## Division of Mathematical Modeling and its Mathematical Analysis

Director | Keiichi Kato: Professor, Department of Mathematics, Faculty of Science Division Ⅰ |
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Research Content | Interdisciplinary researches between mathematical analysis, numerical analysis, physics and technology |

Objetcitves | We, those who study mathematical analysis, numerical analysis, physics or technology, all together study for interdisciplinary researches |

This division is established on the April of 2015. We introduce our plans and our seeds of future researches in the following.

### Application of the representation of solutions to Schrödinger equations via wave packet transform

・ Using our representation of solutions to Schrödinger equations via wave packet transform, we will establish a method to compute the energy levels and its eigenstates for given potentials. We will apply this method to physical situations via numerical analysis. (Keiichi Kato)

### Time-dependent density functional theory (TDDFT) simulations of ultrafast electron-ion correlated dynamics under high external fields

We recently applied the TDDFT to laser-assisted field evaporation of nanostructures to elucidate the microscopic mechanism of electronic excitations and ion detachment. We also develop the TDDFT program code to enable the long- time simulations of multi-component quantum dynamics.

### Stochastic analysis associated with tree structures and hierarchical phenomena

Eligible probabilists are also taking membership of this division. From the fields of p-adic numbers to tree models in various practical studies, crucial importance of hierarchical structures are observed and related mathematical models are applied in cognitive science and DNA analysis, etc. We will work out analytic methods and statistical methods to reveal probabilistic significance in such theoretical frameworks. Potential impacts to mathematical finance and data analysis will be focused on. (Hiroshi Kaneko)

### Asymptotic behavior of solutions to generalized Keller-Segel systems

As a model describing chemotaxis, the Keller-Segel system is well known and studied. From both mathematical and biological point of view it is an important problem whether a solution to the Keller-Segel system exists and is uniformly-in-time bounded or not. Recently Ishida-Yokota found a method to solve the boundedness problem in a slightly generalized model, which is open still now. We will solve the boundedness problem in more generalized model and study the asymptotic behavior of the solution. (Tomomi Yokota, Sachiko Ishida)

### Variational problems for p(x)-growth functionals and its application

A functional with p(x)-growth first appeared in the mathematical model of thermistor, and more generally partial differential equations having terms with variable exponents appear in several models including, for example, rheology. Continuing mathematical analysis on variational problems for p(x)-growth functionals, I would like to try to find a new approach for some applications.（Atsushi Tachikawa）

### Blind separation of multi-reflected signals in a convex polygonal room

The purpose of this study is to present and apply a mathematical formula to a numerical experiment for blind separation of multi-reflected signals in an unknown convex polygonal room. In recent studies, formula for a one-reflection model based on Blind Source Separation (BSS) have been proposed in which the main purpose is to identify a source signal and a one-wall location from observed signals. In practical applications, however, it is often essential to consider multi-reflected signals, and then a one-reflection model requires review to take these into account. In this study, we propose a new iterative method for the multi-reflection issue and apply the method to typical cases in which a one-source signal is multi-reflected by the walls of a room. The basic assumption in our method is that the locations of the observation points are known, while the one-source signal, the locations of the source point and the walls of the room are unknown.(Fumio Sasaki)

### Mathematical analysis on nonlinear elasticity with application to fracture phenomena in mind

Brittle fracture under an assumption of linear elasticity has been systemized as linear fracture mechanics and its simulation software has also been developed. However there are a lot of engineering hypothesis, so it’s difficult to construct mathematical model covered general fracture phenomena. In order to treat wide variety of fracture phenomena it is important to analyze nonlinear elastic model which is physically meaningful. Then, in this research we deal with mathematical analysis on nonlinear elastic model suitable on real fracture phenomena. (Hiromichi Itou)

### Mathematical analysis on inverse problems for nondestructive testing

・ Nondestructive testing is a technique for evaluating specimens embedded defects without destruction. This has a lot of application not only in material mechanics, but also in medical imaging such as computed tomography(CT) and Magnetic resonance imaging(MRI) and geophysics (determination of inner structure of the earth). In the mathematical model, the problems are often described as inverse boundary value problems and we have considered reconstruction problems for cracks, polygonal cavities in linear (visco)elasticity and for welding area in electric conductive body. In the future we will study inverse crack problems in (visco)elasticity for nondestructive testing and inverse problem for evaluation of material constants.(Hiromichi Itou)

### Singularity and large time behavior of solutions to nonlinear partial differential equations

The purpose of this study is to give a sufficient condition for the occurrence of the vacuum state for the generalized barotropic model which describes the motion of gas. Especially, we are going to show that the vacuum state can occure , if initial gas pressure is high. In parallel with this study, I progress in studies of the solvability and the large time behavior for the drift diffusion equation which describes the motion of electron in semiconductor, together with Masakazu Yamamoto in Niigata university.

### Future Development Goals

In this year, we discuss each other on the researches of each member and determine how to make our interdisciplinary researches possible.

### Message

This division has been established on the April of 2015. Our aim is to make interdisciplinary researches between mathematical analysis, numerical analysis, physics, chemistry, biology and technology. The members of our division are willing to cooperate to those who need to techniques of mathematical analysis or numerical analysis.

Research Division

- Division of Advanced Urbanism and Architecture
- Academic Detailing Database Division
- Division of Medical-Science- Engineering Cooperation
- Division of Mathematical Modeling and its Mathematical Analysis
- Fusion of Regenerative Medicine with DDS
- Photovoltaic Science and Technology Research Division
- Advanced EC Device Research Division
- Division of Agri-biotechnology
- Division of Things and Systems
- Atmospheric Science Research Division
- Division of Super Distributed Intelligent Systems
- Brain Interdisciplinary Research Division
- Division of Intelligent System Engineering
- Advanced Agricultural Energy Science and Technology Research Division
- Division of Modern Algebra and Cooperation with Engineering
- Research Division of Multiscale Interfacial Thermofluid Dynamics
- Division of Nanocarbon Research
- Division of Colloid and Interface Science
- Chemical Biology Division Supported by Practical Organic Synthesis
- CAE Advanced Composite Materials and Structures Research Division

Research Center

Joint Usage / Research Center