FDE_VO_Exponential

Variable-order FDEs: solve fractional differential equations (FDEs) of variable-order (VO) with order transition of exponential-type

FDE_VO_Exponential.m solves an initial value problem for a linear or non-linear fractional differential equation (FDE) of fractional variable-order (VO), where the VO fractional derivative is defined according to the approach discussed in [1] and the VO function realizes an exponential transition from alpha1 to alpha2 with rate c, namely alpha(t) = alpha2 + (alpha1-alpha2)*exp(-ct). Systems of VO-FDEs are possible but the order function is the same for each equation.

The code implements an implicit first-order convolution quadrature rule discussed and implemented in [2] according to the theory previously developed in [3]

USAGE:

[T,Y] = FDE_VO_Exponential(ALPHA,F_FUN,J_FUN,T0,TFINAL,Y0,H) integrates the initial value problem for the variable-order fractional differential equation (FDE), or the system of variable-order FDEs,

D^ALPHA(t) Y(t) = F_FUN(T,Y(T))

Y(T0) = Y0

where the variable order function ALPHA(t) realizes the exponential transition from 0<alpha1<1 to 0<alpha2<1 with rate c>0 given by ALPHA(t) = alpha2 + (alpha1-alpha2)*exp(-ct)

DESCRIPTION OF PARAMETERS

ALPHA: is a vector with three entries describing alpha1, alpha2 and c; alpha1 and alpha2 must be in (0,1) and c must be > 0.

F_FUN: function handle describing the vector field F_FUN(t,Y(t)). It can be a vector-valued function if a system of variable-order fractional differential equations is solved

J_FUN: function handle describing the Jacobian of the vector field. If F_FUN is a vactor-valued function, J_FUN must be a matrix-valued function

T0, TFINAL: intial and final time of integration

Y0: initial condition. It must be a vector of the same size of F_FUN

H: positive step-size

REFERENCES:

[1] Garrappa, R., Giusti, A., Mainardi, F.: Variable-order fractional calculus: a change of perspective. Commun. Nonlinear Sci. Numer. Simul. 102, 105904 (2021)

[2] Garrappa, R., Giusti: A computational approach to exponential-type variable-order fractional differential equations. J. Sci. Comp., 2023, 96, Art. no. 63

[3] Lubich, C.: Convolution quadrature and discretized operational calculus. I. Numer. Math. 52, 129–145 (1988)

In case of use of this code in scientific works, please cite it as [2]

Copyright (c) 2023, Roberto Garrappa, University of Bari, Italy

roberto dot garrappa at uniba dot it

Revision: 1.0.0 - Date: August, 3 2023