The numerical treatment of fractional differential equations in an accurate way is more
difficult to tackle than the standard integer-order counterpart, and occasionally non …

We review recent advances in algorithms for quadrature, transforms, differential equations
and singular integral equations using orthogonal polynomials. Quadrature based on …

We develop a rapid and accurate contour method for the solution of time-fractional PDEs.
The method inverts the Laplace transform via an optimised stable quadrature rule, suitable …

It has recently been demonstrated that both regular derivatives and contour integrals of
analytic functions can be numerically evaluated to very high orders of accuracy utilizing only …

We extend the ultraspherical spectral method to solving nonlinear ordinary differential
equation (ODE) boundary value problems. Naive ultraspherical Newton implementations …

We present a spectral method for one-sided linear fractional integral equations on a closed
interval that achieves exponentially fast convergence for a variety of equations, including …

We introduce a method to numerically compute equilibrium measures for problems with
attractive-repulsive power law kernels of the form $ K (xy)=\frac {| xy|^\alpha}{\alpha}-\frac …

In this chapter we review some recent results for fractional nonlinear Schrödinger (FNLS)
and fractional complex Ginzburg-Landau (FCGL) models. In particular, one-and two …

In this contribution we deal with Eta functions and their representations as fractional
derivatives and fractional integrals. A class of fractional Sturm-Liouville eigenvalue problems …

Orthogonal polynomials on quadratic curves in the plane are studied. These include
orthogonal polynomials on ellipses, parabolas, hyperbolas and two lines. For an integral …