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Nonlinear Partial Differential Equations
This book presents recent and very elementary developments of a theory of multiplication of distributions in the field of explicit and numerical solutions of systems of PDEs of physics (nonlinear elasticity, elastoplasticity, hydrodynamics, multifluid flows, acoustics). The prerequisites are kept to introductory calculus level so that the book remains accessible at the same time to pure mathematicians (as a smoothand somewhat heuristic introdcution to this theory) and to applied mathematicians, numerical engineers and theoretical physicists (as a tool to treat problems involving products of distributions).
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.
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.
An accessible yet rigorous introduction to partial differential equations This textbook provides beginning graduate students and advanced undergraduates with an accessible introduction to the rich subject of partial differential equations (PDEs). It presents a rigorous and clear explanation of the more elementary theoretical aspects of PDEs, while also drawing connections to deeper analysis and applications. The book serves as a needed bridge between basic undergraduate texts and more advanced books that require a significant background in functional analysis. Topics include first order equations and the method of characteristics, second order linear equations, wave and heat equations, Laplace and Poisson equations, and separation of variables. The book also covers fundamental solutions, Green's functions and distributions, beginning functional analysis applied to elliptic PDEs, traveling wave solutions of selected parabolic PDEs, and scalar conservation laws and systems of hyperbolic PDEs. Provides an accessible yet rigorous introduction to partial differential equations Draws connections to advanced topics in analysis Covers applications to continuum mechanics An electronic solutions manual is available only to professors An online illustration package is available to professors
A massive transition of interest from solving linear partial differential equations to solving nonlinear ones has taken place during the last two or three decades. The availability of better computers has often made numerical experimentations progress faster than the theoretical understanding of nonlinear partial differential equations. The three most important nonlinear phenomena observed so far both experimentally and numerically, and studied theoretically in connection with such equations have been the solitons, shock waves and turbulence or chaotical processes. In many ways, these phenomena have presented increasing difficulties in the mentioned order. In particular, the latter two phenomena necessarily lead to nonclassical or generalized solutions for nonlinear partial differential equations.
This volume presents lectures given at the Summer School Wisła 18: Nonlinear PDEs, Their Geometry, and Applications, which took place from August 20 - 30th, 2018 in Wisła, Poland, and was organized by the Baltic Institute of Mathematics. The lectures in the first part of this volume were delivered by experts in nonlinear differential equations and their applications to physics. Original research articles from members of the school comprise the second part of this volume. Much of the latter half of the volume complements the methods expounded in the first half by illustrating additional applications of geometric theory of differential equations. Various subjects are covered, providing readers a glimpse of current research. Other topics covered include thermodynamics, meteorology, and the Monge–Ampère equations. Researchers interested in the applications of nonlinear differential equations to physics will find this volume particularly useful. A knowledge of differential geometry is recommended for the first portion of the book, as well as a familiarity with basic concepts in physics.
During the last few years, several fairly systematic nonlinear theories of generalized solutions of rather arbitrary nonlinear partial differential equations have emerged. The aim of this volume is to offer the reader a sufficiently detailed introduction to two of these recent nonlinear theories which have so far contributed most to the study of generalized solutions of nonlinear partial differential equations, bringing the reader to the level of ongoing research.The essence of the two nonlinear theories presented in this volume is the observation that much of the mathematics concerning existence, uniqueness regularity, etc., of generalized solutions for nonlinear partial differential equations can be reduced to elementary calculus in Euclidean spaces, combined with elementary algebra in quotient rings of families of smooth functions on Euclidean spaces, all of that joined by certain asymptotic interpretations. In this way, one avoids the complexities and difficulties of the customary functional analytic methods which would involve sophisticated topologies on various function spaces. The result is a rather elementary yet powerful and far-reaching method which can, among others, give generalized solutions to linear and nonlinear partial differential equations previously unsolved or even unsolvable within distributions or hyperfunctions.Part 1 of the volume discusses the basic limitations of the linear theory of distributions when dealing with linear or nonlinear partial differential equations, particularly the impossibility and degeneracy results. Part 2 examines the way Colombeau constructs a nonlinear theory of generalized functions and then succeeds in proving quite impressive existence, uniqueness, regularity, etc., results concerning generalized solutions of large classes of linear and nonlinear partial differential equations. Finally, Part 3 is a short presentation of the nonlinear theory of Rosinger, showing its connections with Colombeau's theory, which it contains as a particular case.