New directions in spectral methods.
Tobin A. Driscoll (Dept. of Mathematics, University of Delaware)
What if all you had to do to solve a differential equation was to write it down? The development of the Chebfun project for numerical computing of functions has sparked an interest in making this idea a reality. This aim has led to new applications and understanding in spectral methods of automatic differentiation, separation of approximation and residual spaces, and splitting of the domain. For a great variety of interesting problems, the goal of automatic and accurate computation of solutions has been achieved in one dimension. Challenges remain in adaptation for singularly perturbed problems, very large systems of equations, unbounded domains, and time-dependent problems. Progress into multiple dimensions is underway, with many unanswered questions at the outset.