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en: On numerical challenge for the distinction between smooth functions and analytic functions

en: In the paper, we present numerical experiments for investigation of a property of functions.Especially, we focus on the smoothness.Two kinds of methods are used in numerical experiments.One is the interpolation and the other is the Cauchy problem for the Laplace operator.Numerical experiments are performed by using the spectral collocation method and a multiple-precision arithmetic.Numerical results show the possibility of the numerical computation for the distinction between smooth functions and analytic functions.

en: Theoretical and Applied Mechanics Japan

ja: Theoretical and Applied Mechanics Japan


en: ON NUMERICAL COMPUTATION OF RANK DEFICIT OF MATRICES IN MULTIPLE PRECISION

en: Advances in Mathematical Sciences and Applications


en: Numerical Simulation on Local Solutions Partial Differential Equations

en: We present a simple numerical simulation on local solutions of partial differential equations. It is enabled by using IPNS(Infinite-Precision Numerical Simulation). The equation solved here is the Laplace equation. The behaviors of numerical solutions are different corresponding to the existence or nonexistence of the solution. The numerical method used here is not restricted to the Laplace equation. So, our results may be useful for the theoretical proof on the existence or nonexistence of the solution of partial differential equations.


en: NUMERICAL SIMULATION ON NON-EXISTENCE AND NON-UNIQUENESS OF SOLUTIONS FOR THE TRICOMI EQUATION

en: GAKUTO International Series, Mathematical Sciences and Applications


en: Simple Numerical Judgement on the Singularity of the Matrix by Using the Multiple-Precision Arithmetic


en: On Numerical Computation of the Toricomi Equation

en: The Tricomi equation is solved numerically. Boundary value problems (BVPs) and ill-posed Cauchy problems (CPs) are considered. Problems are discretized by using the finite difference method (FDM) or the spectral collocation method (SCM). The numerical computation is carried out in the multiple-precision arithmetic. For BVPs both FDM and SCM work well. When the exact solution is a part of a global and analytic function accuracy of numerical results are expectable. They show that the maximum principle does not hold here. Some other BVPs are solved and numerical results are satisfactory. For CPs SCM works well but FDM does not. When the exact solution is a part of a global and analytic function accuracy of numerical results by SCM is expectable. Some other CPs are solved by SCM. Numerical results suggest that there exist some delicate problems as nonexsistence of the solution. They also show the effectiveness of SCM with the multiple-precision arithmetic in the numerical simulation for delicate problems.


en: Numerical Computation of Continuation Problems in the Annular Domain


en: PARALLEL COMPUTING OF INTERVAL ARITHMETIC IN MULTIPLE PRECISION FOR SIMULTANEOUS LINEAR EQUATIONS

en: 本論文は，離散化手法であるスペクトル選点法の先端的な応用例をいくつか紹介した． その中でも，特に， 直接的な数値計算ができない例として辞典などでとりあげられているFredholm型の積分方程式に対して， スペクトル選点法用いた直接的数値計算法を提案した． 実際の数値計算では，多倍長演算を併用して， 直接的な数値計算が確かにできていることを示した． スペクトル選点法は任意次数の離散化手法であり，これと多倍長演算を併用する手法は， 我々が番号42の論文で名付けた「無限精度数値シミュレーション」そのものである． すなわち，無限精度数値シミュレーションによって， いままで不可能と思われてきたFredholm型の積分方程式の直接数値計算を可能にした．

ja: GAKUTO International Series, Mathematical Sciences and Applications


en: Numerical Computation on Continuation of the Solution for the One-Dimensional Heat Equation

ja: 空間1次元熱伝導方程式の解の接続の数値計算

en: We consider a problem governed by the simple one-dimensional heat equation. In our problem two parameters are introduced. One is for the determination of the initial boundary value problem, the continuation problem or the initial value problem. Another one is for analiticity. Numerical simulations are carried out by using two different numerical methods(FDM⊕EE and IPNS). Numerical results are interesting as follows: Both methods give similar results converging to the smooth solution for the initial value problem which has non-uniqueness of the solution. Moreover, both methods give different results for the continuation problem.

ja: 空間1次元熱伝導方程式の問題を考える.問題にはパラメータが2つあり,1つは,問題を初期値・境界値問題,解の接続問題,初期値問題に設定する.もう1つのパラメータは解析性に関連する.数値計算を2種類の異なる数値解法で行った.解の一意性がない初期値問題では2つの数値解法とも滑らかな解に収束する数値解を与えた.解の接続の数値計算では2つの数値解法が異なる数値計算結果を与えた.

en: 日本応用数理学会論文誌

ja: 日本応用数理学会論文誌


en: Global in Space Simulation for the Black-Scholes Equation Incorporating Transaction Costs


en: Numerical Computation on Analyticity of the Solution of a Cauchy Problem for the Backward Heat Equation

en: Theoretical and Applied Mechanics Japan


en: Global Simulation of a Backward Heat Conduction Problem with a Variable Transform on Time


en: On numerical methods for solving linear systems appearing in infinite precision numerical simulation

en: RIMS Kokyuroku

ja: RIMS Kokyuroku


en: Reaction-diffusion modelling of bacterial colony patterns

en: Physica A: Statistical Mechanics and its Applications


en: Formation of colony patterns by a bacterial cell population

ja: Physica A: Statistical Mechanics and its Applications


en: Interface growth and pattern formation in bacterial colonies


en: Existence and Uniqueness of Quasiperiodic Solutions to Perturbed Nonlinear Oscillators

en: 本論文では摂動項を伴う非線形常微分方程式の準周期解の一意存在性定理を，概周期関数の概念を用いずに確立することを試みた．準周期解の一意性定理が成立するための摂動のパラメータの範囲を正確に示し，また一意性が保証されうる近似解の近傍をも定量的に正確に与えた．得られた結果を摂動項をもつ Duffing 型方程式と Van der Pol 型方程式に適用し，厳密解に対する一意存在性の証明に応用した．

en: Japan Journal of Industrial and Applied Mathematics


en: Breathing and wiggling motions in three-species laterally inhibitory systems

en: Physical Review E, Statistical, Nonlinear, and Soft Matter Physics


en: 熱伝導逆問題の初期値問題に関連する数値計算

ja: 熱伝導逆問題の初期値問題に関連する数値計算

ja: 数理解析研究所講究録


en: 多倍長計算を適用した精度保証数値計算

ja: 多倍長計算を適用した精度保証数値計算


en: On Super Numerical Simulation

ja: On Super Numerical Simulation


en: On numerical methods for solving linear systems appearing in Infinite Precision Numerical Simulation


en: Infinite Precision Numerical Simulation for PDE Systems and Its Applications


en: 偏微分方程式の任意精度数値シミュレーションについて

ja: 偏微分方程式の任意精度数値シミュレーションについて


en: 滑らかな解を持つ偏微分方程式の任意精度数値シミュレーション

ja: 滑らかな解を持つ偏微分方程式の任意精度数値シミュレーション

en: Cooperative Research Report,The Institute of Statistical Mathematics

ja: 統計数理研究所共同リポート