Scientific Computing With Mathematica

by Romano, Antonio; Marasco, Addolorata

• ISBN13: 9780817642051
• ISBN10: 0817642056
• Format: Hardcover
• Copyright: 2001-08-01
• Publisher: Birkhauser

Summary

Many interesting behaviors of real physical, biological, economical, and chemical systems can be described by ordinary differential equations (ODEs). Scientific Computing with Mathematica® 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 of 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 ODEs 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:* Explanation of 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 of the mathematical procedures underlying the twenty-eight 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 engineering and applied mathematics interested in problems of ODEs in both the qualitative and quantitative description of solutions with the Mathematica® program. It is also suitable as a self-study resource for professionals and others seeking an understanding of how to use ODEs in modeling physical, biological, and economic phenomena.

Table of Contents

Preface	xi
Solutions of ODEs and Their Properties	1	(32)
Introduction	1	(4)
Definitions and Existence Theory	5	(3)
Functions DSolve, NDSolve, and DifferentialInvariants	8	(4)
The Phase Portrait	12	(3)
Applications of the Programs Syns, Phase2D, PolarPhase, and Phase3D	15	(14)
Problems	29	(4)
Linear ODEs with Constant Coefficients	33	(16)
Introduction	33	(2)
The General Solution of Linear Differential Systems with Constant Coefficients	35	(2)
The Program LinSys	37	(9)
Problems	46	(3)
Power Series Solutions of ODEs and Frobenius Series	49	(30)
Introduction	49	(1)
Power Series and the Program Taylor	50	(3)
Power Series and Solutions of ODEs	53	(2)
Series Solutions Near Regular Singular Points: Method of Frobenius	55	(4)
The Program SerSol	59	(5)
Other Applications of SerSol	64	(6)
The Program Frobenius	70	(6)
Problems	76	(3)
Poincare's Perturbation Method	79	(20)
Introduction	79	(1)
Poincare's Perturbation Method	80	(2)
How to Introduce the Small Parameter	82	(4)
The Program Poincare	86	(11)
Problems	97	(2)
Problems of Stability	99	(28)
Introduction	99	(1)
Definitions of Stability	100	(3)
Analysis of Stability: The Direct Method	103	(4)
Polynomial Liapunov Functions	107	(4)
The Program Liapunov	111	(8)
Analysis of Stability, the Indirect Method: The Planar Case	119	(2)
The Program LStability	121	(3)
Problems	124	(3)
Stability: The Critical Case	127	(18)
Introduction	127	(1)
The Planar Case and Poincare's Method	128	(3)
The Programs CriticalEqS and CriticalEqN	131	(7)
The Center Manifold	138	(3)
The Program CManifold	141	(2)
Problems	143	(2)
Bifurcation in ODEs	145	(32)
Introduction to Bifurcation	145	(1)
Bifurcation in a Differential Equation Containing One Parameter	146	(7)
The Programs Bif1 and Bif1G	153	(4)
Problems	157	(1)
Bifurcation in a Differential Equation Depending on Two Parameters	158	(6)
The Programas Bif2 and Bif2G	164	(4)
Problems	168	(1)
Hopf's Bifurcation	168	(2)
The Program HopfBif	170	(5)
Problems	175	(2)
The Lindstedt-Poincare Method	177	(24)
Asymptotic Expansions	177	(2)
The Lindstedt-Poincare Method	179	(4)
The Programs LindPoinc and GLindPoinc	183	(16)
Problems	199	(2)
Boundary-Value Problems for Second-Order ODEs	201	(30)
Boundary-Value Problems and Bernstein's Theorem	201	(3)
The Shooting Method	204	(4)
The Program NBoundary	208	(7)
The Finite Difference Method	215	(4)
The Programs NBoundary1 and NBoundary2	219	(10)
Problems	229	(2)
Rigid Body with a Fixed Point	231	(30)
Introduction	231	(1)
Euler's Equations	232	(3)
Free Rotations or Poinsot's Motions	235	(2)
Heavy Gyroscope	237	(2)
The Gyroscopic Effect	239	(2)
The Program Poinsot	241	(8)
The Program Solid	249	(10)
Problems	259	(2)
A How to Use the Package ODE.m	261	(6)
The Notebooks for Using ODE.m	261	(1)
A Brief Introduction to Programming	261	(2)
The Structure of Program Sysn	263	(4)
References	267	(2)
Index	269