Green’s Functions with Applications

DescriptionSince publication of the first edition over a decade ago, Green’s Functions with Applications has provided applied scientists and engineers with a systematic approach to the various methods available for deriving a Green’s function. This fully revised Second Edition retains the same purpose, but has been meticulously updated to reflect the current state of the art.The book opens with necessary background information: a new chapter on the historical development of the Green’s function, coverage of the Fourier and Laplace transforms, a discussion of the classical special functions of Bessel functions and Legendre polynomials, and a review of the Dirac delta function.The text then presents Green’s functions for each class of differential equation (ordinary differential, wave, heat, and Helmholtz equations) according to the number of spatial dimensions and the geometry of the domain. Detailing step-by-step methods for finding and computing Green’s functions, each chapter contains a special section devoted to topics where Green’s functions particularly are useful. For example, in the case of the wave equation, Green’s functions are beneficial in describing diffraction and waves.To aid readers in developing practical skills for finding Green’s functions, worked examples, problem sets, and illustrations from acoustics, applied mechanics, antennas, and the stability of fluids and plasmas are featured throughout the text. A new chapter on numerical methods closes the book.Included solutions and hundreds of references to the literature on the construction and use of Green’s functions make Green’s Functions with Applications, Second Edition a valuable sourcebook for practitioners as well as graduate students in the sciences and engineering.Table of ContentsAcknowledgmentsAuthorPrefaceList of DefinitionsHistorical DevelopmentMr. Green’s EssayPotential EquationHeat EquationHelmholtz’s EquationWave EquationOrdinary Differential EquationsBackground MaterialFourier TransformLaplace TransformBessel FunctionsLegendre PolynomialsThe Dirac Delta FunctionGreen’s FormulasWhat Is a Green’s Function?Green’s Functions for Ordinary Differential EquationsInitial-Value ProblemsThe Superposition IntegralRegular Boundary-Value ProblemsEigenfunction Expansion for Regular Boundary-Value ProblemsSingular Boundary-Value ProblemsMaxwell’s ReciprocityGeneralized Green’s FunctionIntegro-Differential EquationsGreen’s Functions for the Wave EquationOne-Dimensional Wave Equation in an Unlimited DomainOne-Dimensional Wave Equation on the Interval 0 < x < LAxisymmetric Vibrations of a Circular MembraneTwo-Dimensional Wave Equation in an Unlimited DomainThree-Dimensional Wave Equation in an Unlimited DomainAsymmetric Vibrations of a Circular MembraneThermal WavesDiffraction of a Cylindrical Pulse by a Half-PlaneLeaky ModesWater WavesGreen’s Functions for the Heat EquationHeat Equation over Infinite or Semi-Infinite DomainsHeat Equation within a Finite Cartesian DomainHeat Equation within a CylinderHeat Equation within a SphereProduct SolutionAbsolute and Convective InstabilityGreen’s Functions for the Helmholtz EquationFree-Space Green’s Functions for Helmholtz’s and Poisson’s EquationMethod of ImagesTwo-Dimensional Poisson’s Equation over Rectangular and Circular DomainsTwo-Dimensional Helmholtz Equation over Rectangular and Circular DomainsPoisson’s and Helmholtz’s Equations on a Rectangular StripThree-Dimensional Problems in a Half-SpaceThree-Dimensional Poisson’s Equation in a Cylindrical DomainPoisson’s Equation for a Spherical DomainImproving the Convergence Rate of Green’s FunctionsMixed Boundary Value ProblemsNumerical MethodsDiscrete Wavenumber RepresentationLaplace Transform MethodFinite Difference MethodHybrid MethodGalerkin MethodEvaluation of the Superposition IntegralMixed Boundary Value ProblemsAppendix: Relationship between Solutions of Helmholtz’s and Laplace’s Equations in Cylindrical and Spherical CoordinatesAnswers to Some of the ProblemsAuthor IndexSubject IndexAuthor BiographyDean G. Duffy received his bachelor of science in geophysics from Case Institute of Technology, Cleveland, Ohio, USA, and his doctorate of science in meteorology from the Massachusetts Institute of Technology, Cambridge, USA. He served in the US Air Force for four years as a numerical weather prediction officer. After his military service, he began a twenty-five year association with the National Aeronautics and Space Administration’s Goddard Space Flight Center, Greenbelt, Maryland, USA. Widely published, Dr. Duffy has taught courses at the US Naval Academy, Annapolis, Maryland, and the US Military Academy, West Point, New York.