Ordinary Differential Equations: Example-driven, Including Maple Code
234
Ordinary Differential Equations: Example-driven, Including Maple Code
234eBook
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Overview
This introductory text combines models from physics and biology with rigorous reasoning in describing the theory of ordinary differential equations along with applications and computer simulations with Maple. Offering a concise course in the theory of ordinary differential equations, it also enables the reader to enter the field of computer simulations. Thus, it is a valuable read for students in mathematics as well as in physics and engineering. It is also addressed to all those interested in mathematical modeling with ordinary differential equations and systems.
Contents
Part I: Theory
Chapter 1 First-Order Differential Equations
Chapter 2 Linear Differential Systems
Chapter 3 Second-Order Differential Equations
Chapter 4 Nonlinear Differential Equations
Chapter 5 Stability of Solutions
Chapter 6 Differential Systems with Control Parameters
Part II: Exercises
Seminar 1 Classes of First-Order Differential Equations
Seminar 2 Mathematical Modeling with Differential Equations
Seminar 3 Linear Differential Systems
Seminar 4 Second-Order Differential Equations
Seminar 5 Gronwall’s Inequality
Seminar 6 Method of Successive Approximations
Seminar 7 Stability of Solutions
Part III: Maple Code
Lab 1 Introduction to Maple
Lab 2 Differential Equations with Maple
Lab 3 Linear Differential Systems
Lab 4 Second-Order Differential Equations
Lab 5 Nonlinear Differential Systems
Lab 6 Numerical Computation of Solutions
Lab 7 Writing Custom Maple Programs
Lab 8 Differential Systems with Control Parameters
Product Details
| ISBN-13: | 9783110447507 |
|---|---|
| Publisher: | De Gruyter |
| Publication date: | 01/22/2018 |
| Series: | De Gruyter Textbook |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 234 |
| File size: | 14 MB |
| Note: | This product may take a few minutes to download. |
| Age Range: | 18 Years |
About the Author
Radu Precup, Babeș-Bolyai University, Romania.
Table of Contents
Preface vii
Part I Theory
Chapter 1 First-Order Differential Equations 3
1.1 Preliminaries 3
1.2 Classes of First-Order Differential Equations 6
1.2.1 Differential Equations with Separable Variables 6
1.2.2 Differential Equations of Homogeneous Type 7
1.2.3 First-Order Linear Differential Equations 8
1.2.4 Bernoulli Equations 13
1.2.5 Riccati Equations 14
1.3 Mathematical Modeling with First-Order Differential Equations 15
1.3.1 Radioactive Decay 15
1.3.2 Newton's Law of Heat Transfer 16
1.3.3 Chemical Reactions 17
1.3.4 Population Growth of a Single Species 18
1.3.5 The Gompertz Equation 20
Chapter 2 Linear Differential Systems 22
2.1 Preliminaries 22
2.2 Mathematical Modeling with Linear Differential Systems 23
2.3 Matrix Notation for Systems 26
2.4 Superpostition Principle for Linear Systems 27
2.5 Linear Differential Systems with Constant Coefficients 28
2.5.1 The General Solution 28
2.5.2 Structure of Solution Set for Homogeneous Linear Systems 31
2.5.3 The Concept of Fundamental Matrix 32
2.5.4 Method of Eigenvalues and Eigenvectors 35
2.6 Method of Variation of Parameters 40
2.7 Higher-Dimensional Linear Systems 42
2.8 Use of the Jordan Canonical Form of a Matrix 44
2.9 Dynamic Aspects of Differential Systems 46
2.10 Preliminaries of Stability 52
Chapter 3 Second-Order Differential Equations 53
3.1 Newton's Second Law of Motion 53
3.2 Reduction of Order 54
3.3 Equivalence to a First-Order System 57
3.4 The Method of Elimination 58
3.5 Linear Second-Order Differential Equations 59
3.5.1 The Solution Set 59
3.5.2 Homogeneous Linear Equations with Constant Coefficients 59
3.5.3 Variation of Parameters Method 61
3.5.4 The Method of Undetermined Coefficients 62
3.5.5 Euler Equations 64
3.6 Boundary Value Problems 66
3.7 Higher-Order Linear Differential Equations 69
Chapter 4 Nonlinear Differential Equations 72
4.1 Mathematical Models Expressed by Nonlinear Systems 72
4.1.1 The Lotka-Volterra Model 72
4.1.2 The SIR Epidemic Model 73
4.1.3 An Immunological Model 74
4.1.4 A Model in Hematology 75
4.2 Gronwall's Inequality 75
4.3 Uniqueness of Solutions for the Cauchy Problem 77
4.4 Continuous Dependence of Solutions on the Initial Values 81
4.5 The Cauchy Problem for Systems 83
4.6 The Cauchy Problem for Higher-Order Equations 84
4.7 Periodic Solutions 85
4.8 Picard's Method of Successive Approximations 88
4.8.1 Picard's Iteration 88
4.8.2 The Interval of Picard's Iteration 91
4.8.3 Convergence of Picard's Iteration 92
4.9 Existence of Solutions for the Cauchy Problem 93
Chapter 5 Stability of Solutions 97
5.1 The Notion of a Stable Solution 97
5.2 Stability of Linear Systems 99
5.3 Stability of Linear Systems with Constant Coefficients 100
5.4 Stability of Solutions of Nonlinear Systems 101
5.5 Method of Lyapunov Functions 106
5.6 Globally Asymptotically Stable Systems 111
Chapter 6 Differential Systems with Control Parameters 113
6.1 Bifurcations 113
6.2 Hopf Bifurcations 115
6.3 Optimization of Differential Systems 119
6.4 Dynamic Optimization of Differential Systems 122
Part II Exercises
Seminar 1 Classes of First-Order Differential Equations 127
1.1 Solved Exercises 127
1.2 Proposed Exercises 129
1.3 Solutions 130
1.4 Project: Problems of Geometry that Lead to Differential Equations 131
Seminar 2 Mathematical Modeling with Differential Equations 133
2.1 Solved Exercises 133
2.2 Proposed Exercises 135
2.3 Hints and Answers 136
2.4 Project: Influence of External Actions over the Evolution of Some Processes 136
Seminar 3 Linear Differential Systems 139
3.1 Solved Exercises 139
3.2 Proposed Exercises 143
3.3 Hints and Solutions 145
3.4 Project: Mathematical Models Represented by Linear Differential Systems 146
Seminar 4 Second-Order Differential Equations 148
4.1 Solved Exercises 148
4.2 Proposed Exercises 150
4.3 Solutions 151
4.4 Project: Boundary Value Problems for Second-Order Differential Equations 152
Seminar 5 Gronwall's Inequality 154
5.1 Solved Exercises 154
5.2 Proposed Exercises 157
5.3 Hints and Solutions 157
5.4 Project: Integral and Differential Inequalities 158
Seminar 6 Method of Successive Approximations 163
6.1 Solved Exercises 163
6.2 Proposed Exercises 164
6.3 Hints and Solutions 165
6.4 Project: The Vectorial Method for the Treatment of Nonlinear Differential Systems 165
Seminar 7 Stability of Solutions 170
7.1 Solved Exercises 170
7.2 Proposed Exercises 172
7.3 Hints and Solutions 173
7.4 Project: Stable and Unstable Invariant Manifolds 173
Part III Maple Code
Lab 1 Introduction to Maple 179
1.1 Numerical Calculus 179
1.2 Symbolic Calculus 181
Lab 2 Differential Equations with Maple 185
2.1 The DEtools Package 185
2.2 Working Themes 186
Lab 3 Linear Differential Systems 188
3.1 The linalg Package 188
3.2 Linear Differential Systems 188
3.3 Working Themes 190
Lab 4 Second-Order Differential Equations 191
4.1 Spring-Mass Oscillator Equation with Maple 191
4.2 Boundary Value Problems with Maple 193
4.3 Working Themes 193
Lab 5 Nonlinear Differential Systems 195
5.1 The Lotka-Volterra System 195
5.2 A Model from Hematology 196
5.3 Working Themes 197
Lab 6 Numerical Computation of Solutions 198
6.1 Initial Value Problems 198
6.2 Boundary Value Problems 199
6.3 Working Themes 202
Lab 7 Writing Custom Maple Programs 204
7.1 Method of Successive Approximations 204
7.2 Euler's Method 206
7.3 The Shooting Method 208
7.4 Working Themes 211
Lab 8 Differential Systems with Control Parameters 212
8.1 Bifurcations 212
8.2 Optimization with Maple 213
8.3 Working Themes 215
Bibliography 217
Index 219