottingen, 1999, pp. 57-71.

[2] K. Diethelm, N.J. Ford, A.D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms 36 (1) (2004) 31-52.

[3] K. Diethelm, Efficient solution of multi-term fractional differential equations using P(EC)mE methods, Computing 71 (2003), pp. 305-319.

[4] E. Hairer, C. Lubich, M. Schlichte, Fast numerical solution of nonlinear Volterra convolution equations, SIAM J. Sci. Statist. Comput. 6 (3) (1985) 532-541.

[5] R. Garrappa, On linear stability of predictor-corrector algorithms for fractional differential equations, Internat. J. Comput. Math. 87 (10) (2010) 2281-2290.

System Requirements:

MATLAB 7.9 (2009b)
Program Release Status: New Release
Program Install Support: Install and Uninstall

Predictor-corrector PECE method for fractional differential equations (Scripts) Tags:

Click on a tag to find related softwares

Is Predictor-corrector PECE method for fractional differential equations (Scripts) your software?

Manage your software

Most Popular

scripts default icon ASK, OOK, FSK, BPSK, QPSK, 8PSK modulation 1.1
ASK, OOK, FSK, BPSK, QPSK, 8PSK modulation contain several functions for digital modulation simulation
scripts default icon Simulink Communication Labs 1.1
Simulink Communication Labs allows you to learn communication systems in greater depth.
scripts default icon M-QAM modulation and demodulation 1.1
M-QAM modulation and demodulation is the QAM modulation and demodulation tech.
scripts default icon LZW Compression/Decompression 1.1
LZW Compression/Decompression - Updated LZW compressor and decompressor with reasonable performance
scripts default icon InSPIRE utility to plot a 2D displacement field (Scripts) 1.0
This program plots the deformation field (displace vectors) contained in vector.txt.