Introduction to Computation and Modeling for Differential Equations

Uses mathematical, numerical, and programming tools to solve differential equations for physical phenomena and engineering problems

Introduction to Computation and Modeling for Differential Equations, Second Edition features the essential principles and applications of problem solving across disciplines such as engineering, physics, and chemistry. The Second Edition integrates the science of solving differential equations with mathematical, numerical, and programming tools, specifically with methods involving ordinary differential equations; numerical methods for initial value problems (IVPs); numerical methods for boundary value problems (BVPs); partial differential equations (PDEs); numerical methods for parabolic, elliptic, and hyperbolic PDEs; mathematical modeling with differential equations; numerical solutions; and finite difference and finite element methods.

The author features a unique "Five-M" approach: Modeling, Mathematics, Methods, MATLAB, and Multiphysics, which facilitates a thorough understanding of how models are created and preprocessed mathematically with scaling, classification, and approximation and also demonstrates how a problem is solved numerically using the appropriate mathematical methods. With numerous real-world examples to aid in the visualization of the solutions, Introduction to Computation and Modeling for Differential Equations, Second Edition includes:

• New sections on topics including variational formulation, the finite element method, examples of discretization, ansatz methods such as Galerkin's method for BVPs, parabolic and elliptic PDEs, and finite volume methods

• Numerous practical examples with applications in mechanics, fluid dynamics, solid mechanics, chemical engineering, heat conduction, electromagnetic field theory, and control theory, some of which are solved with computer programs MATLAB and COMSOL Multiphysics

• Additional exercises that introduce new methods, projects, and problems to further illustrate possible applications

• A related website with select solutions to the exercises, as well as the MATLAB data sets for ordinary differential equations (ODEs) and PDEs

Introduction to Computation and Modeling for Differential Equations, Second Edition is a useful textbook for upper-undergraduate and graduate-level courses in scientific computing, differential equations, ordinary differential equations, partial differential equations, and numerical methods. The book is also an excellent self-study guide for mathematics, science, computer science, physics, and engineering students, as well as an excellent reference for practitioners and consultants who use differential equations and numerical methods in everyday situations.

Auteur

LENNART EDSBERG, PhD, is Associate Professor in the Numerical Analysis section within the Department of Mathematics at KTH-The Royal Institute of Technology in Stockholm, Sweden, where he has also been Director of the International Master Program in Scientific Computing since 1998-2008. Dr. Edsberg has over 30 years of academic experience and is the author of over 20 journal articles in the areas of numerical methods and differential equations.

Contenu
Preface xi

1 Introduction 1

1.1 What is a Differential Equation? 1

1.2 Examples of an Ordinary and a Partial Differential Equation, 2

1.3 Numerical Analysis, a Necessity for Scientific Computing, 5

1.4 Outline of the Contents of this Book, 8

Bibliography, 10

2 Ordinary Differential Equations 11

2.1 Problem Classification, 11

2.2 Linear Systems of ODEs with Constant Coefficients, 16

2.3 Some Stability Concepts for ODEs, 19

2.3.1 Stability for a Solution Trajectory of an ODE System, 20

2.3.2 Stability for Critical Points of ODE Systems, 23

2.4 Some ODE models in Science and Engineering, 26

2.4.1 Newton's Second Law, 26

2.4.2 Hamilton's Equations, 27

2.4.3 Electrical Networks, 27

2.4.4 Chemical Kinetics, 28

2.4.5 Control Theory, 29

2.4.6 Compartment Models, 29

2.5 Some Examples from Applications, 30

Bibliography, 36

3 Numerical Methods for Initial Value Problems 37

3.1 Graphical Representation of Solutions, 38

3.2 Basic Principles of Numerical Approximation of ODEs, 40

3.3 Numerical Solution of IVPs with Euler's method, 41

3.3.1 Euler's Explicit Method: Accuracy, 43

3.3.2 Euler's Explicit Method: Improving the Accuracy, 46

3.3.3 Euler's Explicit Method: Stability, 48

3.3.4 Euler's Implicit Method, 53

3.3.5 The Trapezoidal Method, 55

3.4 Higher Order Methods for the IVP, 56

3.4.1 RungeKutta Methods, 56

3.4.2 Linear Multistep Methods, 60

3.5 Special Methods for Special Problems, 62

3.5.1 Preserving Linear and Quadratic Invariants, 62

3.5.2 Preserving Positivity of the Numerical Solution, 64

3.5.3 Methods for Newton's Equations of Motion, 64

3.6 The Variational Equation and Parameter Fitting in IVPs, 66

Bibliography, 69

4 Numerical Methods for Boundary Value Problems 71

4.1 Applications, 73

4.2 Difference Methods for BVPs, 78

4.2.1 A Model Problem for BVPs, Dirichlet's BCs, 79

4.2.2 A Model Problem for BVPs, Mixed BCs, 83

4.2.3 Accuracy, 86

4.2.4 Spurious Solutions, 87

4.2.5 Linear Two-Point BVPs, 89

4.2.6 Nonlinear Two-Point BVPs, 91

4.2.7 The Shooting Method, 92

4.3 Ansatz Methods for BVPs, 94

4.3.1 Starting with the ODE Formulation, 95

4.3.2 Starting with the Weak Formulation, 96

4.3.3 The Finite Element Method, 100

Bibliography, 103

5 Partial Differential Equations 105

5.1 Classical PDE Problems, 106

5.2 Differential Operators Used for PDEs, 110

5.3 Some PDEs in Science and Engineering, 114

5.3.1 NavierStokes Equations for Incompressible Flow, 114

5.3.2 Euler's Equations for Compressible Flow, 115

5.3.3 The ConvectionDiffusionReaction Equations, 116

5.3.4 The Heat Equation, 117

5.3.5 The Diffusion Equation, 117

5.3.6 Maxwell's Equations for the Electromagnetic Field, 117

5.3.7 Acoustic Waves, 118

5.3.8 Schrödinger's Equation in Quantum Mechanics, 119

5.3.9 Navier's Equations in Structural Mechanics, 119

5.3.10 BlackScholes Equation in Financial Mathematics, 120

5.4 Initial and Boundary Conditions for PDEs, 121

5.5 Numerical Solution of PDEs, Some General Comments, 121

Bibliography, 122

6 Numerical Methods for Parabolic Partial Differential Equations 123

6.1 Applications, 125

6.2 An Introductory Example of Discretization, 127

6.3 The Method of Lines for Parabolic PDEs, 130

6.3.1 Solving the Test Problem with MoL, 130

6.3.2 Various Types of Boundary Conditions, 134

6.3.3 An Example of the Use of MoL for a Mixed Boundary Condition, 135

6.4 Generalizations of the Heat Equation, 136 6.4.1 The He...