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- Graduate School of Science and Engineering : Science of Environment and Mathematical Modeling (Laboratories : Functional Equations Laboratory)

## Graduate School of Science and Engineering

Science of Environment and Mathematical Modeling

### Functional Equations Laboratory

#### Staff

#### Research Contents

Mathematics is a basic science that uses precise theories to yield results and thus has a wide range of applications in solving problems of many disciplines, such as chemistry, physics, and engineering. Our laboratory conducts analyses of various problems using mathematics and studies relevant applications.

(1) Research on Differential Equations

The equations of conservation laws are nonlinear partial differential equations that describe phenomena in physics such as fluid dynamics or phenomena in chemistry and engineering. Unfortunately, the existence and uniqueness as well as structures of solutions to such equations remain unclear. For example, the compressible Euler equations, which are popular equations describing the motion of air, involve many uncertainties in the existence or uniqueness of their solutions, although analyses of such equations would enable understanding of phenomena including shock waves and rarefaction waves. We conduct theoretical analyses of the equations of conservation laws and also conduct numerical analyses.

(2) Research on Wavelet Analysis

Wavelet analysis has recently emerged as an advanced theory of Fourier analysis. Wavelet analysis are similar to Fourier analysis but can retain positional information. This analysis can yield sharper data than Fourier analysis under certain conditions. Having the roots of Wavelet analysis in various fields including signal theory, time-frequency analysis, and image processing, Mayer constructed the precise mathematical theory about 15 years ago. The theory has later evolved with the development of computers and expanded its use from mathematics to problem-solving in signal processing, image processing, statistics, and to other fields of engineering. In addition to the theoretical study of Wavelet analysis, we also conduct signal analysis and image processing using computers.

(1) Research on Differential Equations

The equations of conservation laws are nonlinear partial differential equations that describe phenomena in physics such as fluid dynamics or phenomena in chemistry and engineering. Unfortunately, the existence and uniqueness as well as structures of solutions to such equations remain unclear. For example, the compressible Euler equations, which are popular equations describing the motion of air, involve many uncertainties in the existence or uniqueness of their solutions, although analyses of such equations would enable understanding of phenomena including shock waves and rarefaction waves. We conduct theoretical analyses of the equations of conservation laws and also conduct numerical analyses.

(2) Research on Wavelet Analysis

Wavelet analysis has recently emerged as an advanced theory of Fourier analysis. Wavelet analysis are similar to Fourier analysis but can retain positional information. This analysis can yield sharper data than Fourier analysis under certain conditions. Having the roots of Wavelet analysis in various fields including signal theory, time-frequency analysis, and image processing, Mayer constructed the precise mathematical theory about 15 years ago. The theory has later evolved with the development of computers and expanded its use from mathematics to problem-solving in signal processing, image processing, statistics, and to other fields of engineering. In addition to the theoretical study of Wavelet analysis, we also conduct signal analysis and image processing using computers.