Mathematics for Modeling and Scientific Computing [Hardcover]

This book provides the mathematical basis for investigating numerically equations from physics, life sciences or engineering. Tools for analysis and algorithms are confronted to a large set of relevant examples that show the difficulties and the limitations of the most naïve approaches. These examples not only provide the opportunity to put into practice mathematical statements, but modeling issues are also addressed in detail, through the mathematical perspective.

Preface ix

Chapter 1. Ordinary Differential Equations 1

1.1. Introduction to the theory of ordinary differential equations  1

1.1.1. Existence–uniqueness of first-order ordinary differential equations 1

1.1.2. The concept of maximal solution  11

1.1.3. Linear systems with constant coefficients  16

1.1.4. Higher-order differential equations 20

1.1.5. Inverse function theorem and implicit function theorem  21

1.2. Numerical simulation of ordinary differential equations, Euler schemes, notions of convergence, consistence and stability  27

1.2.1. Introduction  27

1.2.2. Fundamental notions for the analysis of numerical ODE methods 29

1.2.3. Analysis of explicit and implicit Euler schemes  33

1.2.4. Higher-order schemes 50

1.2.5. Leslie’s equation (Perron–Frobenius theorem, power method)  51

1.2.6. Modeling red blood cell agglomeration 78

1.2.7. SEI model 87

1.2.8. A chemotaxis problem  93

1.3. Hamiltonian problems 102

1.3.1. The pendulum problem  106

1.3.2. Symplectic matrices; sympllsè