Problems in Distributions and Partial Differential Equations : Problems in Distributions and Partial Differential Equations

The aim of this book is to provide a comprehensive introduction to the theory of distributions, by the use of solved problems. Although written for mathematicians, it can also be used by a wider audience, including engineers and physicists. The first six chapters deal with the classical theory, with special emphasis on the concrete aspects. The reader will find many examples of distributions and learn how to work with them. At the beginning of each chapter the relevant theoretical material is briefly recalled. The last chapter is a short introduction to a very wide and important field in analysis which can be considered as the most natural application of distributions, namely the theory of partial differential equations. It includes exercises on the classical differential operators and on fundamental solutions, hypoellipticity, analytic hypoellipticity, Sobolev spaces, local solvability, the Cauchy problem, etc. The aim of this book is to provide a comprehensive introduction to the theory of distributions, by the use of solved problems. Although written for mathematicians, it can also be used by a wider audience, including engineers and physicists.

The first six chapters deal with the classical theory, with special emphasis on the concrete aspects. The reader will find many examples of distributions and learn how to work with them. At the beginning of each chapter the relevant theoretical material is briefly recalled.

The last chapter is a short introduction to a very wide and important field in analysis which can be considered as the most natural application of distributions, namely the theory of partial differential equations. It includes exercises on the classical differential operators and on fundamental solutions, hypoellipticity, analytic hypoellipticity, Sobolev spaces, local solvability, the Cauchy problem, etc.

Content: Edited by Page 5 Copyright page Page 6 Introduction Page 7 Chapter 1 Preliminaries Page 11 Basics Chapter 1 Pages 13-24 Chapter 2 The Distributions Page 25 Basics Chapter 2 Pages 27-50 Chapter 3 Differentiation of Distributions Page 51 Basics Chapter 3 Pages 53-86 Chapter 4 Convergence of Distributions Page 87 Basics Chapter 4 Pages 89-109 Chapter 5 Convolution Of Distributions Page 111 Basics Chapter 5 Pages 113-134 Chapter 6 Fourier and Laplace Transforms of Distributions Page 135 Basics Chapter 6 Pages 137-181 Chapter 7 Applications Page 183 Basics Chapter 7 Pages 185-240 Bibliography Page 241 Index of Words Pages 243-244 Index of Notations Page 245