In this talk we will discuss high order methods for some classes of differential equations including: traditional elliptic equations, integro-differential equations, fractional differential equations, and so on. One of the main features of these problems is their non-locality or singularities, which makes the design of numerical methods difficult. We will present some efficient numerical methods specially adapted to these problems, including fast high order time stepping schemes with low storage for the time fractional differential equation, an extended spectral method for fractional integro-differential equations.

In particular, some new methods making use of Muntz polynomials will be analyzed for problems with solutions of low regularity.