روش‌های ریاضی برای پدیده‌های موجی (علوم کامپیوتر و ریاضیات کاربردی)

Computer Science and Applied Mathematics: Mathematical Methods for Wave Phenomena focuses on the methods of applied mathematics, including equations, wave fronts, boundary value problems, and scattering problems. The publication initially ponders on first-order partial differential equations, Dirac delta function, Fourier transforms, asymptotics, and second-order partial differential equations. Discussions focus on prototype second-order equations, asymptotic expansions, asymptotic expansions of Fourier integrals with monotonic phase, method of stationary phase, propagation of wave fronts, and variable index of refraction. The text then examines wave equation in one space dimension, as well as initial boundary value problems, characteristics for the wave equation in one space dimension, and asymptotic solution of the Klein-Gordon equation. The manuscript offers information on wave equation in two and three dimensions and Helmholtz equation and other elliptic equations. Topics include energy integral, domain of dependence, and uniqueness, scattering problems, Green's functions, and problems in unbounded domains and the Sommerfeld radiation condition. The asymptotic techniques for direct scattering problems and the inverse methods for reflector imaging are also elaborated. The text is a dependable reference for computer science experts and mathematicians pursuing studies on the mathematical methods of wave phenomena. Preface......Page 6 Contents......Page 3 1.1 First-Order Quasi-Linear Differential Equations......Page 11 1.2 An Illustrative Example......Page 17 1.3 First-Order Nonlinear Differential Equations......Page 22 1.4 Examples-The Eikonal Equation-and More Theory......Page 28 1.5 Propagation of Wave Fronts......Page 37 1.6 Variable Index of Refraction......Page 48 1.7 Higher DimensionsFirst-Order Quasi-Linear Differential Equations References......Page 52 References......Page 54 2.1 The Dirac Delta Function and Related Distributions......Page 55 2.2 Fourier Transforms......Page 62 2.3 Fourier Transforms of Distributions......Page 68 2.4 Multidimensional Fourier Transforms......Page 71 2.5 Asymptotic Expansions......Page 76 2.6 Asymptotic Expansions of Fourier Integrals with Monotonic Phase......Page 83 2.7 The Method of Stationary Phase......Page 87 2.8 Multidimensional Fourier Integrals......Page 92 References......Page 99 3.1 Prototype Second-Order Equations......Page 102 3.2 Some Simple Examples......Page 104 References......Page 109 4 THE WAVE EQUATION IN ONE SPACE DIMENSION......Page 0 4.1 Characteristics for the Wave Equation in One Space Dimension......Page 110 4.2 The Initial Boundary Value Problem......Page 115 4.3 The Initial Boundary Value Problem Continued......Page 119 4.4 The Adjoint Equation and the Riemann Function......Page 130 4.5 The Green?ˉs Function......Page 140 4.6 Asymptotic Solution of the Klein-Gordon Equation......Page 143 4.7 More on Asymptotic Solutions......Page 146 References......Page 154 5.1 Characteristics and 111-Posed Cauchy Problems......Page 155 5.2 The Energy Integral, Domain of Dependence, and Uniqueness......Page 160 5.3 The Green?ˉs Function......Page 163 5.4 Scattering Problems......Page 168 References......Page 173 6.1 Green?ˉs Identities and Uniqueness Results......Page 175 6.2 Some Special Features of Laplace?ˉs Equation......Page 180 6.3 Green?ˉs Functions......Page 184 6.4 Problems in Unbounded Domains and the Sommerfeld Radiation Condition......Page 190 6.5 Some Exact Solutions......Page 202 References......Page 212 7.1 Watson?ˉs Lemma......Page 214 7.2 The Method of Steepest Descents: Preliminary Results......Page 221 7.3 Formulas for the Method of Steepest Descents......Page 235 7.4 The Method of Steepest Descents: Implementation......Page 239 References......Page 250 8.1 Scattering by a Half-Space: Analysis by Steepest Descents......Page 251 8.2 Introduction to Ray Methods......Page 267 8.3 Determination of Ray Data......Page 278 8.4 The Kirchhoff Approximation......Page 291 References......Page 309 9.1 The Singular Function and the Characteristic Function......Page 312 9.2 Physical Optics Far-Field Inverse Scattering (POFFIS)......Page 322 9.3 The Seismic Inverse Problem......Page 331 References......Page 347 Index......Page 348