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In this week's seminar Martin Stynes (University College Cork) will speak on
A finite difference method for a two-point boundary value problem with a Caputo fractional derivative This talk assumes no prior knowledge of fractional-order derivatives, which will be introduced gently. A two-point boundary value problem whose highest-order term is a Caputo fractional derivative of order δ ∈ (1,2) is considered. We discuss a suitable comparison/maximum principle for this problem and describe sharp a priori bounds on the derivatives of its solution u. These show that u''(x) may be unbounded at the interval endpoint x=0 which hints that the numerical analysis of this problem will not be routine. We describe a finite difference method for the problem, in which linear algebra considerations lead us to discretize the convective term using simple upwinding in order to get a stable method for all values of δ. A pointwise convergence result is stated for this method and numerical results are presented to illustrate its performance.
While a basic knowledge of finite difference methods and their analysis would be helpful in following the later stages of the talk, most of the material should be accessible to everyone..