## Applied Integral TransformsThis 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 |

253 | |

About the Authors | 261 |

263 | |

### Other editions - View all

Applied Integral Transforms M. Ya Antimirov,A. A. Kolyshkin,Rémi Vaillancourt No preview available - 1993 |

Applied Integral Transforms M. Ya Antimirov,A. A. Kolyshkin,Rémi Vaillancourt No preview available - 1993 |

### Common terms and phrases

a2uxx Applying the inverse Applying the Laplace base rock boundary conditions boundary value problem cap rock chapter closed channel coefficients computed consider convective convergent cosh cylindrical d'Alembert's formula delta function determined differential equations dimensionless variables Dirac delta function eddy-current eddy-current testing Efros eigenfunctions erfc exact formulation example finite interval flaw fluid formula Fourier sine transform Fourier transform function f(x given Green's function half-space Hankel transform heat contents heat equation heat transfer Hence hot-fluid injection i-th region initial boundary value integral transforms integral with respect inverse Laplace transform inverse transform ip(x Lauwerier linear lumped formulation mathematical mechanical equilibrium method of integral multi-layer stratum numerical one-dimensional order of integration parameter perturbations radial second kind semi-infinite interval sinh solution to problem solve stability boundary stratum Sturm-Liouville problem Substituting temperature field term thermal wave equation zero