报告题目：On mathematical models by (variable-order) time-fractional diffusion equations
主持人： 羊丹平 教授
报告人：王宏教授 University of South Carolina
王宏，美国南卡罗来纳大学数学系终身教授、长江学者讲座教授，分别于1982年和1984年获山东大学数学学士学位和计算数学硕士学位，1992年获美国怀俄明大学数学博士学位。主要从事油气田勘探开发、环境污染的预测与治理和二氧化碳埋存等领域的数学模型、数值模拟与大规模科学计算的理论及应用方面的研究；迄今为止已在美国工业与应用数学会的多种期刊（SIAM J Numer. Anal.、 SIAM Sci. Comput.）、计算物理杂志（J Comput Phys）、Numer. Methods PDEs和英国IMA J. Numer. Anal.等国际权威学术杂志发表论文百余篇。王宏教授还是Numer. Methods PDEs、Computing and Visualization in Sciences、Int J. Numer. Anal. Modeling等国际知名杂志的编委。王宏教授的研究得到了美国国家自然科学基金会、挪威自然科学基金会、南卡州以及世界排名前列的石油公司等的多项基金资助。
Recently, Stynes et al proved that time-fractional diffusion equations (tFDEs) generate solutions with singularity near the initial time t=0, which makes the error estimates in the literature that were proved under full regularity assumptions of the true solutions inappropriate.
From a modeling point of view, the singularities of the solutions to tFDEs at t=0 do not seem physically relevant to the diffusive transport the tFDEs model. The fundamental reason lies between the incompatibility between the nonlocality of tFDEs and the locality of the initial condition.
To eliminate the incompatibility, we propose a modified tFDE model in which the fractional order will vary near the time t=0, which naturally leads to variable-order tFDEs. We will also show that variable-order tFDEs occur naturally in applications. Finally, we briefly discuss the mathematical difficulties in the analysis of variable-order tFDEs, since many widely used Laplace transform based techniques do not apply here.
报告题目：Wellposedness and regularity of variable order time fractional diffusion equations
报告人：Xiangcheng Zheng, Department of Mathematics, University of South Carolina
摘要：We prove the wellposedness of a nonlinear variable-order fractional ordinary differential equation and the regularity of its solutions, which is determined by the values of the variable order and its high-order derivatives at time t=0. More precisely, we prove that its solutions have full regularity like its integer-order analogue if the variable order has an integer limit at t=0 or exhibits singular behaviors at t=0 like in the case of the constant-order fractional differential equations if the variable order has a non-integer value at time t=0.
We then extend the developed techniques to prove the wellposedness of a variable-order linear time-fractional diffusion equation in multiple space dimensions and the regularity of its solutions, which depends on the behavior of the variable order at t=0 in the similar manner to that of the fractional ordinary differential equations.
报告人：贾金红, 山东师范大学 数学与统计学院
报告题目：Time-fractional Allen-Cahn and Cahn-Hilliard phase-field models and their numerical investigation
报告人: 刘欢，山东大学 数学学院
摘要：In this talk, we introduce the time-fractional Allen?Cahn and Cahn?Hilliard phase-field models to account for the anomalously subdiffusive transport behavior in heterogeneous porous materials or memory effect of certain materials. We develop an efficient finite difference scheme and a Fourier spectral scheme to effectively treat the significantly increased memory requirement and computational complexity, which arise due to the nonlocal behavior of the time-fractional models.
For time fractional Cahn?Hilliard model, we observe from the numerical results that the bigger the fractional order α is, the faster the energy decays. However, for time fractional Allen?Cahn model, we derived an opposite conclusion. Moreover, we also study the coarsening dynamics for time fractional Cahn?Hilliard model, numerical results reveal that the scaling law for the energy decays as O(t^(-α/3)), which is consistent with the well-known result O(t^(-1/3)) for integer-order Cahn?Hilliard model.