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Doshisha University (Private)

Doshisha University (Private)

Prof. Yorimasa OSHIME

Prof. Yorimasa OSHIME Analysis Laboratory http://istc.doshisha.ac.jp/course/environment/labo_63.html Research Topics Spectra and Green's function properties of differential operators, with a focus on Schrodinger operators (particularly non-self-adjoint operators) and Sturm-Liouville operators Asymptotic behavior of solutions of Lotka-Volterra and other equations Research Contents At our laboratory, we are researching differential equations particularly by applying functional analysis. Many problems that appear in physics and engineering can be solved using mathematics. A large number of these problems can also be solved using calculus. For example, Newton used calculus to solve problems in differential equations to demonstrate Kepler's law, that is, "the planets orbit the sun in an elliptical path with the sun at a focus." There are also many types of differential equations, and our research deals with such differential equations. Keywords Differential equation Integral kernel Functional analysis Banach space Spectrum

Prof. Taketomo MITSUI Computational Mathematics Laboratory http://istc.doshisha.ac.jp/course/environment/labo_66.html Research Topics At the Computational Mathematics Laboratory, the primary focus of our research is numerical analysis of differential equations. To be more precise, we are interested in the individual topics and related fields described below. Discrete-variable methods for large-scale, stiff differential equation systems Time-dependent partial differential equations, including Navier-Stokes equations, can be reduced to large-scale time-marching ordinary differential equations by discretization of space variables (with finite difference or finite element methods, etc.). These large-scale systems actually exhibit stiffness, and numerical solutions are difficult to find for many of them. For stiff systems we are studying the feasibility of parallelism for Rosenbrock formulas, which is equipped with an automatic generation of the Jacobian matrix of the system, as well as for Runge-Kutta formulas (described below). Parallel algorithms Obtaining significant numerical results in mathematical modeling yields an enormous amount of calculations. To overcome these difficulties, an introduction of parallel computation can be a breakthrough. Manipulating implicit Runge-Kutta formulas for large-scale stiff systems can attain parallelism, and we are actually studying this performance using various supercomputers. Parallel algorithms require a fundamental look-up on basic theoretical properties such as convergence, stability and so on, and their computational complexity must be re-examined. Geometric discrete-variable methods The Hamiltonian systems that often appear in the formulation of physical phenomena feature to be symplectic, that is, they preserve an intrinsic quantity called the symplecticness. The feature is also expected to be kept in discrete-variable methods, but the frequently-used numerical solutions are readily shown not to be symplectic. The fact inspires a development of new methods. This structure-preserving method is now called the geometric method. Since many symplectic methods are implicit, an in-depth study related to (2) above is also necessary. The figure below shows the simulation results for the motion of a vortex appearing in the analysis of fluid motion on a plane called the vortex method. The equation of motion is a special Hamiltonian system, and if the symplectic discrete-variable method is not used, it cannot be tracked for an extended period of time. We constructed discrete-variable methods with the G-symplecticness and checking their calculation results. Discrete numerical solutions of stochastic differential equations

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