Solves initial value problems for fractional differential equations
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].
USAGE:
[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
D^ALPHA Y(t) = FDEFUN(T,Y(T))
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.
REFERENCES:
[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.
1.4 | A bug has been corrected (many thanks to prof. S. Ramanathan for his feedback). |
fuchang wang (view profile)
asma br (view profile)
we cannot solve Backward Integration using fde12
is there any suggestion how we could calculate Backward Integration for fractional order nonlinear systems
fuchang wang (view profile)
a = 1 ; mu = 4 ;
fdefun = @(t,y) [ a-(mu+1)*y(1)+y(1)^2*y(2) ; mu*y(1)-y(1)^2*y(2) ] ;
Jfdefun = @(t,y) [ -(mu+1)+2*y(1)*y(2) , y(1)^2 ; mu-2*y(1)*y(2) , -y(1)^2 ] ;
alpha = 0.8 ;
t0 = 0 ; tfinal = 100 ; y0 = [ 0.2 ; 0.03] ;
h = 2^(-6) ;
[t, y_flmm2] = flmm2(alpha,fdefun,Jfdefun,t0,tfinal,y0,h) ;
[t, y_fde12] = fde12(alpha,fdefun,t0,tfinal,y0,h) ;
figure(1)
subplot(1,2,1),plot(t,y_flmm2(1,:),t,y_flmm2(2,:)) ;
xlabel('t') ; ylabel('y(t)') ;
legend('y_1(t)','y_2(t)') ;
title('FDE solved by the FLMM2.m code');
subplot(1,2,2),plot(y_flmm2(1,:),y_flmm2(2,:)) ;
xlabel('y1(t)') ; ylabel('y2(t)') ;
title('FDE solved by the FLMM2.m code');
figure(2)
subplot(1,2,1),plot(t,y_fde12(1,:),t,y_fde12(2,:)) ;
xlabel('t') ; ylabel('y(t)') ;
legend('y_1(t)','y_2(t)') ;
title('FDE solved by the FDE12.m code');
subplot(1,2,2),plot(y_fde12(1,:),y_fde12(2,:)) ;
xlabel('y1(t)') ; ylabel('y2(t)') ;
title('FDE solved by the FDE12.m code');
Mohamed Aburakhis (view profile)
When alpha <1, it is easy to use but when alpha >1 and i have more that one state equations, how can i use it
MVOGO Alain (view profile)
Could you please provide an example how to use this code in the case of the fractional Lorentz system for example
James506 (view profile)
Hi. I was interested how/if you could use this to solve a system of mixed-order FDEs? For example, if I had an FDE which also had standard first and second order derivatives in it. Only a single value can be prescribed for the equation order (alpha). Thanks
Roberto Garrappa (view profile)
Dear Sasan Attar, the usage of this code is very similar to those of other Matlab codes for ODEs (for instance, ode15s or ode45). Anyway, on the software page of my website http://www.dm.uniba.it/Members/garrappa/Software you can find a short tutorial to the use of this code and to the use of other codes for FDEs. Thank you for your interest.
sasan attar (view profile)
Could you please provide an example how to use this code
Manashita Borah (view profile)
very helpful
Roberto Garrappa (view profile)
On the software section of my home page http://www.dm.uniba.it/Members/garrappa/ I have uploaded a short tutorial on the use of this code for solving FDEs
qiq (view profile)
how to solve fde system with 2 or more equations?
amaal (view profile)
how i can write the input fdefun?
Mirza (view profile)
Could you please provide an example how to use this code.