MATLAB Differential Equations (eBook)

César Perez Lopez is a Professor at the Department of Statistics and Operations Research at the University of Madrid. César Perez Lopez is also a Mathematician and Economist at the National Statistics Institute (INE) in Madrid, a body which belongs to the Superior Systems and Information Technology Department of the Spanish Government. César also currently works at the Institute for Fiscal Studies in Madrid.

Contents at a Glance 3
Contents 173
About the Author 177
Chapter 1: Introducing MATLAB and the MATLAB Working Environment 4
Introduction 4
Developing Algorithms and Applications 5
Data Access and Analysis 8
Data Visualization 9
Numerical Calculation 12
Publication of Results and Distribution of Applications 13
The MATLAB working environment 14
Help in MATLAB 18
Numerical Computation with MATLAB 21
Symbolic Calculations with MATLAB 22
Graphics with MATLAB 24
General Notation 28
Help with Commands 29
MATLAB and Programming 31
Commands to Escape and Exit to the MS-DOS Environment 32
Chapter 2: First Order Differential Equations. Exact Equations, Separation of Variables, Homogeneous and Linear Equations 35
First Order Differential Equations 35
Separation of Variables 38
Homogeneous Differential Equations 40
Exact Differential Equations 43
Linear Differential Equations 45
Chapter 3: Higher Order Differential Equations. The Laplace Transform and Special Types of Equations 47
Ordinary High-Order Equations 47
Linear Higher-Order Equations. Homogeneous Equations with Constant Coefficients 49
Non-Homogeneous Equations with Constant Coefficients. Variation of Parameters 50
Non-Homogeneous Equations with Variable Coefficients. Cauchy–Euler Equations 54
The Laplace Transform 54
Orthogonal Polynomials 57
Chebychev Polynomials of the First and Second Kind 57
Legendre Polynomials 58
Associated Legendre Polynomials 58
Hermite Polynomials 58
Generalized Laguerre Polynomials 58
Laguerre Polynomials 59
Jacobi Polynomials 59
Gegenbauer Polynomials 59
Bessel and Airy Functions 60
Chapter 4: Differential Equations Via Approximation Methods 62
Higher Order Equations and Approximation Methods 62
The Taylor Series Method 62
The Runge–Kutta Method 65
Chapter 5: Systems of Differential Equations and Finite Difference Equations 67
Systems of Linear Homogeneous Equations with Constant Coefficients 67
Systems of Linear Non-Homogeneous Equations with Constant Coefficients 68
Finite Difference Equations 69
Partial Differential Equations 71
Chapter 6: Numerical Calclus with MATLAB. Applications to Differential Equations 73
MATLAB and Programming 73
Text Editor 73
Scripts 76
Functions and M-Files. Function, Eval and Feval 78
Local and Global Variables 82
Data Types 84
Flow Control: FOR Loops, WHILE and IF ELSEIF 85
The FOR Loop 85
The WHILE Loop 86
IF ELSEIF ELSE END Loops 87
Switch and Case 88
Continue 89
Break 90
Try ... Catch 91
Return 91
Subfunctions 92
Ordinary Differential Equations Using Numerical Analysis 93
Euler’s Method 94
Heun’s Method 94
The Taylor Series Method 95
Chapter 7: Ordinary and Partial Differential Equations with Initial and Boundary Values 101
Numerical Solutions of Differential Equations 101
Ordinary Differential Equations with Initial Values 101
Ordinary Differential Equations with Boundary Values 105
Partial Differential Equations 108
Exercise 7-1 115
Exercise 7-2 117
Exercise 7-3 119
Chapter 8: Symbolic Differential and Integral Calculus 124
Symbolic Computation with MATLAB. Symbolic Variables 124
Symbolic Functions. Substitution and Functional Operations 130
Mathematical Analysis Functions. Limits, Continuity, and Series 134
Derivatives, Integrals and Differential Equations 138
Linear Algebra: Simplifying and Solving Equations 143