Solving Problems in Scientific Computing Using Maple and MATLAB®

1. The Tractrix and Similar Curves.- 1.1 Introduction.- 1.2 The Classical Tractrix.- 1.3 The Child and the Toy.- 1.4 The Jogger and the Dog.- 1.5 Showing the Motions with Matlab.- 1.6 Jogger with Constant Velocity.- 1.7 Using a Moving Coordinate System.- 1.7.1 Transformation for Jogger/Dog.- 1.7.2 Transformation for Child/Toy.- 1.8 Examples.- References.- 2. Trajectory of a Spinning Tennis Ball.- 2.1 Introduction.- 2.2 Maple Solution.- 2.3 Matlab Solution.- 2.4 Simpler Solution for Matlab 5.- References.- 3. The Illumination Problem.- 3.1 Introduction.- 3.2 Finding the Minimal Illumination Point on a Road.- 3.3 Varying h2 to Maximize the Illumination.- 3.4 Optimal Illumination.- 3.5 Conclusion.- References.- 4. Orbits in the Planar Three-Body Problem.- 4.1 Introduction.- 4.2 Equations of Motion in Physical Coordinates.- 4.3 Global Regularization.- 4.4 The Pythagorean Three-Body Problem.- 4.5 Conclusions.- References.- 5. The Internal Field in Semiconductors.- 5.1 Introduction.- 5.2 Solving a Nonlinear Poisson Equation Using Maple.- 5.3 Matlab Solution.- References.- 6. Some Least Squares Problems.- 6.1 Introduction.- 6.2 Fitting Lines, Rectangles and Squares in the Plane.- 6.3 Fitting Hyperplanes.- References.- 7. The Generalized Billiard Problem.- 7.1 Introduction.- 7.2 The Generalized Reflection Method.- 7.2.1 Line and Curve Reflection.- 7.2.2 Mathematical Description.- 7.2.3 Maple Solution.- 7.3 The Shortest Trajectory Method.- 7.3.1 Maple Solution.- 7.4 Examples.- 7.4.1 The Circular Billiard Table.- 7.4.2 The Elliptical Billiard Table.- 7.4.3 The Snail Billiard Table.- 7.4.4 The Star Billiard Table.- 7.5 Conclusions.- References.- 8. Mirror Curves.- 8.1 The Interesting Waste.- 8.2 The Mirror Curves Created by Maple.- 8.3 The Inverse Problem.- 8.3.1 Outflanking Manoeuvre.- 8.3.2 Geometrical Construction of a Point on the Pattern Curve.- 8.3.3 Maple Solution.- 8.3.4 Analytic Solution.- 8.4 Examples.- 8.4.1 The Circle as the Mirror Curve.- 8.4.2 The Line as the Mirror Curve.- 8.5 Conclusions.- References.- 9. Smoothing Filters.- 9.1 Introduction.- 9.2 Savitzky-Golay Filter.- 9.2.1 Filter Coefficients.- 9.2.2 Results.- 9.3 Least Squares Filter.- 9.3.1 Lagrange Equations.- 9.3.2 Zero Finder.- 9.3.3 Evaluation of the Secular Function.- 9.3.4 MEX-Files.- 9.3.5 Results.- References.- 10. The Radar Problem.- 10.1 Introduction.- 10.2 Converting Degrees into Radians.- 10.3 Transformation into Geocentric Coordinates.- 10.4 The Transformations.- 10.5 Final Algorithm.- 10.6 Practical Example.- References.- 11. Conformai Mapping of a Circle.- 11.1 Introduction.- 11.2 Problem Outline.- 11.3 Maple Solution.- 11.4 Matlab Solution.- References.- 12. The Spinning Top.- 12.1 Introduction.- 12.2 Formulation and Basic Analysis of the Solution.- 12.3 The Numerical Solution.- References.- 13. The Calibration Problem.- 13.1 Introduction.- 13.2 The Physical Model Description.- 13.3 Approximation by Splitting the Solution.- 13.4 Conclusions.- References.- 14. Heat Flow Problems.- 14.1 Introduction.- 14.2 Heat Flow through a Spherical Wall.- 14.2.1 A Steady State Heat Flow Model.- 14.2.2 Fourier Model for Steady State.- 14.2.3 Maple Plots.- 14.3 Non Stationary Heat Flow through an Agriculture Field.- 14.3.1 Maple Plots.- References.- 15. Modeling Penetration Phenomena.- 15.1 Introduction.- 15.2 Short description of the penetration theory.- 15.3 The Tate-Alekseevskii model.- 15.3.1 Special case Rt= YP.- 15.3.2 Special case pp= pt = p.- 15.4 The eroding rod penetration model.- 15.5 Numerical Example.- 15.6 Conclusions.- References.- 16. Heat Capacity of System of Bose Particles.- 16.1 Introduction.- 16.2 Maple Solution.- References.- 17. Free Metal Compression.- 17.1 Introduction.- 17.2 The Base expansion.- 17.3 Base Described by One and Several Functions.- 17.4 The Lateral Side Distortion.- 17.5 Non-centered bases.- 17.6 Three Dimensional Graphical Representation of the Distorted Body.- 17.6.1 Centered base.- 17.6.2 Non-centered, Segmented Base.- 17.6.3 Convex Polygon Base.- 17.7 Three Dimensional Animation.- 17.8 Limitations and Conclusions.- References.- 18. Gauss Quadrature.- 18.1 Introduction.- 18.2 Orthogonal Polynomials.- 18.3 Quadrature Rule.- 18.4 Gauss Quadrature Rule.- 18.5 Gauss-Radau Quadrature Rule.- 18.6 Gauss-Lobatto Quadrature Rule.- 18.7 Weights.- 18.8 Quadrature Error.- References.- 19. Symbolic Computation of Explicit Runge-Kutta Formulas.- 19.1 Introduction.- 19.2 Derivation of the Equations for the Parameters.- 19.3 Solving the System of Equations.- 19.3.1 Gröbner Bases.- 19.3.2 Resultants.- 19.4 The Complete Algorithm.- 19.4.1 Example 1:.- 19.4.2 Example 2:.- 19.5 Conclusions.- References.- 20. Transient Response of a Two-Phase Half-Wave Rectifier.- 20.1 Introduction.- 20.2 Problem Outline.- 20.3 Difficulties in Applying Conventional Codes and Software Packages.- 20.4 Solution by Means of Maple.- References.- 21. Circuits in Power Electronics.- 21.1 Introduction.- 21.2 Linear Differential Equations with Piecewise Constant Coefficients.- 21.3 Periodic Solutions.- 21.4 A Matlab Implementation.- 21.5 Conclusions.- References.- 22. Newton's and Kepler's laws.- 22.1 Introduction.- 22.2 Equilibrium of Two Forces.- 22.3 Equilibrium of Three Forces.- 22.4 Equilibrium of Three Forces, Computed from the Potential Energy.- 22.5 Gravitation of the Massive Line Segment.- 22.5.1 Potential and Intensity.- 22.5.2 The Particle Trajectory.- 22.6 The Earth Satellite.- 22.7 Earth Satellite, Second Solution.- 22.8 The Lost Screw.- 22.9 Conclusions.- References.- 23. Least Squares Fit of Point Clouds.- 23.1 Introduction.- 23.2 Computing the Translation.- 23.3 Computing the Orthogonal Matrix.- 23.4 Solution of the Procrustes Problem.- 23.5 Algorithm.- 23.6 Decomposing the Orthogonal Matrix.- 23.7 Numerical Examples.- 23.7.1 First example.- 23.7.2 Second example.- References.- 24. Modeling Social Processes.- 24.1 Introduction.- 24.2 Modeling Population Migration.- 24.2.1 Cyclic Migration without Regulation.- 24.2.2 Cyclic Migration with Regulation.- 24.3 Modeling Strategic Investment.- References.- 25. Contour Plots of Analytic Functions.- 25.1 Introduction.- 25.2 Contour Plots by the contour Command.- 25.3 Differential Equations.- 25.3.1 Contour Lines r = const.- 25.3.2 Contour Lines ? = const.- 25.4 The Contour Lines r = 1 of f = en.- 25.5 The Contour Lines ? = const of f = en.- References.- 26. Non Linear Least Squares: Finding the most accurate location of an aircraft.- 26.1 Introduction.- 26.2 Building the Least Squares Equations.- 26.3 Solving the Non-linear System.- 26.4 Confidence/Sensitivity Analysis.- 27. Computing Plane Sundials.- 27.1 Introduction.- 27.2 Astronomical Fundamentals.- 27.2.1 Coordinate Systems.- 27.2.2 The Gnomonic Projection.- 27.3 Time Marks.- 27.3.1 Local Real Time.- 27.3.2 Mean Time.- 27.3.3 Babylonic and Italic Hours.- 27.4 Sundials on General Planes.- 27.5 A Concluding Example.- References.