We often study the existence and stability for equilibria of the reaction-diffusion system with suitable initial condition to understand the mechanism of phenomena which appear in various fields. In 1952, A. Turing developed a theory of pattern formation which is called Turing instability, by using the 2-component reaction-diffusion system and the linear analysis to determine conditions where spatially homogeneous equilibria are destabilized. Since the Turing instability suggests the multiple existence of equilibria, one important problem is to find out all equilibria of the system and study their stability properties. In this paper, as a first step to approach the problem, we consider a two competing species system with diffusion, and discuss the global bifurcation structure for radially symmetric equilibria of the system, by employing the comparison principle, the bifurcation theory and the numerical verification method.

View full abstract

Realistic retrieval of aircraft wake turbulence at an airport is not straight-forward due to the difficulty in incorporating actual conditions such as an atmospheric boundary layer and a flight trajectory into the numerical simulation. In this study, the numerical simulation of wake turbulence is performed by integrating lidar measurements at Sendai airport. The four-dimensional variational method is employed for the integration with taking an actual measurement process into consideration. The behavior of wake turbulence under realistic wind conditions is shown after the discussion regarding the convergence of a retrieved flow field.

View full abstract

Shifted linear systems arise in many important applications such as lattice quantum chromodynamics, large-scale electronic structure theory, and quadratic optimization problems. Recent strong need for solving the extremely large shifted linear systems enhance the importance of designing efficient solvers. As a candidate to satisfy the need, iterative methods using Krylov subspaces and the shift-invariance property have been attracting much interest. The primary aim of this paper is to survey the successful iterative methods and to classify them in terms of Krylov subspace methods.

View full abstract