Comments on time-varying fractional order

Abstract

The calculus of arbitrary order, known as the fractional calculus, allows for real- and complex-valued orders. Interest in time-varying order is growing in order to study transient behavior under varying environmental conditions. Popular forms currently in use suffer from fundamental flaws of dimensional inconsistency and predict physically implausible behaviors such as violation of causality and/or violation of conservation of energy. This article reviews some of the motivation behind the study of time-varying fractional order and makes suggestions as to how to overcome the flaws in the forms in current usage. A new fractional-order integral operator is proposed that may allow for modeling time-varying fractional-order systems in a dimensionally consistent and physically plausible manner. Possible experimental tests of the revised fractional-order model are proposed.