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El. knyga: Green's Functions and Linear Differential Equations: Theory, Applications, and Computation

(University of New Orleans, Louisiana, USA)
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Greens Functions and Linear Differential Equations: Theory, Applications, and Computation presents a variety of methods to solve linear ordinary differential equations (ODEs) and partial differential equations (PDEs). The text provides a sufficient theoretical basis to understand Greens function method, which is used to solve initial and boundary value problems involving linear ODEs and PDEs. It also contains a large number of examples and exercises from diverse areas of mathematics, applied science, and engineering.

Taking a direct approach, the book first unravels the mystery of the Dirac delta function and then explains its relationship to Greens functions. The remainder of the text explores the development of Greens functions and their use in solving linear ODEs and PDEs. The author discusses how to apply various approaches to solve initial and boundary value problems, including classical and general variations of parameters, Wronskian method, Bernoullis separation method, integral transform method, method of images, conformal mapping method, and interpolation method. He also covers applications of Greens functions, including spherical and surface harmonics.

Filled with worked examples and exercises, this robust, self-contained text fully explains the differential equation problems, includes graphical representations where necessary, and provides relevant background material. It is mathematically rigorous yet accessible enough for readers to grasp the beauty and power of the subject.
Preface xi
Notations and Definitions xvii
1 Some Basic Results
1(17)
1.1 Euclidean Space
1(2)
1.1.1 Metric Space
2(1)
1.1.2 Inner Product
2(1)
1.2 Classes of Continuous Functions
3(1)
1.3 Convergence
3(3)
1.3.1 Convergence of Sequences
3(1)
1.3.2 Weak Convergence
4(1)
1.3.3 Metric
5(1)
1.3.4 Convergence of Infinite Series
5(1)
1.3.5 Tests for Convergence of Positive Series
6(1)
1.4 Functionals
6(2)
1.4.1 Examples of Linear Functionals
8(1)
1.5 Linear Transformations
8(1)
1.6 Cramer's Rule
9(1)
1.7 Green's Identities
10(3)
1.8 Differentiation and Integration
13(1)
1.8.1 Leibniz's Rules
13(1)
1.8.2 Integration by Parts
13(1)
1.9 Inequalities
14(1)
1.9.1 Bessel's Inequality for Fourier Series
14(1)
1.9.2 Bessel's Inequality for Square-Integrable Functions
14(1)
1.9.3 Schwarz's Inequality for Infinite Sequences
15(1)
1.10 Exercises
15(3)
2 The Concept of Green's Functions
18(29)
2.1 Generalized Functions
18(11)
2.1.1 Heaviside Function
26(1)
2.1.2 Delta Function in Curvilinear Coordinates
27(2)
2.2 Singular Distributions
29(2)
2.3 The Concept of Green's Functions
31(3)
2.4 Linear Operators and Inverse Operators
34(7)
2.4.1 Linear Operators and Inverse Operators
34(1)
2.4.2 Adjoint Operators
35(6)
2.5 Fundamental Solutions
41(3)
2.6 Exercises
44(3)
3 Sturm-Liouville Systems
47(37)
3.1 Ordinary Differential Equations
47(4)
3.1.1 Initial and Boundary Conditions
47(1)
3.1.2 General Solution
48(1)
3.1.3 Method of Variation of Parameters
49(2)
3.2 Initial Value Problems
51(4)
3.2.1 One-Sided Green's Functions
51(3)
3.2.2 Wronskian Method
54(1)
3.2.3 Systems of First-Order Differential Equations
55(1)
3.3 Boundary Value Problems
55(9)
3.3.1 Sturm-Liouville Boundary Value Problems
56(2)
3.3.2 Properties of Green's Functions
58(1)
3.3.3 Green's Function Method
59(5)
3.4 Eigenvalue Problem for Sturm-Liouville Systems
64(9)
3.4.1 Eigenpairs
66(1)
3.4.2 Orthonormal Systems
67(2)
3.4.3 Eigenfunction Expansion
69(3)
3.4.4 Data for Eigenvalue Problems
72(1)
3.5 Periodic Sturm-Liouville Systems
73(1)
3.6 Singular Sturm-Liouville Systems
74(5)
3.7 Exercises
79(5)
4 Bernoulli's Separation Method
84(37)
4.1 Coordinate Systems
84(1)
4.2 Partial Differential Equations
85(4)
4.3 Bernoulli's Separation Method
89(6)
4.3.1 Laplace's Equation in a Cube
89(1)
4.3.2 Laplace's Equation in a Cylinder
90(1)
4.3.3 Laplace's Equation in a Sphere
91(1)
4.3.4 Helmholtz's Equation in Cartesian Coordinates
92(1)
4.3.5 Helmholtz's Equation in Spherical Coordinates
93(1)
4.3.6 Wave Equation
94(1)
4.4 Examples
95(21)
4.5 Exercises
116(5)
5 Integral Transforms
121(22)
5.1 Integral Transform Pairs
121(1)
5.2 Laplace Transform
122(4)
5.2.1 Definition of Dirac Delta Function
125(1)
5.3 Fourier Integral Theorems
126(4)
5.3.1 Properties of Fourier Transforms
127(1)
5.3.2 Fourier Transforms of Derivatives of a Function
127(1)
5.3.3 Convolution Theorems for Fourier Transform
127(3)
5.4 Fourier Sine and Cosine Transforms
130(2)
5.4.1 Properties of Fourier Sine and Cosine Transforms
130(1)
5.4.2 Convolution Theorems for Fourier Sine and Cosine Transforms
131(1)
5.5 Finite Fourier Transforms
132(4)
5.5.1 Properties
134(1)
5.5.2 Periodic Extensions
134(1)
5.5.3 Convolution
135(1)
5.6 Multiple Transforms
136(1)
5.7 Hankel Transforms
137(2)
5.8 Summary: Variables of Transforms
139(1)
5.9 Exercises
139(4)
6 Parabolic Equations
143(32)
6.1 1-D Diffusion Equation
144(4)
6.1.1 Sturm-Liouville System for 1-D Diffusion Equation
144(2)
6.1.2 Green's Function for 1-D Diffusion Equation
146(2)
6.2 2-D Diffusion Equation
148(3)
6.2.1 Dirichlet Problem for the General Parabolic Equation in a Square
149(2)
6.3 3-D Diffusion Equation
151(3)
6.3.1 Electrostatic Analog
151(3)
6.4 Schrodinger Diffusion Operator
154(2)
6.5 Min-Max Principle
156(1)
6.6 Diffusion Equation in a Finite Medium
156(1)
6.7 Axisymmetric Diffusion Equation
157(1)
6.8 1-D Heat Conduction Problem
158(2)
6.9 Stefan Problem
160(3)
6.10 1-D Fractional Diffusion Equation
163(3)
6.10.1 1-D Fractional Diffusion Equation in Semi-Infinite Medium
165(1)
6.11 1-D Fractional Schrodinger Diffusion Equation
166(1)
6.12 Eigenpairs and Dirac Delta Function
167(3)
6.13 Exercises
170(5)
7 Hyperbolic Equations
175(34)
7.1 1-D Wave Equation
175(5)
7.1.1 Sturm-Liouville System for 1-D Wave Equation
175(2)
7.1.2 Vibrations of a Variable String
177(2)
7.1.3 Green's Function for 1-D Wave Equation
179(1)
7.2 2-D Wave Equation
180(1)
7.3 3-D Wave Equation
180(2)
7.4 2-D Axisymmetric Wave Equation
182(1)
7.5 Vibrations of a Circular Membrane
182(1)
7.6 3-D Wave Equation in a Cube
183(3)
7.7 Schrodinger Wave Equation
186(1)
7.8 Hydrogen Atom
187(3)
7.8.1 Harmonic Oscillator
190(1)
7.9 1-D Fractional Nonhomogeneous Wave Equation
190(3)
7.10 Applications of the Wave Operator
193(5)
7.10.1 Cauchy Problem for 2-D and 3-D Wave Equation
193(1)
7.10.2 d'Alembert Solution of the Cauchy Problem for Wave Equation
194(2)
7.10.3 Free Vibration of a Large Circular Membrane
196(1)
7.10.4 Hyperbolic or Parabolic Equations in Terms of Green's Functions
196(2)
7.11 Laplace Transform Method
198(3)
7.12 Quasioptics and Diffraction
201(4)
7.12.1 Diffraction of Monochromatic Waves
201(1)
(a) Fraunhofer Approximation
202(2)
(b) Fresnel Approximation
204(1)
7.13 Exercises
205(4)
8 Elliptic Equations
209(42)
8.1 Green's Function for 2-D Laplace's Equation
209(2)
8.2 2-D Laplace's Equation in a Rectangle
211(1)
8.3 Green's Function for 3-D Laplace's Equation
212(5)
8.3.1 Laplace's Equation in a Rectangular Parallelopiped
213(4)
8.4 Harmonic Functions
217(1)
8.5 2-D Helmholtz's Equation
218(2)
8.5.1 Closed-Form Green's Function for Helmholtz's Equation
219(1)
8.6 Green's Function for 3-D Helmholtz's Equation
220(1)
8.7 2-D Poisson's Equation in a Circle
221(5)
8.8 Method for Green's Function in a Rectangle
226(3)
8.9 Poisson's Equation in a Cube
229(2)
8.10 Laplace's Equation in a Sphere
231(4)
8.11 Poisson's Equation and Green's Function in a Sphere
235(2)
8.12 Applications of Elliptic Equations
237(7)
8.12.1 Dirichlet Problem for Laplace's Equation
237(1)
8.12.2 Neumann Problem for Laplace's Equation
237(2)
8.12.3 Robin Problem for Laplace's Equation
239(1)
8.12.4 Dirichlet Problem for Helmholtz's Equation
239(1)
8.12.5 Dirichlet Problem for Laplace's Equation in the Half-Plane
240(1)
8.12.6 Dirichlet Problem for Laplace's Equation in a Circle
241(1)
8.12.7 Dirichlet Problem for Laplace's Equation in the Quarter Plane
241(2)
8.12.8 Vibration Equation for the Unit Sphere
243(1)
8.13 Exercises
244(7)
9 Spherical Harmonics
251(30)
9.1 Historical Sketch
251(1)
9.2 Laplace's Solid Spherical Harmonics
252(9)
9.2.1 Orthonormalization
254(2)
9.2.2 Condon-Shortley Phase Factor
256(1)
9.2.3 Spherical Harmonics Expansion
257(1)
9.2.4 Addition Theorem
258(1)
9.2.5 Laplace's Coefficients
259(2)
9.3 Surface Spherical Harmonics
261(15)
9.3.1 Poisson Integral Representation
266(2)
9.3.2 Representation of a Function f(θ,Φ)
268(1)
9.3.3 Addition Theorem for Spherical Harmonics
269(2)
9.3.4 Discrete Energy Spectrum
271(3)
9.3.5 Further Developments
274(2)
9.4 Exercises
276(5)
10 Conformal Mapping Method
281(33)
10.1 Definitions and Theorems
281(4)
10.1.1 Cauchy-Riemann Equations
281(1)
10.1.2 Conformal Mapping
282(1)
10.1.3 Symmetric Points
283(1)
10.1.4 Cauchy's Integral Formula
284(1)
10.1.5 Mean-Value Theorem
284(1)
10.2 Dirichlet Problem
285(5)
10.2.1 Dirichlet Problem for a Circle in the (x, y)-Plane
289(1)
10.3 Neumann Problem
290(3)
10.4 Green's and Neumann's Functions
293(10)
10.4.1 Laplacian
293(2)
10.4.2 Green's Function for a Circle
295(2)
10.4.3 Green's Function for an Ellipse
297(2)
10.4.4 Green's Function for an Infinite Strip
299(3)
10.4.5 Green's Function for an Annulus
302(1)
10.5 Computation of Green's Functions
303(6)
10.5.1 Interpolation Method
304(5)
10.6 Exercises
309(5)
A Adjoint Operators
314(3)
B List of Fundamental Solutions
317(5)
B.1 Linear Ordinary Differential Operator with Constant Coefficients
317(1)
B.2 Fundamental Solutions for the Operators
317(1)
B.3 Elliptic Operator
317(1)
B.4 Helmholtz Operator
318(1)
B.5 Fundamental Solution for the Cauchy-Riemann Operator
319(1)
B.6 Fundamental Solution for the Diffusion Operator
319(1)
B.7 Schrodinger Operator
320(1)
B.8 Fundamental Solution for the Wave Operator
321(1)
B.9 Fundamental Solution for the Fokker-Plank Operator
321(1)
B.10 Klein-Gordon Operator
321(1)
C List of Spherical Harmonics
322(7)
C.1 Legendre's Equation
322(1)
C.2 Associated Legendre's Equation
323(1)
C.3 Relations with or without Condon-Shortley Phase Factor
324(2)
C.4 Laguerre's Equation
326(1)
C.5 Associated Laguerre's Equation
327(2)
D Tables of Integral Transforms
329(9)
D.1 Laplace Transform Pairs
329(3)
D.2 Fourier Cosine Transform Pairs
332(1)
D.3 Fourier Sine Transform Pairs
333(1)
D.4 Complex Fourier Transform Pairs
334(1)
D.5 Finite Sine Transform Pairs
335(1)
D.6 Finite Cosine Transform Pairs
336(1)
D.7 Zero-Order Hankel Transform Pairs
337(1)
E Fractional Derivatives
338(3)
F Systems of Ordinary Differential Equations
341(4)
Bibliography 345(4)
Index 349
Prem K. Kythe is a professor emeritus of mathematics at the University of New Orleans. Dr. Kythe is the co-author of Handbook of Computational Methods for Integration (CRC Press, December 2004) and Partial Differential Equations and Boundary Value Problems with Mathematica, Second Edition (CRC Press, November 2002). His research encompasses complex function theory, boundary value problems, wave structure, and integral transforms.
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