Generalized Characteristics Of First Order Pdes
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Generalized Characteristics Of First Order Pdes
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Author : Arik Melikyan
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Generalized Characteristics Of First Order Pdes written by Arik Melikyan and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
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.
Generalized Characteristics Of First Order Pdes
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Author : Arik Melikyan
language : en
Publisher: Springer Science & Business Media
Release Date : 1998-05-15
Generalized Characteristics Of First Order Pdes written by Arik Melikyan and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998-05-15 with Mathematics categories.
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.
Generalized Characteristics Of First Order Pdes
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Author : Arik A. Melikyan
language : en
Publisher:
Release Date : 1998-01-01
Generalized Characteristics Of First Order Pdes written by Arik A. Melikyan and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998-01-01 with Control theory categories.
Generalized Solutions Of First Order Pdes
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Author : Andrei I. Subbotin
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-06-29
Generalized Solutions Of First Order Pdes written by Andrei I. Subbotin and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-06-29 with Mathematics categories.
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].
The Characteristic Method And Its Generalizations For First Order Nonlinear Partial Differential Equations
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Author : Tran Duc Van
language : en
Publisher: CRC Press
Release Date : 1999-06-25
The Characteristic Method And Its Generalizations For First Order Nonlinear Partial Differential Equations written by Tran Duc Van and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1999-06-25 with Mathematics categories.
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.
Handbook Of First Order Partial Differential Equations
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Author : Andrei D. Polyanin
language : en
Publisher: CRC Press
Release Date : 2001-11-15
Handbook Of First Order Partial Differential Equations written by Andrei D. Polyanin and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001-11-15 with Mathematics categories.
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.
Generalized Characteristics Of First Order Pdes
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Author : Arik Artavazdovich Melikiï ̧ a︡n
language : en
Publisher:
Release Date : 1998
Generalized Characteristics Of First Order Pdes written by Arik Artavazdovich Melikiï ̧ a︡n and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998 with Control theory categories.
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.
A Method Of Generalized Characteristics
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Author : Marc A. Berger
language : en
Publisher: American Mathematical Soc.
Release Date : 1982
A Method Of Generalized Characteristics written by Marc A. Berger and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1982 with Mathematics categories.
The classical and Brownian methods of characteristics are generalized to analyze evolution equations of arbitrary order. Calculi of higher orders, analogous to first order classical calculus and second order Ito calculus, are constructed. Solutions of differential equations in the calculi become characteristic propagators of higher order partial differential equations. The solutions of these partial differential equations are then represented as averages of random samples of initial data based on these characteristic flows, in a general sense.
Exact Methods For Nonlinear Pdes
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Author : Andrei D. Polyanin
language : en
Publisher: CRC Press
Release Date : 2025-08-13
Exact Methods For Nonlinear Pdes written by Andrei D. Polyanin and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-08-13 with Mathematics categories.
Exact Methods for Nonlinear PDEs describes effective analytical methods for finding exact solutions to nonlinear differential equations of mathematical physics and other partial differential equations and also demonstrates the practical applications of these methods. It covers the methods of generalized separation of variables, methods of functional separation of variables, the classical method of symmetry reductions, the direct method of symmetry reductions, the method of weak symmetry reductions, and the method of differential constraints. The book presents several simple methods for finding exact solutions to nonlinear partial differential equations (PDEs). These methods do not require specialized knowledge and aim to minimize intermediate calculations. For the first time, it discusses the application of nonrigorous, intuitive reasoning in deriving exact solutions to nonlinear PDEs. Each section provides numerous examples, problems, and exercises to help readers develop practical skills in applying the methods. The material is illustrated with equations of mass and heat transfer, hydrodynamics, wave theory, nonlinear optics, and other nonlinear equations of mathematical physics. The key points that distinguish this book from others in the field include: • it presents many methods in a simpler and more visual format; • it describes a number of simple methods for constructing exact solutions to nonlinear PDEs and delay PDEs; • it emphasizes and details the practical use of non-rigorous reasoning to derive exact solutions for nonlinear PDEs. The book is intended for a diverse audience, including researchers, university professors, engineers, postgraduates, and students specializing in applied mathematics, theoretical physics, and engineering sciences.
Numerical Methods For Engineers And Scientists
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Author : Joe D. Hoffman
language : en
Publisher: CRC Press
Release Date : 2018-10-03
Numerical Methods For Engineers And Scientists written by Joe D. Hoffman and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-10-03 with Mathematics categories.
Emphasizing the finite difference approach for solving differential equations, the second edition of Numerical Methods for Engineers and Scientists presents a methodology for systematically constructing individual computer programs. Providing easy access to accurate solutions to complex scientific and engineering problems, each chapter begins with objectives, a discussion of a representative application, and an outline of special features, summing up with a list of tasks students should be able to complete after reading the chapter- perfect for use as a study guide or for review. The AIAA Journal calls the book "...a good, solid instructional text on the basic tools of numerical analysis."