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CLASSIFICATION OF SECOND-ORDER EQUATIONS 8.5 EXAMPLES OF TWO-FACTOR MODELS FROM COMPUTATIONAL FINANCE 8.6 SUMMARY AND CONCLUSIONS CHAPTER 9: Transforming Partial Differential Equations to a Bounded Domain 9.1 INTRODUCTION AND OBJECTIVES 9.2 THE DOMAIN IN WHICH A PDE IS DEFINED: PREAMBLE 9.3 OTHER EXAMPLES 9.4 HOTSPOTS 9.5 WHAT HAPPENED TO DOMAIN TRUNCATION? 9.6 ANOTHER WAY TO REMOVE MIXED DERIVATIVE TERMS 9.7 SUMMARY AND CONCLUSIONS CHAPTER 10: Boundary Value Problems for Elliptic and Parabolic Partial Differential Equations 10.1 INTRODUCTION AND OBJECTIVES 10.2 NOTATION AND PREREQUISITES 10.3 THE LAPLACE EQUATION 10.4 PROPERTIES OF THE LAPLACE EQUATION 10.5 SOME ELLIPTIC BOUNDARY VALUE PROBLEMS 10.6 EXTENDED MAXIMUM-MINIMUM PRINCIPLES 10.7 SUMMARY AND CONCLUSIONS CHAPTER 11: Fichera Theory, Energy Inequalities and Integral Relations 11.1 INTRODUCTION AND OBJECTIVES 11.2 BACKGROUND AND PROBLEM STATEMENT 11.3 WELL-POSED PROBLEMS AND ENERGY ESTIMATES 11.4 THE FICHERA THEORY: OVERVIEW 11.5 THE FICHERA THEORY: THE CORE BUSINESS 11.6 THE FICHERA THEORY: FURTHER EXAMPLES AND APPLICATIONS 11.7 SOME USEFUL THEOREMS 11.8 SUMMARY AND CONCLUSIONS CHAPTER 12: An Introduction to Time-Dependent Partial Differential Equations 12.1 INTRODUCTION AND OBJECTIVES 12.2 NOTATION AND PREREQUISITES 12.3 PREAMBLE: SEPARATION OF VARIABLES FOR THE HEAT EQUATION 12.4 WELL-POSED PROBLEMS 12.5 VARIATIONS ON INITIAL BOUNDARY VALUE PROBLEM FOR THE HEAT EQUATION 12.6 MAXIMUM-MINIMUM PRINCIPLES FOR PARABOLIC PDES 12.7 PARABOLIC EQUATIONS WITH TIME-DEPENDENT BOUNDARIES 12.8 UNIQUENESS THEOREMS FOR BOUNDARY VALUE PROBLEMS IN TWO DIMENSIONS 12.9 SUMMARY AND CONCLUSIONS CHAPTER 13: Stochastics Representations of PDEs and Applications 13.1 INTRODUCTION AND OBJECTIVES 13.2 BACKGROUND, REQUIREMENTS AND PROBLEM STATEMENT 13.3 AN OVERVIEW OF STOCHASTIC DIFFERENTIAL EQUATIONS (SDEs) 13.4 AN INTRODUCTION TO ONE-DIMENSIONAL RANDOM PROCESSES 13.5 AN INTRODUCTION TO THE NUMERICAL APPROXIMATION OF SDEs 13.6 PATH EVOLUTION AND MONTE CARLO OPTION PRICING 13.7 TWO-FACTOR PROBLEMS 13.8 THE ITO FORMULA 13.9 STOCHASTICS MEETS PDEs 13.10 FIRST EXIT-TIME PROBLEMS 13.11 SUMMARY AND CONCLUSIONS

      8  PART C: The Foundations of the Finite Difference Method (FDM) CHAPTER 14: Mathematical and Numerical Foundations of the Finite Difference Method, Part I 14.1 INTRODUCTION AND OBJECTIVES 14.2 NOTATION AND PREREQUISITES 14.3 WHAT IS THE FINITE DIFFERENCE METHOD, REALLY? 14.4 FOURIER ANALYSIS OF LINEAR PDES 14.5 DISCRETE FOURIER TRANSFORM 14.6 THEORETICAL CONSIDERATIONS 14.7 FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS 14.8 SUMMARY AND CONCLUSIONS CHAPTER 15: Mathematical and Numerical Foundations of the Finite Difference Method, Part II 15.1 INTRODUCTION AND OBJECTIVES 15.2 A SHORT HISTORY OF NUMERICAL METHODS FOR CDR EQUATIONS 15.3 EXPONENTIAL FITTING AND TIME-DEPENDENT CONVECTION-DIFFUSION 15.4 STABILITY AND CONVERGENCE ANALYSIS 15.5 SPECIAL LIMITING CASES 15.6 STABILITY FOR INITIAL BOUNDARY VALUE PROBLEMS 15.7 SEMI-DISCRETISATION FOR CONVECTION-DIFFUSION PROBLEMS 15.8 PADÉ MATRIX APPROXIMATION 15.9 TIME-DEPENDENT

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