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On mathematical models by (variable-order) time-fractional diffusion equations

王宏 教授(美国南卡罗莱纳大学)
Wednesday, July 29th, 2020, 8:00 PM  Zoom会议 房间号:924 6976 5319
主持人:郑海标 副教授
报告时间:2020年7月29日晚上20:00- 21:00
报告平台:Zoom会议 房间号:924 6976 5319 密码 123456

报告人介绍:王宏,美国南卡罗来纳大学数学系终身教授,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。迄今为止已在国际知名期刊SIAM J Numer. Anal.、 SIAM J Sci. Comput.、 J Comput Phys、和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和Int J. Numer. Anal. Modeling等国际知名杂志的编委。

报告内容摘要:Fractional partial differential equations (FPDEs) have shown to provide more accurate descriptions of anomalously diffusive transport of solute in heterogeneous porous media than integer-order PDEs do, because they generate solutions with power law (instead of exponentially) decaying tails that were observed in field tests.

However, Stynes et al recently proved that solutions to time-fractional partial differential equations (tFPDEs) have nonphysical singularity at the initial time t=0, which does not seem physically relevant and makes many error estimates in the literature that were proved under the full regularity assumption of the true solutions in appropriate. It is getting increasingly clear that the reason lies in the incompatibility between the nonlocality of the power law decaying tail of the solutions and the locality of the initial condition. But there is no consensus on how to correct the nonphysical behavior of tFPDEs.

We argue that the order of a physically correct tFPDE model should vary smoothly near the initial time to account for the impact of the locality of the initial condition. Moreover, variable-order tFPDEs themselves also occur in a variety of applications. However, rigorous analysis on variable-order tFPDEs is meager.

We also outline the proof of the wellposedness and smoothing properties of the tFPDEs. More precisely, we prove that their solutions have the similar regularities as their integer-order analogues if the order has an integer limit at the initial time or have the same singularity near the initial time as their constant-order tFPDE analogues if the order has a non-integer limit at the initial time.
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