Mathematics

Prof. Varsha Gejji
Pune University

Fractional calculus (FC) deals with integration and differentiation of arbitrary orders. Though this subject has history of more than three hundred years, it has witnessed rapid development in the last three-four decades as it finds applications in diverse topics in science and engineering. Fractional models give better fit than the conventional integer order models while describing various phenomena, especially which deal with memory effects. This new tool has widened the descriptive power of calculus beyond the familiar integer order concepts of rates of change and area under a curve.
     In the present talk we give history of FC in brief and motivate the basic concepts such as fractional derivative, fractional integral and their algebra. Further we elaborate on fractional differential equations (FDEs), their analysis, existence uniqueness theorems and present numerical methods for solving FDEs; recently developed by our group. These methods are instrumental in the study of fractional ordered dynamical systems (FODS), a subject which is in its infancy.
     Finally we present our recent conjecture regarding chaos in FODS, which generalizes the well-known Poincare-Bendixon theorem for ordinary dynamical systems.