Fundamental Solutions for Differential Operators and Applications

• 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. BoundaryElement 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.

"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