New modified Runge–Kutta–Nyström methods for the numerical integration of the Schrödinger equation

AbstractIn this work we construct new Runge–Kutta–Nyström methods especially designed to integrate exactly the test equation y″=−w2y. We modify two existing methods: the Runge–Kutta–Nyström methods of fifth and sixth order. We apply the new methods to the computation of the eigenvalues of the Schrödinger equation with different potentials such as the harmonic oscillator, the doubly anharmonic oscillator and the exponential potential

An algebraic method to solve the radial Schrödinger equation

AbstractWe propose a method of numerical integration of differential equations of the type x2y″+f(x)y=0 by approximating its solution with solutions of equations of the type x2y″+(ax2+bx+c)y=0. This approximation is performed by segmentary approximation on an interval. We apply the method to obtain approximate solutions of the radial Schrödinger equation on a given interval and test it for two different potentials. We conclude that our method gives a similar accuracy than the Taylor method of higher order

エネルギー関数を持つ発展方程式に対する幾何学的数値計算法

学位の種別: 課程博士審査委員会委員 : (主査)東京大学教授 松尾 宇泰, 東京大学教授 中島 研吾, 東京大学准教授 鈴木 秀幸, 東京大学准教授 長尾 大道, 東京大学准教授 齋藤 宣一University of Tokyo(東京大学

Mathematical Modeling with Differential Equations in Physics, Chemistry, Biology, and Economics

This volume was conceived as a Special Issue of the MDPI journal Mathematics to illustrate and show relevant applications of differential equations in different fields, coherently with the latest trends in applied mathematics research. All the articles that were submitted for publication are valuable, interesting, and original. The readers will certainly appreciate the heterogeneity of the 10 papers included in this book and will discover how helpful all the kinds of differential equations are in a wide range of disciplines. We are confident that this book will be inspirational for young scholars as well

The numerical simulation of nonlinear waves in a hydrodynamic model test basin

This thesis describes the development of a numerical algorithm for the fully nonlinear simulation of free-surface waves. The aim of the research is to develop, implement and investigate an algorithm for the deterministic and accurate simulation of twodimensional nonlinear water waves in a model test basin. The simulated wave field may have a broad-banded spectrum and the simulations should be carried out by an efficient algorithm in order to be applicable in practical situations. The algorithm is based on a combination of Runge-Kutta (for time integration), Finite Element (boundary value problem) and Finite Difference (velocity recovery) methods. The scheme is further refined and investigated using different models for wave generation, propagation and absorption of waves

Many Body Quantum Chaos

This editorial remembers Shmuel Fishman, one of the founding fathers of the research field "quantum chaos", and puts into context his contributions to the scientific community with respect to the twelve papers that form the special issue

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal