On the Generalization of Block Pulse Operational Matrices for Fractional and Operational Calculus

A more rigorous derivation for the generalized block pulse operational matrices is proposed in this paper. The Riemann-Liouville fractional integral for repeated fractional (and operational) integration is integrated exactly, then expanded in block pulse functions to yield the generalized block pulse operational matrices. The generalized block pulse operational matrices perform as s^-α(α\s>;0,α∈R) in the Laplace domain and as fractional (and operational) integrators in the time domain. Also, the generalized block pulse operational matrices of differentiation which correspond to s^α(α\s>;0,α∈R) in the Laplace domain are derived. Based on these results, the inversions of rational and irrational transfer functions are proposed in a simple, accurate and efficient way.