Ordinary differential equations using Matlab

Cover
Title page
Preface
1. Introduction to MATLAB
	Numerical Expressions
	Mathematical Functions
	Output Format
	Comp1ex Arithmetic
	Recording Your Work
	Exercises
2. Plotting in MATLAB
	EZ plotting
	Matrices and Vectors in MATLAB
	Addition, Subtraction, and Multiplication by Scalars
	Array Operations and Array Smart Functions
	Plotting in MALAAS Using the plot Command
	Editing Graphics
	Saving, Printing, and Exporting Your Plot
	Script M-files
	Text strings in MATLAB
	A Little Programming
	Other Issues with M-files
	Handle graphics
	Exercises
3. Introduction to DFIELD6
	Starting DFIELD6
	The Direction Field
	Initial Value Problems
	Existence and Uniqueness
	Qualilative Analysis
	Using MATLAB While DFIELD6 is Open
	Subscripts and Greek Letters
	Editing Ihe Display Window
	Other Features of DFIELD6
	Exercises
	Appendix: Downloading and Installing the Software Used in This Manual
4 The Use of Function M-Files
	New functions in MATLAB
	Function M-files, Computational Exercices and Graphics
	Functions of Several Variables
	Naming and Organizing Files
	Exercises
5. Numerical Methods for ODEs
	Euler's Method
	Euler's Method Versus the Exact So1ution
	Changing the Step Size - Using Script M-Files
	Numerical Error Analysis
	The Second Order Runge-Kutta Method
	The Fourth Order Runge-Kutta Method
	Comparing Euler, RK2, and RK4
	Exercises
	Appendix: M-files for Numerical Methods
6 Advanced Use of DFIELD6
	Step Functions
	Step Functions in DFIELD6
	Using Funetion M-files in DFIELD6
	The Square Wave Function
	Solvers and Solver Settings in DFIELD6
	Exercices
7. Introduction to PPLANE6
	Starting PPLANE6
	Changing the System - Using the PPLANE6 Setup Window
	Plotting Solution Curves
	Other Propertiesof PPLANE6
	Exercices
8. Solving ODEs in MATLAB
	MATLAB's ODE Suite
	Single First Ordcr Differential Equations
	Function M-File Drivers and Subfunctions
	Systems of First Order Equations
	Second Order Differential Equations
	The Lorenz System and Passing Parameters
	Improving Accuracy
	Behavior Near Discontinuities
	Stiff Equations
	Other Possibilities
	Exercices
	Student Projects
9. Introduction to ODESOLVE
	Staning ODESOlVE
	Oplions in ODF.50lVE
	Exercises
10. The Symho!ic Toolbox
	Algebraic Expressions and Symbolic Objects
	Verifying Solutions to Differential Equations
	Solving Ordinary Differential Equations
	Solving Systems of Ordinary Differential Equations
	Interpreting Ihe Output of DSOLVE
	The Solve Command
	Exercices
11. Linear Algebra Using MATLAB
	Systems of Linear Equations
	Linear Combinations of Vectors and the Matrix-Vector Product
	Matrix Multiplication
	Solving Systems of Equations and Row Operations
	MatrixIndexing in MATLAB and Row Operations
	The Rational Format
	Elimination and Backsolving
	Determined Systems of Equations
	The Determinant and Systems
	The Nullspaçe of a Matrix
	Linear Dependence and Independence
	The Nullspace and Dependence
	Exercices
12. Homogeneous Linear Systems of ODEs
	Eigenvalues Using MATLAB
	Eigenvectors Using MATLAB
	MATLAB's eig Command
	Tying It All Together - Solving Systems of ODEs
	Complex Eigenvalues
	The Exponential or a Matrix
	Solving Systems with Repeated Eigenvalues
	Exercices
13. Advanced Use of PPLANE6
	Nullclines
	Equilibrium Points
	Linear Systemw
	Nonlinear Systems
	Separatrices
	Limit Sets or Solution Curves
	Other Options in PPLANE6
	Exercises
	Appendix: A Summary of Equilibrium Point Messages