Discrete Fractional Calculus: Non-Equidistant Grids and Variable Step Length

In this paper we further develop Podlubny’s matrix approach to discretization of integrals and derivatives of arbitrary real order. Numerical integration and differentiation on a set of non-equidistant nodes is described and illustrated by several examples of numerical solution of fractional differential equations. In this paper, for the first time, we present a variable step length approach that we call “the method of large steps”, since it is applied in combination with the matrix approach for each “large step”. This new method is also illustrated by an example. The presented approach allows fractional order differentiation and integration of non-uniformly sampled signals, and opens the way to development of variable and adaptive step length techniques for fractional differential equations.