Abstract

In this talk we will present two kinds of spectral methods for fractional differential equations. The first method makes use of traditional polynomials for approximating fractional differential equations having regular solutions. The second new method is constructed for a class of equations having solutions of low regularity. The new method makes use of fractional polynomials, also known as Muntz polynomials. We first present some basic approximation properties of the Muntz polynomials, including error estimates for the weighted projection and interpolation operators. Then we will show how to construct efficient spectral methods by using the Muntz polynomials. A detailed convergence analysis will be provided. The potential application of the new method covers a large number of problems, including classical elliptic equations, integro-differential equations with weakly singular kernels, fractional differential equations, and so on.