## A finite difference method for a two-point boundary value problem with a Caputo fractional derivative

Title: A finite difference method for a two-point boundary value problem with a Caputo fractional derivative.

Speaker : Prof. Martin StynesSchool of Mathematical Sciences, University of College Cork & Beijing Computational Science Research Center

Date: 16:00-17:00pm, April 16, 2014

Place: Conference Room A404, Department of Mathematical Sciences

Abstract: An introduction is given to the general ideas and properties of fractional derivatives. A two-point boundary value problem whose highest-order term is a Caputo fractional derivative of order $\delta \in (1,2)$ is considered. Al-Refai's comparison principle is improved and modified to fit our problem. Sharp a priori bounds on derivatives of the solution $u$ of the boundary value problem are established, showing that $u''(x)$ may be unbounded at the interval endpoint $x=0$. These bounds and a discrete comparison principle are used to prove point wise convergence of a finite difference method for the problem, where the convective term is discretized using simple upwinding to yield stability on coarse meshes for all values of $\delta$. Numerical results are presented to illustrate the performance of the method.

ContactZhongyi Huang