There are several mathematical research areas in the CREST and PRESTO programs of JST,^*1 and researchers in various specialized areas, such as pure and applied mathematics, have engaged in their research. PRESTO researchers have had few chances to involve in the research subjects of CREST, and CREST researchers have had limited opportunities to discuss with the PRESTO researchers. Due to this, the researchers of mathematics-related areas of the programs requested to hold the place to enhance a collaboration beyond the boundaries of research area and specialized fields.

Thereafter, a specific plan was proposed for researchers in the field of mathematics to work together in groups. In the workshop, researchers discussed unsolved problems considered as current problems, explained the background and the significance of each problem, worked on the problems with young mathematicians who participated, shared their knowledge, conducted joint research; including simulations on the spot, and found clues to the solutions. Hence, the JST Math Open Problem Workshop has been held every year since 2017 in the form of a two-night camp. The photo shows an active discussion at the actual camp in 2017.

However, in 2020, it became difficult to hold face-to-face discussions like before due to the coronavirus outbreak. Nevertheless, we managed to conduct this workshop online through various online tools. Thereafter, the members of the workshop and subsequent discussions were able to solve the problems and produce concrete outputs such as the publication of a coauthored paper.

We are pleased to introduce a paper^*2 that was published in the Journal of Computational and Applied Mathematics as a successful example of the output of the first workshop.

In the solutions of an equation called a reaction-diffusion system, there is a traveling spot solution that can propagate while keeping its shape constant. The question of how the motion of this solution is affected by the shape of the region has been of interdisciplinary importance. Professor Shin-Ichiro EI, Hokkaido University, elucidated the relationship between the motion of a traveling spot inside a domain and the shape of the domain through the equation of motion. The solution of an equation, called the modified Helmholtz equation, is required to solve the equation of motion; however, it has been unsolved except for certain domains. A question of how to obtain a solution of the modified Helmholtz equation to the given shape was presented at the Open Problem Workshop and a joint research was launched.

A team led by Associate Professor Yoshitaro Tanaka, Future University Hakodate, applied the method of fundamental solutions (MFS). This method obtains an approximate solution of a partial differential equation to solve the problem. They succeeded in showing that an approximate solution can be theoretically constructed in a disc domain, and that the convergence of the approximate solution using the MFS to the exact solution with exponential order. This enabled to estimate the error between exact and approximate solutions under certain conditions beforehand. It seems to be world’s first that the existence of the approximate solution using the MFS for the Neumann boundary condition of the modified Helmholtz equation is demonstrated, particularly obtaining the mathematical results of exponential convergence. It was not possible to achieve this outcome without this workshop.

(*1)

CREST and PRESTO: Alliance for Breakthrough between Mathematics and Sciences

CREST: Modeling Methods allied with Modern Mathematics

CREST: Creating information utilization platform by integrating mathematical and information sciences, and development to society

PRESTO: Collaborative Mathematics for Real World Issues

PRESTO: Elucidating mathematical structures in real/virtual world objects and their utilization

(*2)

Title: Method of fundamental solutions for Neumann problems of the modified Helmholtz equation in disk domains

Author: Shin-Ichiro Ei (Hokkaido University), Hiroyuki Ochiai (Kyusyu University), Yoshitaro Tanaka (Future University Hakodate)

Journal: Journal of Computational and Applied Mathematics 402 (2022) 113795

https://www.sciencedirect.com/science/article/pii/S0377042721004179