Table of Contents
1 Cover
4 Preface
6 PART A: Mathematical Foundation for One-Factor Problems CHAPTER 1: Real Analysis Foundations for this Book 1.1 INTRODUCTION AND OBJECTIVES 1.2 CONTINUOUS FUNCTIONS 1.3 DIFFERENTIAL CALCULUS 1.4 PARTIAL DERIVATIVES 1.5 FUNCTIONS AND IMPLICIT FORMS 1.6 METRIC SPACES AND CAUCHY SEQUENCES 1.7 SUMMARY AND CONCLUSIONS CHAPTER 2: Ordinary Differential Equations (ODEs), Part 1 2.1 INTRODUCTION AND OBJECTIVES 2.2 BACKGROUND AND PROBLEM STATEMENT 2.3 DISCRETISATION OF INITIAL VALUE PROBLEMS: FUNDAMENTALS 2.4 SPECIAL SCHEMES 2.5 FOUNDATIONS OF DISCRETE TIME APPROXIMATIONS 2.6 STIFF ODEs 2.7 INTERMEZZO: EXPLICIT SOLUTIONS 2.8 SUMMARY AND CONCLUSIONS CHAPTER 3: Ordinary Differential Equations (ODEs), Part 2 3.1 INTRODUCTION AND OBJECTIVES 3.2 EXISTENCE AND UNIQUENESS RESULTS 3.3 OTHER MODEL EXAMPLES 3.4 EXISTENCE THEOREMS FOR STOCHASTIC DIFFERENTIAL EQUATIONS (SDEs) 3.5 NUMERICAL METHODS FOR ODES 3.6 THE RICCATI EQUATION 3.7 MATRIX DIFFERENTIAL EQUATIONS 3.8 SUMMARY AND CONCLUSIONS CHAPTER 4: An Introduction to Finite Dimensional Vector Spaces 4.1 SHORT INTRODUCTION AND OBJECTIVES 4.2 WHAT IS A VECTOR SPACE? 4.3 SUBSPACES 4.4 LINEAR INDEPENDENCE AND BASES 4.5 LINEAR TRANSFORMATIONS 4.6 SUMMARY AND CONCLUSIONS CHAPTER 5: Guide to Matrix Theory and Numerical Linear Algebra 5.1 INTRODUCTION AND OBJECTIVES 5.2 FROM VECTOR SPACES TO MATRICES 5.3 INNER PRODUCT SPACES 5.4 FROM VECTOR SPACES TO MATRICES 5.5 FUNDAMENTAL MATRIX PROPERTIES 5.6 ESSENTIAL MATRIX TYPES 5.7 THE CAYLEY TRANSFORM 5.8 SUMMARY AND CONCLUSIONS CHAPTER 6: Numerical Solutions of Boundary Value Problems 6.1 INTRODUCTION AND OBJECTIVES 6.2 AN INTRODUCTION TO NUMERICAL LINEAR ALGEBRA 6.3 DIRECT METHODS FOR LINEAR SYSTEMS 6.4 SOLVING TRIDIAGONAL SYSTEMS 6.5 TWO-POINT BOUNDARY VALUE PROBLEMS 6.6 ITERATIVE MATRIX SOLVERS 6.7 EXAMPLE: ITERATIVE SOLVERS FOR ELLIPTIC PDEs 6.8 SUMMARY AND CONCLUSIONS CHAPTER 7: Black–Scholes Finite Differences for the Impatient 7.1 INTRODUCTION AND OBJECTIVES 7.2 THE BLACK–SCHOLES EQUATION: FULLY IMPLICIT AND CRANK–NICOLSON METHODS 7.3 THE BLACK–SCHOLES EQUATION: TRINOMIAL METHOD 7.4 THE HEAT EQUATION AND ALTERNATING DIRECTION EXPLICIT (ADE) METHOD 7.5 ADE FOR BLACK–SCHOLES: SOME TEST RESULTS 7.6 SUMMARY AND CONCLUSIONS
7 PART B: Mathematical Foundation for Two-Factor Problems CHAPTER 8: Classifying and Transforming Partial Differential Equations 8.1 INTRODUCTION AND OBJECTIVES 8.2 BACKGROUND AND PROBLEM STATEMENT 8.3 INTRODUCTION TO ELLIPTIC EQUATIONS