Doshisha University (Private)

Phenomena that involve time-dependent probability events are mathematically formulated with stochastic differential equations,
which raise many issues that must be solved in their simulation. We are studying topics such as pseudorandom number generators
that simulate the standard stochastic process known as the Wiener process on a computer, convergence of the discrete-variable
method and its convergence speed, and numerical stability of discrete-variable solutions.
Discrete-variable solutions of delay-differential equations
Analytic solutions of delay-differential equations that appear in control engineering, population dynamics theory, and other fields
present more difficulties than typical differential equations, and discrete approximation solutions are required in simulations. We are
studying discrete-variable methods, particularly criteria of their numerical stability, and engaged in research on implementation
issues and phenomena modeled by difference-differential equations.
Research Contents
We are currently studying computational mathematics with a focus on numerical analysis. Numerical analysis is a field of
mathematical science that can be defined as "the mathematic theory of designing, analyzing, and evaluating numerical algorithms."
In many aspects of science and technology, the process of mathematical modeling is repeated, which involves building a
mathematical model for analyzing a certain phenomenon using the power of computers, and obtaining qualitative and quantitative
data about these problems from the simulation results to open up new knowledge. Most mathematical models are expressed in
terms of differential equations, but it is impossible to directly solve differential equations on a computer, and therefore solutions
through discretization or approximation are necessary. Creating new algorithms enables us to solve previously unsolvable problems
or solve problems thousands of times faster than before, and for this reason, numerical algorithms are critically important for model
simulations.
Newton, Euler, and other prominent mathematicians in history have contributed to numerical analysis, and with the widespread use
of computers today, many more problems are being challenged. Our research ranges from the foundation of this field to its real
applications.
Keywords
Numerical analysis
Ordinary differential equation
Mathematical modeling
Numerical algorithm
Parallel computation