PDE Seminar (2018/4/20): Inverse problems in time fractional equations and on fluorescence imaging by diffusion process

2018-4-20 16:30 - 2018-4-20 17:30
Faculty of Science Building #3, Room 309
Sun Chunlong (Southeast University/Hokkaido University)
Part I: Inverse problems in time fractional equations
The fractional diffusion equations have played an important role in modeling of the anomalous diffusion phenomena and in the theory of the complex systems during the last few decades. However, for real problems, the part of the boundary, or the initial data, or the diffusion coefficient, or the source term cannot be obtained directly and we have to determine them by some additional measurements, which yields to inverse problems arising in the fractional diffusion models.

Part II: On fluorescence imaging by diffusion process
Fluorescence imaging is a type of wave spectroscopy that analyzes fluorescence property from some measurable data of the sample. This process in the randomly inhomogeneous medium is governed by the radiative transfer equation for excitation and emission filed. By introducing the average of angularly reserved wave energy density, we derive an imaging model by a coupled diffusion system for the average filed. Such a new model is efficient for imaging the fluorescence density, since the random scattering from different directions are averaged, and we establish the explicit analytical expression for both excitation and emission filed, which provides the fundamentals for efficient implementations. We present some numerical simulations to show the validity of the proposed scheme.