《NUMERICAL ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS》求取 ⇩

Chapter 1.Contemporary state of Numerical Analysis3

1WHAT IS NUMERICAL ANALYSIS?3

2 AREAS OF NUMERICAL ANALYSIS7

3 AN OLDER SOVIET VIEW OF NUMERICAL ANALYSIS8

4 A MORE PECENT SOVIET VIEW OF NUMERICAL ANALYSIS10

5 AUTOMATIC DIGITAL COMPUTERS IN THE WORLD11

6 TWO SOVIET COMPUTERS14

7 LITERATURE ON NUMERICAL ANALYSIS15

8 NUMERICAL INTEGRATION AND INTERPOLATION16

9 APPROXIMATIONS OF FUNCTIONS18

10 SOLVING LINEAR ALGEBRAIC EQUATIONS21

11 SOLVING MATRIX EIGENVALUE PROBLEMS27

12 DIFFERENCE METHODS FOR LAPLACE'S EQUATION31

Bibliography38

Introduction45

Chapter 1Partial Differential Equations in the Complex Domain46

The Cauchy Problem46

1TRANSFORMATION TO NORMAL FORM46

2 BASIC EXISTENCE THEOREMS48

The Leray-Fantappié Operational Calculus50

1PRELIMINARIES50

Equations oF Higher Order55

Chapter 2General Theory in the Real Domain59

Uniqueness and Domains of Dependence for the Cauchy Problem59

Preliminaries on Function Spaces62

1TESTING SPACES62

2 DUAL SPACES AND DISTRIBUTIONS64

3 BANACH AND HILBERT SPACES66

4 LINEAR OPERATORS67

5 APPLICATIONS TO DIFFERENTIAL EQUATIONS70

Fourier and Laplace Transforms.Other Operational Calculi75

1FOURIER TRANSFORMS75

2 LAPLACE TRANSFORMS77

3 SYSTEMS OF PARTIAL-DIFFERENTIAL EQUATIONS AND POTENTIALS79

4 GENERAL REMARKS ON OPERATIONAL CALCULI82

5 THEORY OF SEMIGROUPS85

Chapter 3General Theory of Equations with Constant Coefficients88

The Work of Ehrenpreis,Malgrange,and Hormander88

1GENERAL EXISTENCE THEOREMS88

2 GENERAL REGULARITY THEOREMS92

The Method of Gelfand and Silov97

Chapter 4Equations of Parabolic Type102

Equations with Constant Coefficients102

1THE CAUCHY PRQBLEM FOR HOMOGENEOUS EQUATIONS102

2 NONHOMOGENEOUS EQUATIONS105

Parabolic Equations with Variable Coefficents108

1PERTURBATION AMD PARAMETRIX METHODS108

2 APPLICATIONS OF SEMIGROUP THEORY113

3 BOUNDARY VALUE PROBLEMS115

4 THE METHOD OF VISIK117

5 VARIATIONAL PRINCIPLES119

6 REGULARITY PROPERTTES121

7 BARRAR'S a priori ESTIMATES124

Chapter 5Equations of Elliptic Type126

General Theory126

1CLASSIFICATION126

2 JOHN'S FUNDAMENTAL SOLUTION127

3 STRONGLY ELLIPTIC SYSTEMS132

4 REGULARITY SYSTEMS134

5 GARDING'S INEQUALITY136

6 THE DOUGLIS-NIRENBERG ESTIMATES138

7 THE ASSUMPTION OF BOUNDARY VALUES139

8 GREEN'S AND NEUMANN FUNCTIONS.THE BERGMAN RERNEL142

9 CONSTRUCTION OF SOLUTIONS145

Elliptic Equations of Second Order150

1MAXIMUM PRINCIPLE150

2 LAPLACE EQUATION IN A SPHERE152

3 LOCAL EXISTENCE THEOREMS154

4 PERRON'S METHOD159

5 CONCLUDING REMARKS161

Bibliography164

Index197

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