Solving Problems in Scientific Computing Using Maple and MATLAB 9

Transcription

1 Walter Gander Jiri Hirebicek Solving Problems in Scientific Computing Using Maple and MATLAB 9 Third, Expanded and Revised Edition 1997 With 161 Figures and 12 Tables Springer

2 r Contents Chapter 1. The Tractrix and Similar Curves Introduction The Classical Tractrix The Child and the Toy The Jogger and the Dog Showing the Motions with MATLAB Jogger with Constant Velocity Using a Moving Coordinate System Transformation for Jogger/Dog Transformation for Child/Toy Examples 22 References 26 Chapter 2. Trajectory of a Spinning Tennis Ball Introduction MAPLE Solution MATLAB Solution Simpler Solution for MATLAB 5 35 References 37 Chapter 3. The Illumination Problem Introduction Finding the Minimal Illumination Point on a Road Varying h 2 to Maximize the Illumination Optimal Illumination Conclusion 49 References 49 Chapter 4. Orbits in the Planar Three-Body Problem Introduction Equations of Motion in Physical Coordinates Global Regularization The Pythagorean Three-Body Problem Conclusions 70 References 72

3 xiv CONTENTS Chapter 5. The Internal Field in Semiconductors Introduction Solving a Nonlinear Poisson Equation Using MAPLE MATLAB Solution 80 References 81 Chapter 6. Some Least Squares Problems 83 C.I Introduction Fitting Lines, Rectangles and Squares in the Plane Fitting Hyperplanes 95 References 101 Chapter 7. The Generalized Billiard Problem Introduction The Generalized Reflection Method Line and Curve Reflection Mathematical Description MAPLE Solution The Shortest Trajectory Method MAPLE Solution Examples The Circular Billiard Table The Elliptical Billiard Table The Snail Billiard Table The Star Billiard Table Conclusions 119 References 121 Chapter 8. Mirror Curves The Interesting Waste The Mirror Curves Created by MAPLE The Inverse Problem Outflanking Manoeuvre Geometrical Construction of a Point on the Pattern Curve MAPLE Solution Analytic Solution Examples The Circle as the Mirror Curve The Line as the Mirror Curve Conclusions 132 References 134

4 r CONTENTS xv Chapter 9. Smoothing Filters Introduction Savitzky-Golay Filter Filter Coefficients Results Least Squares Filter Lagrange Equations Zero Finder Evaluation of the Secular Function MEX-Files Results 150 References 152 Chapter 10. The Radar Problem Introduction Converting Degrees into Radians Transformation into Geocentric Coordinates The Transformations Final Algorithm Practical Example 162 References 164 Chapter 11. Conformal Mapping of a Circle Introduction Problem Outline MAPLE Solution MATLAB Solution 170 References 172 Chapter 12. The Spinning Top Introduction Formulation and Basic Analysis of the Solution The Numerical Solution 180 References 182 Chapter 13. The Calibration Problem Introduction The Physical Model Description Approximation by Splitting the Solution Conclusions 191 References 192 Chapter 14. Heat Flow Problems Introduction Heat Flow through a Spherical Wall A Steady State Heat Flow Model 194

5 xvi CONTENTS Fourier Model for Steady State MAPLE Plots Non Stationary Heat Flow through an Agriculture Field MAPLE Plots 201 References 201 Chapter 15. Modeling Penetration Phenomena Introduction Short description of the penetration theory The Tate-Alekseevskii model Special case R t -Y p Special case p p = p t = p The eroding rod penetration model Numerical Example Conclusions 218 References 218 Chapter 16. Heat Capacity of System of Bose Particles Introduction MAPLE Solution 223 References 227 Chapter 17. Free Metal Compression Introduction The Base Expansion Base Described by One and Several Functions The Lateral Side Distortion Non-centered Bases Three Dimensional Graphical Representation of the Distorted Body Centered base Non-centered, Segmented Base Convex Polygon Base Three Dimensional Animation Limitations and Conclusions 250 References 252 Chapter 18. Gauss Quadrature Introduction Orthogonal Polynomials Quadrature Rule Gauss Quadrature Rule Gauss-Radau Quadrature Rule 270

6 CONTENTS xvii 18.6 Gauss-Lobatto Quadrature Rule Weights Quadrature Error 277 References 280 Chapter 19. Symbolic Computation of Explicit Runge-Kutta Formulas Introduction Derivation of the Equations for the Parameters Solving the System of Equations Grobner Bases Resultants The Complete Algorithm Example 1: Example 2: Conclusions 298 References 299 Chapter 20. Transient Response of a Two-Phase Half-Wave Rectifier Introduction Problem Outline Difficulties in Applying Conventional Codes and Software Packages Solution by Means of MAPLE 306 References 312 Chapter 21. Circuits in Power Electronics Introduction Linear Differential Equations with Piecewise Constant Coefficients Periodic Solutions A MATLAB Implementation Conclusions 324 References 324 Chapter 22. Newton's and Kepler's laws Introduction Equilibrium of Two Forces Equilibrium of Three Forces Equilibrium of Three Forces, Computed from the Potential Energy Gravitation of the Massive Line Segment Potential and Intensity The Particle Trajectory 333

7 xviii CONTENTS 22.6 The Earth Satellite Earth Satellite, Second Solution The Lost Screw Conclusions 339 References 340 Chapter 23. Least Squares Fit of Point Clouds Introduction Computing the Translation Computing the Orthogonal Matrix Solution of the Procrustes Problem Algorithm Decomposing the Orthogonal Matrix Numerical Examples First example Second example 350 References 351 Chapter 24. Modeling Social Processes Introduction Modeling Population Migration Cyclic Migration without Regulation Cyclic Migration with Regulation Modeling Strategic Investment 358 References 360 Chapter 25. Contour Plots of Analytic Functions Introduction Contour Plots by the contour Command Differential Equations Contour Lines r = const Contour Lines <p = const The Contour Lines r = 1 of / = e n The Contour Lines tp = const of / = e n 372 References 373 Chapter 26. Non Linear Least Squares: Finding the most accurate location of an aircraft Introduction Building the Least Squares Equations Solving the Non-linear System Confidence/Sensitivity Analysis 381

8 CONTENTS xix Chapter 27. Computing Plane Sundials Introduction Astronomical Fundamentals Coordinate Systems The Gnomonic Projection Time Marks Local Real Time Mean Time Babylonic and Italic Hours Sundials on General Planes A Concluding Example 398 References 400 Index 401 Index of used MAPLE Commands 407 Index of used MATLAB Commands 411