Part I : Applications to Ordinary Differential Equations ,

1.INTRODUCTION, 2

Operators; linearity; superposition

2. THE ADJOINT OPERATOR, 6

Formal adjoint; adjoint; formal self-adjointness; self-adjointness; inner product

3. THE DELTA FUNCTION, 11

Introduction to generalized functions; delts function; Heaviside function

4. THE GREEN’S FUNCTION METHOD, 21

Development of Green’s function method; symmetry property;Fourier transform; generalized Green’s function; integral equations

Example 1. Loaded String, 22

Example 2. A more Complicated Operator, 27

Example 3. Infinite Beam on Elastic Foundation, 30

Example 4. A Bessel Equation, 33

Example 5. The Generalized Green’s Function, 36

5. THE EIGENFUNCTION METHOD, 42

Eigenvalue problem; Sturm-Liouville systems; orthogonality;completeness; Fourier series; expansion of Green’s function

Application of Eigenfunction Method, 46

6. SUMMARY, 50

Summary of the Green’s function procedure for ordinary differential equations

PART II Application to Partial Differential Equations

1. INTRODUCTION, 52

General second order linear equation with two independent variables; classification; examples

2. THE ADJOINT OPERATOR, 56

Formal adjoint; adjoint; formal self- adjointness; self-adjointness; inner product

3. THE DELTA FUNCTION, 60

Two-dimensional delta function

4. THE GREEN’S FUNCTION METHOD, 61

Outline of method; principal solutions; “splitting” technique

5. PRINCIPAL SOLUTIONS, 63

Calculation of principal solutions; Fourier transform

Laplace Operator, 63

Helmholtz Operator, 65

Diffusion Operator, 66

Wave Operator, 67

6. GREEN’S FUNCTION METHOD FOR THE LAPLACE OPERATOR, 71

Images; conformal mapping; Poisson integral formula; symmetry; Dirichlet, Neumann, and mixed boundary conditions

Example 1. Circular Disk, 72

Example 2. Half-Plane, 81

Example 3. Mixed Boundary Conditions, 84

Example 4. Quarter- Plane, 86

7. GREEN’S FUNCTION METHOD FOR THE HELMHOLTZ OPERATOR, 93

Separation of variables; radiation condition; images

Example 1. Vibrating Circular Membrane, 93

Example 2. Acoustic Radiation, 94

8. GREEN’S FUNCTION FOR THE DIFFUSION OPERATOR, 99

Images; iteration

Example 1. Semi-infinite Rod, 99

9. GREEN’S FUNCTION METHOD FOR THE WAVE OPERATOR, 104

D’Alembert formula

Example 1. Doubly- infinite String, 105

10. THE EIGENFUNCTION METHOD, 106

Illustration of the method

Example 1. Poisson Equation for a Rectangle, 106

11. ADDITIONAL EXAMPLES, 112

More than two independent variables; higher order equations; images; Poisson integral formula; Laplace transform; Lienard- Wiechert potential; plate theory

Example 1. Laplace Operator in Three Dimensions, 112

Example 2. Two- and Three- Dimensional Acoustics, 116

Example 3. Biharmonic Equation, 125

12. SUMMARY. 130

Summary of Green’s function procedure for partial differential equations

Errata, 133

Suggested Reading, 135

Index, 137