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AuthorKanwal, Ram P. author
TitleGeneralized Functions Theory and Technique [electronic resource] / by Ram P. Kanwal
ImprintBoston, MA : Birkhรคuser Boston, 1998
Edition Second Edition
Connect tohttp://dx.doi.org/10.1007/978-1-4684-0035-9
Descript 474 p. online resource

SUMMARY

The theory of generalized functions is a fundamental part of the toolkit of mathematicians, physicists, and theoretically inclined engineers. It has become increasingly clear that methods based on this theory, also known as the theory of distributions, not only help us to solve previously unsolved problems but also enalble us to recover known solutions in a very simple manner. 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. This edition has been strengthened in many ways. Various new concepts have been added. Some of the new material has been reorganized to improve the logical flow of ideas. And the set of examples has been expanded considerably to make more of the ideas concrete in the reader's eye


CONTENT

1. The Dirac Delta Function and Delta Sequences -- 1.1. The heaviside function -- 1.2. The Dirac delta function -- 1.3. The delta sequences -- 1.4. A unit dipole -- 1.5. The heaviside sequences -- Exercises -- 2. The Schwartz-Sobolev Theory of distributions -- 2.1. Some introductory definitions -- 2.2. Test functions -- 2.3. Linear functionals and the Schwartz-Sobolev theory of distributions -- 2.4. Examples -- 2.5. Algebraic operations on distributions -- 2.6. Analytic operations on distributions -- 2.7. Examples -- 2.8. The support and singular support of a distribution -- Exercises -- 3. Additional Properties of Distributions -- 3.1. Transformation properties of the delta distribution -- 3.2. Convergence of distributions -- 3.3. Delta sequences with parametric dependence -- 3.4. Fourier series -- 3.5. Examples -- 3.6. The delta function as a Stieltjes integral -- Exercises -- 4. Distributions Defined by Divergent Integrals -- 4.1. Introduction -- 4.2. The pseudofunction H(x)/xn, n = 1, 2, 3,... -- 4.3. Functions with algebraic singularity of order m -- 4.4. Examples -- Exercises -- 5. Distributional Derivatives of Functions with Jump Discontinuities -- 5.1. Distributional derivatives in R1 -- 5.2. Moving surfaces of discontinuity in Rn, n ? 2 -- 5.3. Surface distributions -- 5.4. Various other representations -- 5.5. First-order distributional derivatives -- 5.6. Second-order distributional derivatives -- 5.7. Higher-order distributional derivatives -- 5.8. The two-dimensional case -- 5.9. Examples -- 5.10. The function Pf(1/r) and its derivatives -- 6. Tempered Distributions and the Fourier Transform -- 6.1. Preliminary concepts -- 6.2. Distributions of slow growth (tempered distributions) -- 6.3. The Fourier transform -- 6.4. Examples -- Exercises -- 7. Direct Products and Convolutions of Distributions -- 7.1. Definition of the direct product -- 7.2. The direct product of tempered distributions -- 7.3. The Fourier transform of the direct product of tempered distributions -- 7.4. The convolution -- 7.5. The role of convolution in the regularization of the distributions -- 7.6. The dual spaces E and E? -- 7.7. Examples -- 7.8. The Fourier transform of a convolution -- 7.9. Distributional solutions of integral equations -- Exercises -- 8. The Laplace Transform -- 8.1. A brief discussion of the classical results -- 8.2. The Laplace transform distributions -- 8.3. The Laplace transform of the distributional derivatives and vice versa -- 8.4. Examples -- Exercises -- 9. Applications to Ordinary Differential Equations -- 9.1. Ordinary differential operators -- 9.2. Homogeneous differential equations -- 9.3. Inhomogeneous differentational equations: the integral of a distribution -- 9.4. Examples -- 9.5. Fundamental solutions and Greenโ{128}{153}s functions -- 9.6. Second-order differential equations with constant coefficients -- 9.7. Eigenvalue problems -- 9.8. Second-order differential equations with variable coefficients -- 9.9. Fourth-order differential equations -- 9.10. Differential equations of nth order -- 9.11. Ordinary differential equations with singular coefficients -- Exercises -- 10. Applications to Partial Differential Equations -- 10.1. Introduction -- 10.2. Classical and generalized solutions -- 10.3. Fundamental solutions -- 10.4. The Cauchy-Riemann operator -- 10.5. The transport operator -- 10.6. The Laplace operator -- 10.7. The heat operator -- 10.8. The Schrรถdinger operator -- 10.9. The Helmholtz operator -- 10.10. The wave operator -- 10.11. The inhomogeneous wave equation -- 10.12. The Klein-Gordon operator -- Exercises -- 11. Applications to Boundary Value Problems -- 11.1. Poissonโ{128}{153}s equation -- 11.2. Dumbbell-shaped bodies -- 11.3. Uniform axial distributions -- 11.4. Linear axial distributions -- 11.5. Parabolic axial distributions, n = 5 -- 11.6. The fourth-order polynomial distribution, n = 7; spheroidal cavities -- 11.7. The polarization tensor for a spheroid -- 11.8. The virtual mass tensor for a spheroid -- 11.9. The electric and magnetic polarizability tensors -- 11.10. The distributional approach to scattering theory -- 11.11. Stokes flow -- 11.12. Displacement-type boundary value problems in elastostatistics -- 11.13. The extension to elastodynamics -- 11.14. Distributions on arbitrary lines -- 11.15. Distributions on plane curves -- 11.16. Distributions on a circular disk -- 12. Applications to Wave Propagation -- 12.1. Introduction -- 12.2. The wave equation -- 12.3. First-order hyperbolic systems -- 12.4. Aerodynamic sound generation -- 12.5. The Rankine-Hugoniot conditions -- 12.6. Wave fronts that carry infinite singularities -- 12.7. Kinematics of wavefronts -- 12.8. Derivation of the transport theorems for wave fronts -- 12.9. Propagation of wave fronts carrying multilayer densities -- 12.10. Generalized functions with support on the light cone -- 12.11. Examples -- 13. Interplay Between Generalized Functions and the Theory of Moments -- 13.1. The theory of moments -- 13.2. Asymptotic approximation of integrals -- 13.3. Applications to the singular perturbation theory -- 13.4. Applications to number theory -- 13.5. Distributional weight functions for orthogonal polynomials -- 13.6. Convolution type integral equation revisited -- 13.7. Further applications -- 14. Linear Systems -- 14.1. Operators -- 14.2. The step response -- 14.3. The impulse response -- 14.4. The response to an arbitrary input -- 14.5. Generalized functions as impulse response functions -- 14.6. The transfer function -- 14.7. Discrete-time systems -- 14.8. The sampling theorem -- 15. Miscellaneous Topics -- 15.1. Applications to probability and random processes -- 15.2. Applications to economics -- 15.3. Periodic distributions -- 15.4. Applications to microlocal theory -- References


Mathematics Partial differential equations Functions of real variables Special functions Applied mathematics Engineering mathematics Mathematics Applications of Mathematics Partial Differential Equations Real Functions Special Functions



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