Differential Equations with Maple: An Interactive Approach

ISBN-13: 9781461271208 / Angielski / Miękka / 2013 / 411 str.

John H. Davis; Jon Davis

Opis

Differential equations is a subject of wide applicability, and knowledge of dif- Differential equations is a subject of wide applicability, and knowledge of dif- ferential ferential equations equations topics topics permeates permeates all all areas areas of of study study in in engineering engineering and and applied applied mathematics. mathematics. Some Some differential differential equations equations are are susceptible susceptible to to analytic analytic means means of of so- so- lution, lution, while while others others require require the the generation generation of of numerical numerical solution solution trajectories trajectories to to see see the the behavior behavior of of the the system system under under study. study. For For both both situations, situations, the the software software package package Maple Maple can can be be used used to to advantage. advantage. To To the the student student Making Making effective effective use use of of differential differential equations equations requires requires facility facility in in recognizing recognizing and and solving solving standard standard "tractable" "tractable" problems, problems, as as well well as as having having the the background background in in the the subject subject to to make make use use of of tools tools for for dealing dealing with with situations situations that that are are not not amenable amenable to to simple simple analytic analytic approaches. approaches.

Recenzje

"A very novel approach to the teaching/learning of all basic aspects of differential equations. ... The book, therefore, provides an introduction to MAPLE as well as standard material on differential equations written in a friendly style." -Aslib Book Guide

"The course in differential equations starts with fairly well-explained examples from different fields of application. The theory is presented in a student-friendly, understandable form.... The problems presented at the end of each chapter contain material for using the theory or the tools presented in the frame of MAPLE. ... The textbook is recommended for those who start solving differential equation problems by MAPLE and also for those who need an introduction in (mainly ordinary) differential equations." -Applications of Mathematics

Spis treści

I Maple Use and Programming.- 1 Introduction to Maple.- 1.1 Calculator or What?.- 1.1.1 First Contact.- 1.2 Programming in Maple.- 1.3 Maple Variables.- 1.3.1 Variable Types.- 1.3.2 Maple Expressions.- 1.3.3 Aggregate Types.- 1.3.4 Maple Tables.- 1.4 Maple Syntax.- 1.4.1 Function Declarations.- 1.4.2 Assignments.- 1.4.3 Conditionals.- 1.4.4 Loops.- 1.4.5 Maple Evaluation.- 1.4.6 Details about Maple Procedures.- 1.4.7 Procedure Arguments.- 1.4.8 Programming Gotchas.- 1.4.9 Maple Program Development.- II Differential Equations.- 2 Introduction to Differential Equations.- 2.1 Example Problems.- 2.1.1 Simple Rate Equations.- 2.1.2 Mechanical Problems.- 2.1.3 Electrical Models.- 2.1.4 Kinetics.- 2.1.5 Beam Models.- 2.2 Order, Dimension and Type.- 2.2.1 First and Higher Order.- 2.2.2 Partial Derivative Problems.- 2.3 Problems.- 3 First Order Equations.- 3.1 Separable Problems.- 3.1.1 Separable Examples.- 3.2 Level Sets and Exact Equations.- 3.2.1 Solving Exact Problems.- 3.2.2 Exact Equation Examples.- 3.3 Linear Equations.- 3.3.1 Linear Examples.- 3.4 Tricky Substitutions.- 3.4.1 Lagrange.- 3.4.2 Riccati.- 3.4.3 Mechanics.- 3.5 Do Solutions Exist?.- 3.5.1 The Issue.- 3.5.2 Existence Theorems.- 3.6 Problems.- 4 Introduction to Numerical Methods.- 4.1 The Game and Euler’s Method.- 4.2 Solution Taylor Series.- 4.3 Runge-Kutta Methods.- 4.4 General Runge-Kutta Methods.- 4.5 Maple Numeric Routines.- 4.6 Calling all RK4’s (and relatives).- 4.7 Variable Step Size Methods.- 4.7.1 Running RK4 With RK3.- 4.7.2 Adjusting h.- 4.8 Serious Methods.- 4.9 Problems.- 5 Higher Order Differential Equations.- 5.1 Equations of Order N.- 5.1.1 Vector Equations.- 5.1.2 Equivalent Formulations.- 5.2 Linear Independence and Wronskians.- 5.2.1 Linear Independence.- 5.2.2 Wronskians.- 5.3 Fundamental Solutions.- 5.3.1 Existence of Fundamental Solutions.- 5.3.2 Constructing Fundamental Solutions.- 5.4 General and Particular Solutions.- 5.5 Constant Coefficient Problems.- 5.5.1 Homogeneous Case.- 5.5.2 Undetermined Coefficients.- 5.6 Problems.- 6 Laplace Transform Methods.- 6.1 Basic Definition.- 6.1.1 Examples.- 6.2 New Wine From Old.- 6.2.1 Algebra and Tables.- 6.3 Maple Facilities.- 6.4 Derivatives and Laplace.- 6.5 High Order Problems by Laplace.- 6.5.1 Fundamental Solutions.- 6.5.2 Undetermined Coefficients Redux.- 6.6 Convolutions.- 6.6.1 Inhomogeneous Problems.- 6.7 Problems.- 7 Systems of Equations.- 7.1 Linear Systems.- 7.2 Bases and Eigenvectors.- 7.3 Diagonalization.- 7.3.1 Using Maple.- 7.4 Jordan Forms.- 7.4.1 Using Maple.- 7.5 Matrix Exponentials.- 7.5.1 Laplace Transforms.- 7.5.2 Using Maple.- 7.6 Problems.- 8 Stability.- 8.1 Second Order Problems.- 8.2 Harmonic Oscillator Again.- 8.3 Nodes and Spirals.- 8.4 Higher Order Problems.- 8.4.1 The Easy Cases.- 8.4.2 Jordan Form.- 8.5 Linearization.- 8.6 Introduction to Lyapunov Theory.- 8.6.1 Linear Problems.- 8.6.2 Applications.- 8.7 Problems.- 9 Periodic Problems.- 9.1 Periodic Inputs.- 9.2 Phasors.- 9.3 Fourier Series.- 9.4 Time Domain Methods.- 9.5 Periodic Coefficients.- 9.6 Fundamental Matrices.- 9.7 Stability.- 9.8 Floquet Representation.- 9.9 Problems.- 10 Impedances and Differential Equations.- 10.1 Introduction to Impedances.- 10.2 AC Circuits.- 10.3 Transient Circuits.- 10.4 Circuit Impedance Examples.- 10.5 Loops and Nodes.- 10.6 Problems.- 11 Partial Differential Equations.- 11.1 Basic Problems.- 11.1.1 Heat Equation.- 11.1.2 Wave Equation.- 11.1.3 Laplace’s Equation.- 11.1.4 Beam Vibrations.- 11.2 Boundary Conditions.- 11.2.1 Heat and Diffusion Problems.- 11.2.2 Wave Equations.- 11.2.3 Laplace’s Equation.- 11.3 Separation of Variables.- 11.3.1 Cooling of a Slab.- 11.3.2 Standing Wave Solutions.- 11.3.3 Steady State Heat Conduction.- 11.4 Problems.- III Maple Application Topics.- 12 Introduction to Maple Applications.- 13 Plotting With Maple.- 13.1 Maple Plotting Structures.- 13.2 Remember Tables.- 13.3 Plotting Numerical Results.- 13.4 Plotting Vector Variables.- 13.5 Further Plotting.- 14 Maple and Laplace Transforms.- 14.1 Walking Maple Through Problems.- 14.2 Coding a Beam Problem Solver.- 15 Maple Linear Algebra Applications.- 15.1 Canonical Forms.- 15.1.1 Eigenvalues and Eigenvectors.- 15.1.2 Jordan Form Calculations.- 15.2 Matrix Exponentials.- 15.3 Stability Examples.- 15.4 Periodic Solutions with Maple.- 16 Runge-Kutta Designs.- 16.1 Orientation and Overview.- 16.2 Notational Issues.- 16.3 True Solution Derivatives.- 16.4 Runge-Kutta Derivatives.- 16.5 Traps and Tricky Bits.- 16.6 A Sample Run.- 16.7 Order Condition Code.- 17 Maple Packages.- 17.1 Introduction.- 17.2 Constructing Packages.- 17.3 A Control Design Package.- 17.3.1 Root Locus.- 17.3.2 Nyquist Locus.- 17.3.3 Bode Plots.- 17.3.4 Classical vs. State Space.- 17.3.5 LQR Utility Code.- 17.3.6 Least Squares Optimal Control.- 17.3.7 Control Equation Generation.- 17.3.8 LTI Realization Calculations.- 17.3.9 Initialization.- 17.4 Package Creation and Installation.- 17.5 How About Help?.- 17.5.1 Maple Libraries.- 17.5.2 Creating and Installing Help Files.- 17.6 Control Package Demo.- IV Appendices.- A Review Problems.- B Laplace Transform Table.- References.