Ｍａｔｈｅｍａｔｉｃａを用いた常微分方程式の科学計算<br>Scientific Computing with Mathematica : Mathematical Problems for Ordinary Differential Equations (Modeling and Simulation in Science, Engineering & T

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Ｍａｔｈｅｍａｔｉｃａを用いた常微分方程式の科学計算
Scientific Computing with Mathematica : Mathematical Problems for Ordinary Differential Equations (Modeling and Simulation in Science, Engineering & T

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  • 製本 Hardcover:ハードカバー版／ページ数 270 p.
  • 言語 ENG
  • 商品コード 9780817642051
  • DDC分類 515.350285

Full Description

Many interesting behaviors of real physical, biological, economical,
and chemical systems can be described by ordinary differential
equations (ODEs).
Scientific Computing with Mathematica for Ordinary Differential
Equations
provides a general framework useful for the applications, on the
conceptual aspects of the theory of ODEs, as well as a sophisticated
use of Mathematica software for the solutions of problems related to
ODEs. In particular, a chapter is devoted to the use ODEs and
Mathematica in the Dynamics of rigid bodies.
Mathematical methods and scientific computation are dealt with jointly
to supply a unified presentation. The main problems of ordinary
differential equations such
as, phase portrait, approximate solutions, periodic orbits, stability,
bifurcation, and
boundary problems are covered in an integrated fashion with numerous
worked examples and computer program demonstrations using
Mathematica.
Topics and Features:*Explains how to use the Mathematica package ODE.m
to support qualitative and quantitative problem solving *End-of-
chapter exercise sets incorporating the use of Mathematica programs
*Detailed description and explanation of the mathematical procedures
underlying the programs written in Mathematica *Appendix describing
the use of ten notebooks to guide the reader through all the
exercises.
This book is an essential text/reference for students, graduates and
practitioners in applied mathematics and engineering interested in
ODE's problems in both the qualitative and quantitative description of
solutions with the Mathematica program. It is also suitable as a self-

Contents

1 Solutions of ODEs and Their Properties.- 1.1 Introduction.- 1.2 Definitions and Existence Theory.- 1.3 Functions DSolve, NDSolve, and Differentiallnvariants.- 1.4 The Phase Portrait.- 1.5 Applications of the Programs Sysn, Phase2D, PolarPhase, and Phase3D.- 1.6 Problems.- 2 Linear ODEs with Constant Coefficients.- 2.1 Introduction.- 2.2 The General Solution of Linear Differential Systems with Constant Coefficients.- 2.3 The Program LinSys.- 2.4 Problems.- 3 Power Series Solutions of ODEs and Frobenius Series.- 3.1 Introduction.- 3.2 Power Series and the Program Taylor.- 3.3 Power Series and Solutions of ODEs.- 3.4 Series Solutions Near Regular Singular Points: Method of Frobenius.- 3.5 The Program SerSol.- 3.6 Other Applications of SerSol.- 3.7 The Program Frobenius.- 3.8 Problems.- 4 Poincaré's Perturbation Method.- 4.1 Introduction.- 4.2 Poincaré's Perturbation Method.- 4.3 How to Introduce the Small Parameter.- 4.4 The Program Poincare.- 4.5 Problems.- 5 Problems of Stability.- 5.1 Introduction.- 5.2 Definitions of Stability.- 5.3 Analysis of Stability: The Direct Method.- 5.4 Polynomial Liapunov Functions.- 5.5 The Program Liapunov.- 5.6 Analysis of Stability, the Indirect Method: The Planar Case.- 5.7 The Program LStability.- 5.8 Problems.- 6 Stability: The Critical Case.- 6.1 Introduction.- 6.2 The Planar Case and Poincaré's Method.- 6.3 The Programs CriticalEqS and CriticalEqN.- 6.4 The Center Manifold.- 6.5 The Program CManifold.- 6.6 Problems.- 7 Bifurcation in ODEs.- 7.1 Introduction to Bifurcation.- 7.2 Bifurcation in a Differential Equation Containing One Parameter.- 7.3 The Programs Bifl and Bif1G.- 7.4 Problems.- 7.5 Bifurcation in a Differential Equation Depending on Two Parameters.- 7.6 The Programs Bif2 and Bif2G.- 7.7 Problems.- 7.8 Hopf'sBifurcation.- 7.9 The Program HopfBif.- 7.10 Problems.- 8 The Lindstedt-Poincaré Method.- 8.1 Asymptotic Expansions.- 8.2 The Lindstedt-Poincaré Method.- 8.3 The Programs LindPoinc and GLindPoinc.- 8.4 Problems.- 9 Boundary-Value Problems for Second-Order ODEs.- 9.1 Boundary-Value Problems and Bernstein's Theorem.- 9.2 The Shooting Method.- 9.3 The Program NBoundary.- 9.4 The Finite Difference Method.- 9.5 The Programs NBoundaryl and NBoundary2.- 9.6 Problems.- 10 Rigid Body with a Fixed Point.- 10.1 Introduction.- 10.2 Euler's Equations.- 10.3 Free Rotations or Poinsot's Motions.- 10.4 Heavy Gyroscope.- 10.5 The Gyroscopic Effect.- 10.6 The Program Poinsot.- 10.7 The Program Solid.- 10.8 Problems.- A How to Use the Package ODE.m.- References.