Applied Integral Transforms

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American Mathematical Soc., Apr 24, 2007 - Mathematics - 265 pages
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This book constructs the kernels of integral transforms by solving the generalized Sturm-Liouville problems associated with the partial differential equations at hand. In the first part of the book, the authors construct the kernels and use them to solve elementary problems of mathematical physics. This part requires little mathematical background and provides an introduction to the subject of integral transforms as it proceeds mainly by examples and includes a variety of exercises. In the second part of the book, the method of integral transforms is used to solve modern applied problems in convective stability, temperature fields in oil strata, and eddy-current testing. The choice of topics reflects the authors' research experience and involvement in industrial applications. The first part of the book is accessible to undergraduates, while the second part is aimed at graduate students and researchers. Because of the applications, the book will interest engineers (especially petroleum engineers) and physicists.
 

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Contents

Introduction
3
12 General References
5
Basic Integral Transforms
9
21 The Laplace Transform
10
22 The Fourier Transform
13
23 Fourier sine and cosine Transforms
14
24 Hankel Transforms
15
25 Finite Fourier sine and cosine Transforms
16
Problems
124
Convective Stability Problems
127
72 NavierStokes Equations
132
73 Convection in a Horizontal Layer
134
74 Vertical Circular Cylinder
139
75 Vertical Cylinder in a Magnetic Field
149
Temperature Fields in Oil Strata
157
82 The Incomplete Lumped Formulation
169

26 Finite Hankel Transforms
18
Kernels of Integral Transforms
21
32 Kernels on a Finite Interval
26
33 Kernels on Circular Regions
34
34 Discrete to Continuous Spectrum
42
35 Kernel on a SemiInfinite Interval
50
36 Kernel on a Radially Symmetric Plane
58
Problems
62
Hyperbolic Equations
65
42 The Telegraph and Other Equations
79
Problems
88
Parabolic Equations
89
52 The Fundamental Solution
97
53 The OneDimensional Heat Equation
101
54 Greens Functions by the Laplace Transform
107
The ThreeDimensional Wave Equation
113
62 The NonHomogeneous Wave Equation
119
63 The Method of Descent
120
83 The Lumped Radial Case
177
84 MultiLayer Stratum with Thermal Injection
185
85 Computation in a MultiLayer Stratum
195
86 Exact Heat Contents with Asymmetry
203
87 The Lauwerier Solution
207
88 Finite Heat Transfer in a SingleLayer
215
EddyCurrent Testing Problems
227
92 The Perturbation Method
228
93 Formulation of the Problem
230
94 Flaws of Rectangular CrossSections
231
95 Numerical Results
238
96 Finite Circular Cylindrical Flaws
239
97 Comparison With a TwoLayer Medium
245
Answers to the Problems
249
References
253
About the Authors
261
Index
263
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