Scientific Computing with Mathematica®

Provides a general framework useful for the application of the theory of ODEs, as well as a sophisticated use of Mathematica software for the solution of problems related to ODEs. Through extensive worked examples and computer program demonstrations using Mathematica, the authors cover phase portrait, approximate solutions, periodic orbits, stability, bifurcation, and boundary problems. For graduate students, researchers, and practitioners in applied mathematics and engineering seeking an understanding of using ODEs in modeling physical, biological, and economic phenomena.

Includes supplementary material: sn.pub/extras

Inhalt
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.