Table of contents 1. The heaviside function. The Dirac delta function. The delta sequences. A unit dipole. The heaviside sequences. The Schwartz-Sobolev Theory of distributions. Some introductory definitions. Test functions. Linear functionals and the Schwartz-Sobolev theory of distributions. Algebraic operations on distributions.
Analytic operations on distributions. The support and singular support of a distribution. Additional Properties of Distributions. Transformation properties of the delta distribution. Convergence of distributions.
Delta sequences with parametric dependence. Fourier series.
The delta function as a Stieltjes integral. Distributions Defined by Divergent Integrals. Functions with algebraic singularity of order m.
Distributional Derivatives of Functions with Jump Discontinuities. Distributional derivatives in R1. Moving surfaces of discontinuity in Rn, n? Surface distributions. Various other representations. First-order distributional derivatives. Second-order distributional derivatives. Higher-order distributional derivatives. The two-dimensional case.
Tempered Distributions and the Fourier Transform. Preliminary concepts. Distributions of slow growth tempered distributions. The Fourier transform.
Direct Products and Convolutions of Distributions. Definition of the direct product. The direct product of tempered distributions. The Fourier transform of the direct product of tempered distributions. The convolution. The role of convolution in the regularization of the distributions. The dual spaces E and E?. The Fourier transform of a convolution.
Distributional solutions of integral equations.
The Laplace Transform. A brief discussion of the classical results. The Laplace transform distributions. The Laplace transform of the distributional derivatives and vice versa. Applications to Ordinary Differential Equations. Ordinary differential operators. Homogeneous differential equations. Inhomogeneous differentational equations: the integral of a distribution. Fundamental solutions and Green's functions. Second-order differential equations with constant coefficients.
Eigenvalue problems. Second-order differential equations with variable coefficients. Fourth-order differential equations. Differential equations of nth order. Ordinary differential equations with singular coefficients. Applications to Partial Differential Equations. Classical and generalized solutions. Fundamental solutions. The Cauchy-Riemann operator. The transport operator.
Many operations which are defined on smooth functions with compact support can also be defined for distributions. Biagioni A Nonlinear Theory of Ge Preliminary concepts. Namespaces Article Talk. Kinematics of wavefronts. About This Item We aim to show you accurate product information. For the concept of distributions in probability theory, see Probability distribution.
The Laplace operator. The heat operator. The Schroedinger operator.
The Helmholtz operator. The wave operator. The inhomogeneous wave equation. The Klein-Gordon operator. Applications to Boundary Value Problems. Poisson's equation. Dumbbell-shaped bodies. Uniform axial distributions. Linear axial distributions. The polarization tensor for a spheroid. The virtual mass tensor for a spheroid.
The electric and magnetic polarizability tensors. The distributional approach to scattering theory. This book contains both the theory and applications of generalized functions with a significant feature being the quantity and variety of applications. Definitions and theorems are stated precisely, but rigor is minimized in favor of comprehension of techniques.
Most of the material is easily accessible to senior undergraduate and graduate students in mathematical, physical and engineering sciences. The background required is limited to the standard courses in advanced calculus, ordinary and partial differential equations, and boundary value problems.
The chapters that are suitable as a one semester course are furnished with sets of exercises. Accordingly, almost all the chapters have been revised. The bibliography has been enlarged considerably. Some of the material has been reorganized. For example, Chapters 12 and 13 of the first edition have been consolidated into Chapter 12 of this edition by a judicious process of elimination and addition of the subject matter.