Matrix approach to discrete fractional calculus III: non-equidistant grids, variable step length and distributed orders

I. Podlubny, T. Skovranek, B. M. Vinagre Jara, I. Petras, V. Verbitsky, Y. Chen
2013 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences  
In this paper, we further develop Podlubny's matrix approach to discretization of integrals and derivatives of non-integer order. Numerical integration and differentiation on non-equidistant grids is introduced and illustrated by several examples of numerical solution of differential equations with fractional derivatives of constant orders and with distributedorder derivatives. In this paper, for the first time, we present a variable-step-length approach that we call 'the method of large
more » ... because it is applied in combination with the matrix approach for each 'large step'. This new method is also illustrated by an easyto-follow example. The presented approach allows fractional-order and distributed-order differentiation and integration of non-uniformly sampled signals, and opens the way to development of variable-and adaptive-step-length techniques for fractional-and distributed-order differential equations.
doi:10.1098/rsta.2012.0153 pmid:23547230 fatcat:4qllyzrvuvhd7fporvrdotsjsm