Differential and Difference Equations through Computer Experiments - Kocak, Hüseyin

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Résumé :
Phaser is a sophisticated program for IBM personal com- puters, developed atBrown University by the author and some of his students, which enables usersto experiment with differential and difference equations and dynamical systems in an interactive environment using graphics. This book begins with a brief discussion of the geometric inter- pretation of differential equations and numerical methods, and proceeds to guide the student through the use of the program. To run Phaser, you need an IBM PC, XT, AT, or PS/2 with an IBM Color GRaphics Board (CGB), Enhanced Graphics Adapter (VGA). A math coprocessor is supported; however, one is not required for Phaser to run on the above hardware.

Sommaire:
I: Mathematical Synopsis.- 1. What is a Differential Equation?.- 1.1. Systems of Differential Equations.- 1.2. Geometrical Interpretation.- 1.3. Higher Order Equations.- 1.4. Non-autonomous Equations.- 2. Numerical Methods.- 2.1. Overview of Numerics.- 2.2. A Comparison of Algorithms.- 2.3. Practical Guidelines.- 3. What is a Difference Equation?.- 3.1. One-dimensional Difference Equations.- 3.2. Application: Calculating dollars dollars\sqrt{2} dollars dollars.- 3.3. Systems of Difference Equations.- 3.4. Planar Poincare Maps.- II: Handbook of PHASER.- 4. Learning to Use PHASER.- 4.1. The Startup Screen Layout.- 4.2. Interacting with PHASER.- 5. Lessons with PHASER.- Lesson 1. In and out of PHASER.- Lesson 2. Changing menus.- Lesson 3. Direction field.- Lesson 4. Changing views.- Lesson 5. Graphing variables.- Lesson 6. Initial conditions and parameters.- Lesson 7. Making transcripts for demonstrations.- Lesson 8. Browsing through the library.- Lesson 9. Entering your equations.- Lesson 10. Euler vs. Runge-Kutta.- Lesson 11. Stair step diagrams.- Lesson 12. Three-dimensional graphics.- Lesson 13. Advanced graphics: A 4D example.- Lesson 14. Planar sections and Poincare maps.- Lesson 15. Grand finale: Demo.- 6. Reference Guide to Menus.- 6.1. The NUMERICS Menu.- 6.2. The UTILITIES Menu.- 6.3. The VISUALAID Menu.- 6.4. The Graphical VIEWS.- III: Library of Equations.- 7. Differential Equations.- 7.1. One-dimensional Equations cubicld.- 7.2. Two-dimensional Equations linear2d, pendulum, vanderpol, predprey, saddlenod, pitchfork, transcrit, hopf, dzerol, dzero2, dzero3, hilbert2, hilbert4, averfvdp, gradient.- 7.3. Three-dimensional Equations lorenz, linear3d, vibration, bessel, euler, laguerre, legendre, forcevdp, forcepen, mathieu, forceduf, rossler, zeroim, silnikov, silnikov2.- 7.4. Four-dimensional Equations harmoscil, kepler, r3body, henheile, coplvdpl, coplvdp2, couplosc, reson21.- 8. Difference Equations.- 8.1. One-dimensional Equations logistic, dislinld, discubic, newton, tent, singer, mod, arnold.- 8.2. Two-dimensional Equations dislin2d, gauss, quadl, quad2, henon, lozi, dellogis, dispprey, bounball, anosov, Cremona, icremona, gingerman.- 8.3. Three-dimensional Equations dislin3d, act.- What remains to be done?.- Appendix A: PHASER Quick Reference.- Appendix B: Library of Equations.- References.