Numerical approximation of time-fractional differential equations

Abstract

Fractional calculus, as a generalization of ordinary calculus, has been an interesting topic since the end of the 17th century. In comparison with ordinary derivatives and integrals, the fractional derivatives and fractional integrals are introduced in various kinds of ways, and possess some interesting mathematical properties. Recently, there arose some attractive applications in the fields of physics, chemistry and engineering by applying fractional integrals and derivatives to construct mathematical models that describe anomalous diffusion processes, for instance, subdiffusion, which is slower than the Brownian diffusion orL´ evyflight. As a consequence, a variety of differential-integral equations have been derived such as the fractional diffusion equation, the fractional diffusion-advection equation, the fractional Fokker-Planck equation and the fractional Klein-Kramers equation. These fractional models provide a straightforward way of implementing some phenomena in real world. In this thesis, the main ob-jective is to investigate the numerical approximation for the fractional equations with respect to time, which is used to describe the sub-diffusive phenomenon.