**MACSI at the department of Mathematics and Statistics at the University of Limerick invites you to a seminar**

**Date:** Friday October 18, 2019, Room A2-002 @ 3p.m.

**Speaker:** Prof. Martin Stynes (Beijing Computational Science Research Center).

**Title:** Time-fractional differential equations and their numerical solution.

**Abstract:** First, an extended introduction to fractional derivatives and some of their properties is presented. The regularity of solutions to Caputo fractional initial-value problems in one dimension is then discussed; it is shown that typical solutions have a weak singularity at the initial time t=0. This singularity has to be taken into account when designing and analysing numerical methods for the solution of such problems. (But many published papers pretend the singularity is not present! Some sharp criticisms will be made.) To address this difficulty we use graded meshes, which cluster mesh points near t=0, and answer the question: how exactly should the mesh grading be chosen? Finally, initial-boundary value problems in one space dimension are considered, where the time derivative is a Caputo fractional derivative. (This is a fractional-derivative generalisation of the classical parabolic heat equation.) Once again a weak singularity appears at t=0, and the mesh in the time coordinate should be graded to compute satisfactory numerical solutions.

Although the speaker’s main interest is the numerical solution of these problems, at most one-third of this talk is devoted to numerical analysis, and it won’t get very heavy. No knowledge of fractional derivatives is assumed. But some basic familiarity with differential equations is helpful.

**Further Information**: If you have any questions regarding this seminar, please direct them to Romina Gaburro (061 2131930, email romina.gaburro@ul.ie or Clifford Nolan (061 202766), __clifford.nolan@ul.ie__).