Chapter Fourteen - Introduction of new kernels and new models to solve the drawbacks of fractional integration/differentiation operators and classical fractional-order models

Abstract

This chapter first highlights the drawbacks associated to classical fractional operators (integration and differentiation) and to classical fractional-order models (fractional pseudostate-space descriptions or fractional differential equations). These drawbacks mainly are induced by the doubly infinite dimension of fractional operators (they can be viewed as distributed parameter models defined on an infinite spatial domain). This results in initialization and physical consistency problems of these operators and models. To solve these drawbacks, it is then proposed:•to replace the singular kernel used in the definition of the fractional-order integration operator by new nonsingular kernels, but which still make it possible to generate power-law behaviors;•to use new models such as Volterra equations of the first kind, nonlinear models, distributed time delay models, or partial differential equations with spatially variable coefficients.