NUMERICAL ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONSPDF电子书下载

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Chapter 1．Contemporary state of Numerical Analysis 3

1 WHAT IS NUMERICAL ANALYSIS? 3

2 AREAS OF NUMERICAL ANALYSIS 7

3 AN OLDER SOVIET VIEW OF NUMERICAL ANALYSIS 8

4 A MORE PECENT SOVIET VIEW OF NUMERICAL ANALYSIS 10

5 AUTOMATIC DIGITAL COMPUTERS IN THE WORLD 11

6 TWO SOVIET COMPUTERS 14

7 LITERATURE ON NUMERICAL ANALYSIS 15

8 NUMERICAL INTEGRATION AND INTERPOLATION 16

9 APPROXIMATIONS OF FUNCTIONS 18

10 SOLVING LINEAR ALGEBRAIC EQUATIONS 21

11 SOLVING MATRIX EIGENVALUE PROBLEMS 27

12 DIFFERENCE METHODS FOR LAPLACE＇S EQUATION 31

Bibliography 38

Introduction 45

Chapter 1 Partial Differential Equations in the Complex Domain 46

The Cauchy Problem 46

1 TRANSFORMATION TO NORMAL FORM 46

2 BASIC EXISTENCE THEOREMS 48

The Leray-Fantappié Operational Calculus 50

1 PRELIMINARIES 50

Equations oF Higher Order 55

Chapter 2 General Theory in the Real Domain 59

Uniqueness and Domains of Dependence for the Cauchy Problem 59

Preliminaries on Function Spaces 62

1 TESTING SPACES 62

2 DUAL SPACES AND DISTRIBUTIONS 64

3 BANACH AND HILBERT SPACES 66

4 LINEAR OPERATORS 67

5 APPLICATIONS TO DIFFERENTIAL EQUATIONS 70

Fourier and Laplace Transforms．Other Operational Calculi 75

1 FOURIER TRANSFORMS 75

2 LAPLACE TRANSFORMS 77

3 SYSTEMS OF PARTIAL-DIFFERENTIAL EQUATIONS AND POTENTIALS 79

4 GENERAL REMARKS ON OPERATIONAL CALCULI 82

5 THEORY OF SEMIGROUPS 85

Chapter 3 General Theory of Equations with Constant Coefficients 88

The Work of Ehrenpreis,Malgrange,and Hormander 88

1 GENERAL EXISTENCE THEOREMS 88

2 GENERAL REGULARITY THEOREMS 92

The Method of Gelfand and Silov 97

Chapter 4 Equations of Parabolic Type 102

Equations with Constant Coefficients 102

1 THE CAUCHY PRQBLEM FOR HOMOGENEOUS EQUATIONS 102

2 NONHOMOGENEOUS EQUATIONS 105

Parabolic Equations with Variable Coefficents 108

1 PERTURBATION AMD PARAMETRIX METHODS 108

2 APPLICATIONS OF SEMIGROUP THEORY 113

3 BOUNDARY VALUE PROBLEMS 115

4 THE METHOD OF VISIK 117

5 VARIATIONAL PRINCIPLES 119

6 REGULARITY PROPERTTES 121

7 BARRAR＇S a priori ESTIMATES 124

Chapter 5 Equations of Elliptic Type 126

General Theory 126

1 CLASSIFICATION 126

2 JOHN＇S FUNDAMENTAL SOLUTION 127

3 STRONGLY ELLIPTIC SYSTEMS 132

4 REGULARITY SYSTEMS 134

5 GARDING＇S INEQUALITY 136

6 THE DOUGLIS-NIRENBERG ESTIMATES 138

7 THE ASSUMPTION OF BOUNDARY VALUES 139

8 GREEN＇S AND NEUMANN FUNCTIONS．THE BERGMAN RERNEL 142

9 CONSTRUCTION OF SOLUTIONS 145

Elliptic Equations of Second Order 150

1 MAXIMUM PRINCIPLE 150

2 LAPLACE EQUATION IN A SPHERE 152

3 LOCAL EXISTENCE THEOREMS 154

4 PERRON＇S METHOD 159

5 CONCLUDING REMARKS 161

Bibliography 164

Index 197

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