Generalized Solutions Of First Order Pdes

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Generalized Solutions of First Order PDEs

Author : Andrei I. Subbotin
Publisher : Springer Science & Business Media
ISBN 13 : 1461208475
Total Pages : 324 pages
Book Rating : 4.4/5 (612 download)

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Book Synopsis Generalized Solutions of First Order PDEs by : Andrei I. Subbotin

Download or read book Generalized Solutions of First Order PDEs written by Andrei I. Subbotin and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 324 pages. Available in PDF, EPUB and Kindle. Book excerpt: Hamilton-Jacobi equations and other types of partial differential equa tions of the first order are dealt with in many branches of mathematics, mechanics, and physics. These equations are usually nonlinear, and func tions vital for the considered problems are not smooth enough to satisfy these equations in the classical sense. An example of such a situation can be provided by the value function of a differential game or an optimal control problem. It is known that at the points of differentiability this function satisfies the corresponding Hamilton-Jacobi-Isaacs-Bellman equation. On the other hand, it is well known that the value function is as a rule not everywhere differentiable and therefore is not a classical global solution. Thus in this case, as in many others where first-order PDE's are used, there arises necessity to introduce a notion of generalized solution and to develop theory and methods for constructing these solutions. In the 50s-70s, problems that involve nonsmooth solutions of first order PDE's were considered by Bakhvalov, Evans, Fleming, Gel'fand, Godunov, Hopf, Kuznetzov, Ladyzhenskaya, Lax, Oleinik, Rozhdestven ski1, Samarskii, Tikhonov, and other mathematicians. Among the inves tigations of this period we should mention the results of S.N. Kruzhkov, which were obtained for Hamilton-Jacobi equation with convex Hamilto nian. A review of the investigations of this period is beyond the limits of the present book. A sufficiently complete bibliography can be found in [58, 126, 128, 141].

Generalized Solutions of First-order PDEs

Author : Andreĭ Izmaĭlovich Subbotin
Publisher :
ISBN 13 : 9783764337407
Total Pages : 312 pages
Book Rating : 4.3/5 (374 download)

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Book Synopsis Generalized Solutions of First-order PDEs by : Andreĭ Izmaĭlovich Subbotin

Download or read book Generalized Solutions of First-order PDEs written by Andreĭ Izmaĭlovich Subbotin and published by . This book was released on 1995 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt:

On the Uniqueness of Generalized Solutions of First Order, Quasilinear Partial Differential Equations

Author : Avron Douglis
Publisher :
ISBN 13 :
Total Pages : 16 pages
Book Rating : 4.3/5 (91 download)

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Book Synopsis On the Uniqueness of Generalized Solutions of First Order, Quasilinear Partial Differential Equations by : Avron Douglis

Download or read book On the Uniqueness of Generalized Solutions of First Order, Quasilinear Partial Differential Equations written by Avron Douglis and published by . This book was released on 1960 with total page 16 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Generalized Characteristics of First Order PDEs

Author : Arik Melikyan
Publisher : Springer Science & Business Media
ISBN 13 : 146121758X
Total Pages : 319 pages
Book Rating : 4.4/5 (612 download)

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Book Synopsis Generalized Characteristics of First Order PDEs by : Arik Melikyan

Download or read book Generalized Characteristics of First Order PDEs written by Arik Melikyan and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 319 pages. Available in PDF, EPUB and Kindle. Book excerpt: In some domains of mechanics, physics and control theory boundary value problems arise for nonlinear first order PDEs. A well-known classical result states a sufficiency condition for local existence and uniqueness of twice differentiable solution. This result is based on the method of characteristics (MC). Very often, and as a rule in control theory, the continuous nonsmooth (non-differentiable) functions have to be treated as a solutions to the PDE. At the points of smoothness such solutions satisfy the equation in classical sense. But if a function satisfies this condition only, with no requirements at the points of nonsmoothness, the PDE may have nonunique solutions. The uniqueness takes place if an appropriate matching principle for smooth solution branches defined in neighboring domains is applied or, in other words, the notion of generalized solution is considered. In each field an appropriate matching principle are used. In Optimal Control and Differential Games this principle is the optimality of the cost function. In physics and mechanics certain laws must be fulfilled for correct matching. A purely mathematical approach also can be used, when the generalized solution is introduced to obtain the existence and uniqueness of the solution, without being aimed to describe (to model) some particular physical phenomenon. Some formulations of the generalized solution may meet the modelling of a given phenomenon, the others may not.

The Characteristic Method and Its Generalizations for First-Order Nonlinear Partial Differential Equations

Author : Tran Duc Van
Publisher : CRC Press
ISBN 13 : 9781584880165
Total Pages : 256 pages
Book Rating : 4.8/5 (81 download)

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Book Synopsis The Characteristic Method and Its Generalizations for First-Order Nonlinear Partial Differential Equations by : Tran Duc Van

Download or read book The Characteristic Method and Its Generalizations for First-Order Nonlinear Partial Differential Equations written by Tran Duc Van and published by CRC Press. This book was released on 1999-06-25 with total page 256 pages. Available in PDF, EPUB and Kindle. Book excerpt: Despite decades of research and progress in the theory of generalized solutions to first-order nonlinear partial differential equations, a gap between the local and the global theories remains: The Cauchy characteristic method yields the local theory of classical solutions. Historically, the global theory has principally depended on the vanishing viscosity method. The authors of this volume help bridge the gap between the local and global theories by using the characteristic method as a basis for setting a theoretical framework for the study of global generalized solutions. That is, they extend the smooth solutions obtained by the characteristic method. The authors offer material previously unpublished in book form, including treatments of the life span of classical solutions, the construction of singularities of generalized solutions, new existence and uniqueness theorems on minimax solutions, differential inequalities of Haar type and their application to the uniqueness of global, semi-classical solutions, and Hopf-type explicit formulas for global solutions. These subjects yield interesting relations between purely mathematical theory and the applications of first-order nonlinear PDEs. The Characteristic Method and Its Generalizations for First-Order Nonlinear Partial Differential Equations represents a comprehensive exposition of the authors' works over the last decade. The book is self-contained and assumes only basic measure theory, topology, and ordinary differential equations as prerequisites. With its innovative approach, new results, and many applications, it will prove valuable to mathematicians, physicists, and engineers and especially interesting to researchers in nonlinear PDEs, differential inequalities, multivalued analysis, differential games, and related topics in applied analysis.

Partial Differential Equations

Author : Walter A. Strauss
Publisher : John Wiley & Sons
ISBN 13 : 0470054565
Total Pages : 467 pages
Book Rating : 4.4/5 (7 download)

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Book Synopsis Partial Differential Equations by : Walter A. Strauss

Download or read book Partial Differential Equations written by Walter A. Strauss and published by John Wiley & Sons. This book was released on 2007-12-21 with total page 467 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Linear and Nonlinear Waves

Author : G. B. Whitham
Publisher : John Wiley & Sons
ISBN 13 : 1118031202
Total Pages : 660 pages
Book Rating : 4.1/5 (18 download)

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Book Synopsis Linear and Nonlinear Waves by : G. B. Whitham

Download or read book Linear and Nonlinear Waves written by G. B. Whitham and published by John Wiley & Sons. This book was released on 2011-10-18 with total page 660 pages. Available in PDF, EPUB and Kindle. Book excerpt: Now in an accessible paperback edition, this classic work is just as relevant as when it first appeared in 1974, due to the increased use of nonlinear waves. It covers the behavior of waves in two parts, with the first part addressing hyperbolic waves and the second addressing dispersive waves. The mathematical principles are presented along with examples of specific cases in communications and specific physical fields, including flood waves in rivers, waves in glaciers, traffic flow, sonic booms, blast waves, and ocean waves from storms.

Generalized Functions and Partial Differential Equations

Author : Avner Friedman
Publisher : Courier Corporation
ISBN 13 : 9780486152912
Total Pages : 352 pages
Book Rating : 4.1/5 (529 download)

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Book Synopsis Generalized Functions and Partial Differential Equations by : Avner Friedman

Download or read book Generalized Functions and Partial Differential Equations written by Avner Friedman and published by Courier Corporation. This book was released on 2011-11-30 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt: This self-contained text details developments in the theory of generalized functions and the theory of distributions, and it systematically applies them to a variety of problems in partial differential equations. 1963 edition.

Partial Differential Equations and Boundary-Value Problems with Applications

Author : Mark A. Pinsky
Publisher : American Mathematical Soc.
ISBN 13 : 0821868896
Total Pages : 545 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Partial Differential Equations and Boundary-Value Problems with Applications by : Mark A. Pinsky

Download or read book Partial Differential Equations and Boundary-Value Problems with Applications written by Mark A. Pinsky and published by American Mathematical Soc.. This book was released on 2011 with total page 545 pages. Available in PDF, EPUB and Kindle. Book excerpt: Building on the basic techniques of separation of variables and Fourier series, the book presents the solution of boundary-value problems for basic partial differential equations: the heat equation, wave equation, and Laplace equation, considered in various standard coordinate systems--rectangular, cylindrical, and spherical. Each of the equations is derived in the three-dimensional context; the solutions are organized according to the geometry of the coordinate system, which makes the mathematics especially transparent. Bessel and Legendre functions are studied and used whenever appropriate throughout the text. The notions of steady-state solution of closely related stationary solutions are developed for the heat equation; applications to the study of heat flow in the earth are presented. The problem of the vibrating string is studied in detail both in the Fourier transform setting and from the viewpoint of the explicit representation (d'Alembert formula). Additional chapters include the numerical analysis of solutions and the method of Green's functions for solutions of partial differential equations. The exposition also includes asymptotic methods (Laplace transform and stationary phase). With more than 200 working examples and 700 exercises (more than 450 with answers), the book is suitable for an undergraduate course in partial differential equations.

Handbook of First-Order Partial Differential Equations

Author : Andrei D. Polyanin
Publisher : CRC Press
ISBN 13 : 9780415272674
Total Pages : 522 pages
Book Rating : 4.2/5 (726 download)

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Book Synopsis Handbook of First-Order Partial Differential Equations by : Andrei D. Polyanin

Download or read book Handbook of First-Order Partial Differential Equations written by Andrei D. Polyanin and published by CRC Press. This book was released on 2001-11-15 with total page 522 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains about 3000 first-order partial differential equations with solutions. New exact solutions to linear and nonlinear equations are included. The text pays special attention to equations of the general form, showing their dependence upon arbitrary functions. At the beginning of each section, basic solution methods for the corresponding types of differential equations are outlined and specific examples are considered. It presents equations and their applications, including differential geometry, nonlinear mechanics, gas dynamics, heat and mass transfer, wave theory and much more. This handbook is an essential reference source for researchers, engineers and students of applied mathematics, mechanics, control theory and the engineering sciences.

First-Order Partial Differential Equations, Vol. 1

Author : Hyun-Ku Rhee
Publisher : Courier Corporation
ISBN 13 : 0486146200
Total Pages : 561 pages
Book Rating : 4.4/5 (861 download)

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Book Synopsis First-Order Partial Differential Equations, Vol. 1 by : Hyun-Ku Rhee

Download or read book First-Order Partial Differential Equations, Vol. 1 written by Hyun-Ku Rhee and published by Courier Corporation. This book was released on 2014-05-05 with total page 561 pages. Available in PDF, EPUB and Kindle. Book excerpt: This first volume of a highly regarded two-volume text is fully usable on its own. After going over some of the preliminaries, the authors discuss mathematical models that yield first-order partial differential equations; motivations, classifications, and some methods of solution; linear and semilinear equations; chromatographic equations with finite rate expressions; homogeneous and nonhomogeneous quasilinear equations; formation and propagation of shocks; conservation equations, weak solutions, and shock layers; nonlinear equations; and variational problems. Exercises appear at the end of most sections. This volume is geared to advanced undergraduates or first-year grad students with a sound understanding of calculus and elementary ordinary differential equations. 1986 edition. 189 black-and-white illustrations. Author and subject indices.

Finite Difference Methods for Ordinary and Partial Differential Equations

Author : Randall J. LeVeque
Publisher : SIAM
ISBN 13 : 9780898717839
Total Pages : 356 pages
Book Rating : 4.7/5 (178 download)

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Book Synopsis Finite Difference Methods for Ordinary and Partial Differential Equations by : Randall J. LeVeque

Download or read book Finite Difference Methods for Ordinary and Partial Differential Equations written by Randall J. LeVeque and published by SIAM. This book was released on 2007-01-01 with total page 356 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces finite difference methods for both ordinary differential equations (ODEs) and partial differential equations (PDEs) and discusses the similarities and differences between algorithm design and stability analysis for different types of equations. A unified view of stability theory for ODEs and PDEs is presented, and the interplay between ODE and PDE analysis is stressed. The text emphasizes standard classical methods, but several newer approaches also are introduced and are described in the context of simple motivating examples.

Nonlinear Partial Differential Equations

Author : Gui-Qiang Chen
Publisher : American Mathematical Soc.
ISBN 13 : 0821811967
Total Pages : 323 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Nonlinear Partial Differential Equations by : Gui-Qiang Chen

Download or read book Nonlinear Partial Differential Equations written by Gui-Qiang Chen and published by American Mathematical Soc.. This book was released on 1999 with total page 323 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is a collection of original research papers and expository articles stemming from the scientific program of the Nonlinear PDE Emphasis Year held at Northwestern University (Evanston, IL) in March 1998. The book offers a cross-section of the most significant recent advances and current trends and directions in nonlinear partial differential equations and related topics. The book's contributions offer two perspectives. There are papers on general analytical treatment of the theory and papers on computational methods and applications originating from significant realistic mathematical models of natural phenomena. Also included are articles that bridge the gap between these two perspectives, seeking synergistic links between theory and modeling and computation. The volume offers direct insight into recent trends in PDEs. This volume is also available on the Web. Those who purchase the print edition can gain free access by going to www.ams.org/conm/.

Generalized Solutions of Nonlinear Partial Differential Equations

Author : E.E. Rosinger
Publisher : Elsevier
ISBN 13 : 9780080872575
Total Pages : 406 pages
Book Rating : 4.8/5 (725 download)

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Book Synopsis Generalized Solutions of Nonlinear Partial Differential Equations by : E.E. Rosinger

Download or read book Generalized Solutions of Nonlinear Partial Differential Equations written by E.E. Rosinger and published by Elsevier. This book was released on 1987-11-01 with total page 406 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Waves and Structures in Nonlinear Nondispersive Media

Author : Sergey Nikolaevich Gurbatov
Publisher : Springer Science & Business Media
ISBN 13 : 3642236170
Total Pages : 477 pages
Book Rating : 4.6/5 (422 download)

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Book Synopsis Waves and Structures in Nonlinear Nondispersive Media by : Sergey Nikolaevich Gurbatov

Download or read book Waves and Structures in Nonlinear Nondispersive Media written by Sergey Nikolaevich Gurbatov and published by Springer Science & Business Media. This book was released on 2012-03-23 with total page 477 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Waves and Structures in Nonlinear Nondispersive Media: General Theory and Applications to Nonlinear Acoustics” is devoted completely to nonlinear structures. The general theory is given here in parallel with mathematical models. Many concrete examples illustrate the general analysis of Part I. Part II is devoted to applications to nonlinear acoustics, including specific nonlinear models and exact solutions, physical mechanisms of nonlinearity, sawtooth-shaped wave propagation, self-action phenomena, nonlinear resonances and engineering application (medicine, nondestructive testing, geophysics, etc.). This book is designed for graduate and postgraduate students studying the theory of nonlinear waves of various physical nature. It may also be useful as a handbook for engineers and researchers who encounter the necessity of taking nonlinear wave effects into account of their work. Dr. Gurbatov S.N. is the head of Department, and Vice Rector for Research of Nizhny Novgorod State University. Dr. Rudenko O.V. is the Full member of Russian Academy of Sciences, the head of Department at Moscow University and Professor at BTH (Sweden). Dr. Saichev A.I. is the Professor at the Faculty of Radiophysics of Nizhny Novgorod State University, Professor of ETH Zürich.

Advances in Nonlinear Dynamics and Control: A Report from Russia

Author : Alexander B. Kurzhanski
Publisher : Springer Science & Business Media
ISBN 13 : 146120349X
Total Pages : 286 pages
Book Rating : 4.4/5 (612 download)

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Book Synopsis Advances in Nonlinear Dynamics and Control: A Report from Russia by : Alexander B. Kurzhanski

Download or read book Advances in Nonlinear Dynamics and Control: A Report from Russia written by Alexander B. Kurzhanski and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 286 pages. Available in PDF, EPUB and Kindle. Book excerpt: The purpose of this volume is to present a coherent collection of overviews of recent Russian research in Control Theory and Nonlinear Dynamics written by active investigators in these fields. It is needless to say that the contribution of the scientists of the former Soviet Union to the development of nonlinear dynamics and control was significant and that their scientific schools and research community have highly evolved points of view, accents and depth which complemented, enhanced and sometimes inspired research directions in the West. With scientific exchange strongly increasing, there is still a consider able number of Eastern publications unknown to the Western community. We have therefore encouraged the authors to produce extended bibliogra phies in their papers. The particular emphasis of this volume is on the treatment of uncer tain systems in a deterministic setting-a field highly developed in the former Soviet Union and actively investigated in the West. The topics are concentrated around the three main branches of un certain dynamics which are the theory of Differential Games, the set membership approach to Evolution, Estimation and Control and the the ory of Robust Stabilization. The application of these techniques to non linear systems as well as the global optimization of the latter are also among the issues treated in this volume.

Generalized Solutions of a System of Differential Equations of the First Order and Elliptic Type with Discontinuous Coefficients

Author : Bogdan Bojarski
Publisher :
ISBN 13 : 9789513934866
Total Pages : 72 pages
Book Rating : 4.9/5 (348 download)

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Book Synopsis Generalized Solutions of a System of Differential Equations of the First Order and Elliptic Type with Discontinuous Coefficients by : Bogdan Bojarski

Download or read book Generalized Solutions of a System of Differential Equations of the First Order and Elliptic Type with Discontinuous Coefficients written by Bogdan Bojarski and published by . This book was released on 2009 with total page 72 pages. Available in PDF, EPUB and Kindle. Book excerpt: