Fundamental Solutions for Differential Operators and Applications

ISBN-13: 9780817638696 / Angielski / Twarda / 1996 / 414 str.

Prem K. Kythe

Opis

Overview Many problems in mathematical physics and applied mathematics can be reduced to boundary value problems for differential, and in some cases, inte grodifferential equations. These equations are solved by using methods from the theory of ordinary and partial differential equations, variational calculus, operational calculus, function theory, functional analysis, probability theory, numerical analysis and computational techniques. Mathematical models of quantum physics require new areas such as generalized functions, theory of distributions, functions of several complex variables, and topological and al gebraic methods. The main purpose of this book is to provide a self contained and system atic introduction to just one aspect of analysis which deals with the theory of fundamental solutions for differential operators and their applications to boundary value problems of mathematical physics, applied mathematics, and engineering, with the related applicable and computational features. The sub ject matter of this book has its own deep rooted theoretical importance since it is related to Green's functions which are associated with most boundary value problems. The application of fundamental solutions to a recently devel oped area of boundary element methods has provided a distinct advantage in that an integral equation representation of a boundary value problem is often x PREFACE more easily solved by numerical methods than a differential equation with specified boundary and initial conditions. This situation makes the subject more attractive to those whose interest is primarily in numerical methods."

Recenzje

"Kythe explores fundamental solutions of both linear and nonlinear differential operators from mathematical physics, elasticity, fluid dynamics, piezoelectrics, and cosmology. The book begins with a historical development of the subject.... Three appendices provide background on Fourier and Laplace transforms, details about several computer programs that can be used in conjunction with the book, and a listing of 76 differential operators. Extensive bibliography."   -Choice

Spis treści

1. Historical Background.- 2. Modern Developments.- 1: Some Basic Concepts.- 1.1. Definitions.- 1.2. Green’s Identities.- 1.3. Distributions.- 1.4. Fundamental Solutions.- 2: Linear Elliptic Operators.- 2.1. Constant Coefficients.- 2.2. Laplace Operator.- 2.3. Helmholtz Operator.- 2.4. Cauchy-Riemann Operator.- 2.5. Nonhomogeneous Operator.- 2.6. Maximum Principle.- 2.7. Method of Images.- 3: Linear Parabolic Operators.- 3.1. Diffusion Operator.- 3.2. Heat Potentials.- 3.3. Cauchy Problem.- 3.4. Maximum Principle.- 3.5. Schrödinger Operator.- 3.6. Method of Images.- 4: Linear Hyperbolic Operators.- 4.1. Wave Operator.- 4.2. Harmonie Oscillators.- 4.3. Wave Potentials.- 4.4. Cauchy Problem.- 4.5. Wave Propagation.- 4.6. Maxwell’s Equations.- 5: Nonlinear Operators.- 5.1. Einstein-Kolmogorov Operator.- 5.2. Fokker-Plank Operator.- 5.3. Klein-Gordon Operator.- 5.4. Dirac’s Operator.- 5.5. Transport Equation.- 5.6. Transport Operator.- 5.7. Biharmonic Operator.- 5.8. Nonlinear Wave Equations.- 5.9. The ?-function.- 5.10. Quasihyperbolic Operator.- 6: Elastostatics.- 6.1. Basic Relations.- 6.2. Cauchy-Navier Operator.- 6.3. Half-Space Solutions.- 6.4. Axisymmetric Solutions.- 6.5. Somigliana’s Identity.- 7: Elastodynamics.- 7.1. Elastodynamic Operator.- 7.2. Wave Structures.- 7.3. Bernoulli-Euler Operator.- 7.4. Elastoplasticity.- 7.5. Anisotropic Medium.- 8: Fluid Dynamics.- 8.1. Navier-Stokes Equations.- 8.2. Aerodynamic Flows.- 8.3. Non-Newtonian Flows.- 8.4. Porous Media.- 8.5. Underwater Acoustic Scattering.- 9: Piezoelectrics.- 9.1. Green’s Functions.- 9.2. Dynamic Piezoelectric Operator.- 9.3. Fundamental Solutions.- 9.4. Steady-State Solutions.- 9.5. Piezocrystal Waves.- 10: Boundary Element Methods.- 10.1. Boundary Integral Equations.- 10.2. Boundary Element Method.- 10.3. Poisson Equation.- 10.4. Transient Fourier Equation.- 10.5. Laplace Transform BEM.- 10.6. Elastostatic BEM.- 10.7. Fracture Mechanics.- 11: Domain Integrals.- 11.1. Dual Reciprocity Method.- 11.2. Multiple Reciprocity Method.- 11.3. Transient DRM.- 11.4. Transient MRM.- 11.5. Fourier Series Method.- 11.6. Complex Variable BEM.- 12: Finite Deflection of Plates.- 12.1. von Karman Equations.- 12.2. Boundary Integral Equations.- 12.3. Large Deflections.- 12.4. Singularities in Biharmonic Problems.- 13: Miscellaneous Topics.- 13.1. Poroelasticity.- 13.2. Heat Conduction.- 13.3. Thermoelasticity.- 13.4. Neutron Diffusion.- 13.5. Biomechanics.- 14: Quasilinear Elliptic Operators.- 14.1. p-Laplacian.- 14.2. Lane-Emden Equation.- 14.3. Emden-Fowler Equation.- 14.4. Black Hole Solutions.- 14.5. Einstein-Yang-Mills Equation.- Appendix A: Transforms of Distributions.- A.1. Fourier Transform.- A.2. Laplace Transform.- A.3. Inverse Laplace Transform.- Appendix B: Computational Aspects.- Appendix C: List of Differential Operators.