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Index
Cover
Contents
Title Page
Copyright
Dedication
List of Figure
Preface
1 Introduction
1.1 WHAT IS A DIFFERENTIAL EQUATION?
1.2 EXAMPLES OF AN ORDINARY AND A PARTIAL DIFFERENTIAL EQUATION
1.3 NUMERICAL ANALYSIS, A NECESSITY FOR SCIENTIFIC COMPUTING
1.4 OUTLINE OF THE CONTENTS OF THIS BOOK
Bibliography
2 Ordinary Differential Equations
2.1 PROBLEM CLASSIFICATION
2.2 LINEAR SYSTEMS OF ODEs WITH CONSTANT COEFFICIENTS
2.3 SOME STABILITY CONCEPTS FOR ODEs
2.4 SOME ODE MODELS IN SCIENCE AND ENGINEERING
2.5 SOME EXAMPLES FROM APPLICATIONS
Bibliography
3 Numerical Methods for Initial Value Problems
3.1 GRAPHICAL REPRESENTATION OF SOLUTIONS
3.2 BASIC PRINCIPLES OF NUMERICAL APPROXIMATION OF ODEs
3.3 NUMERICAL SOLUTION OF IVPs WITH EULER’S METHOD
3.4 HIGHER-ORDER METHODS FOR THE IVP
3.5 THE VARIATIONAL EQUATION AND PARAMETER FITTING IN IVPs
Bibliography
4 Numerical Methods for Boundary Value Problems
4.1 APPLICATIONS
4.2 DIFFERENCE METHODS FOR BVPs
4.3 ANSATZ METHODS FOR BVPS
Bibliography
5 Partial Differential Equations
5.1 CLASSICAL PDE PROBLEMS
5.2 DIFFERENTIAL OPERATORS USED FOR PDEs
5.3 SOME PDEs IN SCIENCE AND ENGINEERING
5.4 INITIAL AND BOUNDARY CONDITIONS FOR PDES
5.5 NUMERICAL SOLUTION OF PDES, SOME GENERAL COMMENTS
Bibliography
6 Numerical Methods for Parabolic Partial Differential Equations
6.1 APPLICATIONS
6.2 AN INTRODUCTORY EXAMPLE OF DISCRETIZATION
6.3 THE METHOD OF LINES FOR PARABOLIC PDEs
6.4 GENERALIZATIONS OF THE HEAT EQUATION
6.5 ANSATZ METHODS FOR THE MODEL EQUATION
Bibliography
7 Numerical Methods for Elliptic Partial Differential Equations
7.1 APPLICATIONS
7.2 THE FINITE DIFFERENCE METHOD
7.3 DISCRETIZATION OF A PROBLEM WITH DIFFERENT BCs
7.4 THE FINITE ELEMENT METHOD
Bibliography
8 Numerical Methods for Hyperbolic Partial Differential Equations
8.1 APPLICATIONS
8.2 NUMERICAL SOLUTION OF HYPERBOLIC PDEs
8.3 INTRODUCTION TO NUMERICAL STABILITY FOR HYPERBOLIC PDEs
Bibliography
9 Mathematical Modeling with Differential Equations
9.1 NATURE LAWS
9.2 CONSTITUTIVE EQUATIONS
9.3 CONSERVATIVE EQUATIONS
9.4 SCALING OF DIFFERENTIAL EQUATIONS TO DIMENSIONLESS FORM
9.4 SCALING OF DIFFERENTIAL EQUATIONS TO DIMENSIONLESS FORM
A. APPENDIX: SOME NUMERICAL AND MATHEMATICAL TOOLS
A.1 NEWTON’S METHOD FOR SYSTEMS OF NONLINEAR ALGEBRAIC EQUATIONS
A.2 SOME FACTS ABOUT LINEAR DIFFERENCE EQUATIONS
A.3 DERIVATION OF DIFFERENCE APPROXIMATIONS
A.4 THE INTERPRETATIONS OF DIV AND CURL
A.5 NUMERICAL SOLUTION OF ALGEBRAIC SYSTEMS OF EQUATIONS
A.6 SOME RESULTS FOR FOURIER TRANSFORMS
B. APPENDIX: SOFTWARE FOR SCIENTIFIC COMPUTING
B.1 MATLAB®
B.2 COMSOL MULTIPHYSICS ®
C. APPENDIX: COMPUTER EXERCISES TO SUPPORT THE CHAPTERS
COMPUTER LAB 1 SUPPORTING CHAPTER 2
COMPUTER LAB 2 SUPPORTING CHAPTER 3
COMPUTER LAB 3 SUPPORTING CHAPTER 4
COMPUTER LAB 4 SUPPORTING CHAPTER 6
COMPUTER LAB 5 SUPPORTING CHAPTER 7
COMPUTER LAB 6 SUPPORTING CHAPTER 8
Index
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