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Talk of Prof. Martin Stynes

Martin Stynes

Martin Stynes,北京计算科学研究中心讲席教授、中组部千人计划外籍专家。 主要研究领域包括奇异摄动微分方程的数值解法和分数阶微分方程数值解法等。Stynes教授曾多次在相关领域的国际大会上作特邀大会报告。1998-2003年期间曾担任SIAM(美国工业与应用数学学会)Journal on Numerical Analysis杂志的编委,2003-2005年期间担任SIAM在英国和爱尔兰分部主管,现任Advances in Computational Mathematics, Applied Numerical Mathematics, Computational Methods in Applied Mathematics and Mathematical Proceedings of the Royal Irish Academy等杂志编委。


Numerical solution of time-fractional problems


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 analyzing numerical methods for the solution of such problems. 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 generalization 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.