Non-linear pseudo-differential evolution equation in fractional time.

Abstract

Fractional calculus is the part of mathematical analysis that studies derivatives and integrals of any arbitrary order, real or complex. We usually refer to it when we want to describe evolution problems with memory. The first known historical record where a fractional derivative is mentioned, can be found in a letter written to Guillaume de l'Hôpital (1661−1704) by Gottfried Wilhelm Leibniz (1646−1716) in 1695. Long after, the 19th and 20th centuries would be the witnesses of just how important is this branch of mathematics. Researchers from various areas has been motivated by the increasing use of fractional calculus in the mathematical modeling of processes in health sciences, natural sciences, economy and engineering (see, e.g. [11], [12], [42] and [48]). This calculus is more reliable in predicting the evolution of some phenomena or processes, such as particle motion, conservation of mass, propagation of acoustic waves and anomalous diffusion in complex media. In this thesis we are particularly interested in the later.