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Research using differential equations that can describe various physical phenomena

Even if you don’t know the word “analysis,” you have probably heard of differential and integral calculus. Differential and integral calculus is a technique for finding how much the value of a function changes when the variables of that function are changed slightly, but it is actually much more complicated. Analysis methods have been developed to solve models and other problems using differential equations in applied fields and are widely used in science and engineering. Professor Kato is an expert in solving partial differential equations with a large number of variables. “Physical phenomena such as vibration and heat transfer can basically be described using partial differential equations,” says Professor Kato. “Although this area has been well studied in physics, there are still many things that we do not understand mathematically, and we are working on them.” On the other hand, Professor Ushijima specializes in nonlinear parabolic partial differential equations and numerical analysis. “I am interested in various mathematical models that describe natural and social phenomena, and I perform mathematical and numerical analyses on them,” says Professor Ushijima. “Specifically, we model traffic flows, epidemics of infectious diseases, and so on.”

Collaborative research by a theory expert and a numerical calculation expert

Professor Kato and Professor Ushijima are currently engaged in collaborative research on the Schrödinger equation, a fundamental equation of quantum mechanics in physics. Professor Kato says, “I have been developing an efficient and long-time stable numerical calculation scheme for the Schrödinger equation. Now that I have largely completed it, I am trying to make the calculation possible on a computer with the help of Professor Ushijima, who is very knowledgeable in numerical calculations.” Here, Professor Kato is studying methods of constructing theoretical solutions and prototyping numerical calculations, and Professor Ushijima is developing specific numerical calculation methods. Since the Schrödinger equation is a fundamental equation that describes the motion of microscopic materials, it could be used in various fields such as condensed matter physics and chemistry if the collaborative research is successful and an efficient calculation method is found.

Division of Research Alliance for Mathematical Analysis, which brings together a large number of researchers in mathematical analysis

There are various types of differential equations. Some researchers study them mathematically, while others work on using them to solve problems. The division of Research Alliance for Mathematical Analysis brings such researchers together and aims to stimulate collaborative research and expand into research in the boundary area between mathematics and science/engineering. “I study partial differential equations mathematically; in architecture, for example, they are used to calculate vibrations in buildings,” says Professor Kato. “What is important in each field is different, but since partial differential equations are common, I believe there is potential for each researcher to expand into other fields.” Professor Ushijima says, “Since our strengths are slightly different, bringing them together should allow us to utilize them in a wide range of fields.” The development of analysis methods at the Tokyo University of Science is worth watching closely.

Faculty of Science Division I, Department of Mathematics
Professor Keiichi Kato

■ Main research themes

Professor Kato specializes in partial differential equations. He is mathematically studying the solution of the Schrödinger equation, which is the fundamental equation of quantum mechanics, by expressing it in an original method using wave packet transformation.

Faculty of Science and Technology, Department of Mathematics
Professor Takeo Ushijima

■ Main research themes

Professor Ushijima specializes in applied mathematics, partial differential equations, and numerical analysis. In particular, he is researching the properties of solutions to nonlinear parabolic partial differential equations from both numerical and theoretical analysis perspectives.

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