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