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

Author: C. Zuily

Publisher: Elsevier

Published: 1988-04-01

Total Pages: 247

ISBN-13: 0080872549

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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.

Principles of Partial Differential Equations
Principles of Partial Differential Equations

Author: Alexander Komech

Publisher: Springer Science & Business Media

Published: 2009-10-05

Total Pages: 165

ISBN-13: 1441910956

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This concise book covers the classical tools of Partial Differential Equations Theory in today’s science and engineering. The rigorous theoretical presentation includes many hints, and the book contains many illustrative applications from physics.

Tools and Problems in Partial Differential Equations
Tools and Problems in Partial Differential Equations

Author: Thomas Alazard

Publisher: Springer Nature

Published: 2020-10-19

Total Pages: 357

ISBN-13: 3030502848

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This textbook offers a unique learning-by-doing introduction to the modern theory of partial differential equations. Through 65 fully solved problems, the book offers readers a fast but in-depth introduction to the field, covering advanced topics in microlocal analysis, including pseudo- and para-differential calculus, and the key classical equations, such as the Laplace, Schrödinger or Navier-Stokes equations. Essentially self-contained, the book begins with problems on the necessary tools from functional analysis, distributions, and the theory of functional spaces, and in each chapter the problems are preceded by a summary of the relevant results of the theory. Informed by the authors' extensive research experience and years of teaching, this book is for graduate students and researchers who wish to gain real working knowledge of the subject.

A Guide to Distribution Theory and Fourier Transforms
A Guide to Distribution Theory and Fourier Transforms

Author: Robert S. Strichartz

Publisher: World Scientific

Published: 2003

Total Pages: 238

ISBN-13: 9789812384300

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This important book provides a concise exposition of the basic ideas of the theory of distribution and Fourier transforms and its application to partial differential equations. The author clearly presents the ideas, precise statements of theorems, and explanations of ideas behind the proofs. Methods in which techniques are used in applications are illustrated, and many problems are included. The book also introduces several significant recent topics, including pseudodifferential operators, wave front sets, wavelets, and quasicrystals. Background mathematical prerequisites have been kept to a minimum, with only a knowledge of multidimensional calculus and basic complex variables needed to fully understand the concepts in the book.A Guide to Distribution Theory and Fourier Transforms can serve as a textbook for parts of a course on Applied Analysis or Methods of Mathematical Physics, and in fact it is used that way at Cornell.

Distributions and Operators
Distributions and Operators

Author: Gerd Grubb

Publisher: Springer Science & Business Media

Published: 2008-10-14

Total Pages: 464

ISBN-13: 0387848940

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This book gives an introduction to distribution theory, based on the work of Schwartz and of many other people. It is the first book to present distribution theory as a standard text. Each chapter has been enhanced with many exercises and examples.

Partial Differential Equations III
Partial Differential Equations III

Author: M. A. Shubin

Publisher: Springer Verlag

Published: 1991

Total Pages: 216

ISBN-13: 9783540520030

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Two general questions regarding partial differential equations are explored in detail in this volume of the Encyclopaedia. The first is the Cauchy problem, and its attendant question of well-posedness (or correctness). The authors address this question in the context of PDEs with constant coefficients and more general convolution equations in the first two chapters. The third chapter extends a number of these results to equations with variable coefficients. The second topic is the qualitative theory of second order linear PDEs, in particular, elliptic and parabolic equations. Thus, the second part of the book is primarily a look at the behavior of solutions of these equations. There are versions of the maximum principle, the Phragmen-Lindel]f theorem and Harnack's inequality discussed for both elliptic and parabolic equations. The book is intended for readers who are already familiar with the basic material in the theory of partial differential equations.

Distributions
Distributions

Author: J.J. Duistermaat

Publisher: Springer Science & Business Media

Published: 2010-08-09

Total Pages: 455

ISBN-13: 0817646752

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This textbook is an application-oriented introduction to the theory of distributions, a powerful tool used in mathematical analysis. The treatment emphasizes applications that relate distributions to linear partial differential equations and Fourier analysis problems found in mechanics, optics, quantum mechanics, quantum field theory, and signal analysis. The book is motivated by many exercises, hints, and solutions that guide the reader along a path requiring only a minimal mathematical background.

Partial Differential Equations
Partial Differential Equations

Author: Walter A. Strauss

Publisher: John Wiley & Sons

Published: 2007-12-21

Total Pages: 467

ISBN-13: 0470054565

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Our understanding of the fundamental processes of the natural world is based to a large extent on partial differential equations (PDEs). The second edition of Partial Differential Equations provides an introduction to the basic properties of PDEs and the ideas and techniques that have proven useful in analyzing them. It provides the student a broad perspective on the subject, illustrates the incredibly rich variety of phenomena encompassed by it, and imparts a working knowledge of the most important techniques of analysis of the solutions of the equations. In this book mathematical jargon is minimized. Our focus is on the three most classical PDEs: the wave, heat and Laplace equations. Advanced concepts are introduced frequently but with the least possible technicalities. The book is flexibly designed for juniors, seniors or beginning graduate students in science, engineering or mathematics.

Theory of Distributions
Theory of Distributions

Author: Svetlin G. Georgiev

Publisher: Springer

Published: 2015-07-13

Total Pages: 217

ISBN-13: 3319195271

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This book explains many fundamental ideas on the theory of distributions. The theory of partial differential equations is one of the synthetic branches of analysis that combines ideas and methods from different fields of mathematics, ranging from functional analysis and harmonic analysis to differential geometry and topology. This presents specific difficulties to those studying this field. This book, which consists of 10 chapters, is suitable for upper undergraduate/graduate students and mathematicians seeking an accessible introduction to some aspects of the theory of distributions. It can also be used for one-semester course.

Distributions, Partial Differential Equations, and Harmonic Analysis
Distributions, Partial Differential Equations, and Harmonic Analysis

Author: Dorina Mitrea

Publisher: Springer

Published: 2018-12-29

Total Pages: 615

ISBN-13: 3030032965

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The aim of this book is to offer, in a concise, rigorous, and largely self-contained manner, a rapid introduction to the theory of distributions and its applications to partial differential equations and harmonic analysis. The book is written in a format suitable for a graduate course spanning either over one-semester, when the focus is primarily on the foundational aspects, or over a two-semester period that allows for the proper amount of time to cover all intended applications as well. It presents a balanced treatment of the topics involved, and contains a large number of exercises (upwards of two hundred, more than half of which are accompanied by solutions), which have been carefully chosen to amplify the effect, and substantiate the power and scope, of the theory of distributions. Graduate students, professional mathematicians, and scientifically trained people with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. Throughout, a special effort has been made to develop the theory of distributions not as an abstract edifice but rather give the reader a chance to see the rationale behind various seemingly technical definitions, as well as the opportunity to apply the newly developed tools (in the natural build-up of the theory) to concrete problems in partial differential equations and harmonic analysis, at the earliest opportunity. The main additions to the current, second edition, pertain to fundamental solutions (through the inclusion of the Helmholtz operator, the perturbed Dirac operator, and their iterations) and the theory of Sobolev spaces (built systematically from the ground up, exploiting natural connections with the Fourier Analysis developed earlier in the monograph).