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Author: Jan Harm Van der Walt
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
In this thesis, we establish a general and type independent theory for the existence and regularity of generalized solutions of large classes of systems of nonlinear partial differential equations (PDEs). In this regard, our point of departure is the Order Completion Method. The spaces of generalized functions to which the solutions of such systems of PDEs belong are constructed as the completions of suitable uniform convergence spaces of normal lower semi-continuous functions. It is shown that large classes of systems of nonlinear PDEs admit generalized solutions in the mentioned spaces of generalized functions. Furthermore, the generalized solutions that we construct satisfy a blanket regularity property, in the sense that such solutions may be assimilated with usual normal lower semi-continuous functions. These fundamental existence and regularity results are obtain as applications of basic topological processes, namely, the completion of uniform convergence spaces, and elementary properties of real valued continuous functions. In particular, those techniques from functional analysis which are customary in the study of nonlinear PDEs are not used at all. The mentioned sophisticated methods of functional analysis are used only to obtain additional regularity properties of the generalized solutions of systems of nonlinear PDEs, and are thus relegated to a secondary role. Over and above the mentioned blanket regularity of the solutions, it is shown that for a large class of equations, the generalized solutions are in fact usual classical solutions of the respective system of equations everywhere except on a closed, nowhere dense subset of the domain of definition of the system of equations. This result is obtained under minimal assumptions on the smoothness of the equations, and is an application of convenient compactness theorems for sets of sufficiently smooth functions with respect to suitable topologies on spaces of such functions. As an application of the existence and regularity results presented here, we obtain for the first time in the literature an extension of the celebrated Cauchy-Kovalevskaia Theorem, on its own general and type independent grounds, to equations that are not analytic.
Author: Jan Harm Van der Walt
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
In this thesis, we establish a general and type independent theory for the existence and regularity of generalized solutions of large classes of systems of nonlinear partial differential equations (PDEs). In this regard, our point of departure is the Order Completion Method. The spaces of generalized functions to which the solutions of such systems of PDEs belong are constructed as the completions of suitable uniform convergence spaces of normal lower semi-continuous functions. It is shown that large classes of systems of nonlinear PDEs admit generalized solutions in the mentioned spaces of generalized functions. Furthermore, the generalized solutions that we construct satisfy a blanket regularity property, in the sense that such solutions may be assimilated with usual normal lower semi-continuous functions. These fundamental existence and regularity results are obtain as applications of basic topological processes, namely, the completion of uniform convergence spaces, and elementary properties of real valued continuous functions. In particular, those techniques from functional analysis which are customary in the study of nonlinear PDEs are not used at all. The mentioned sophisticated methods of functional analysis are used only to obtain additional regularity properties of the generalized solutions of systems of nonlinear PDEs, and are thus relegated to a secondary role. Over and above the mentioned blanket regularity of the solutions, it is shown that for a large class of equations, the generalized solutions are in fact usual classical solutions of the respective system of equations everywhere except on a closed, nowhere dense subset of the domain of definition of the system of equations. This result is obtained under minimal assumptions on the smoothness of the equations, and is an application of convenient compactness theorems for sets of sufficiently smooth functions with respect to suitable topologies on spaces of such functions. As an application of the existence and regularity results presented here, we obtain for the first time in the literature an extension of the celebrated Cauchy-Kovalevskaia Theorem, on its own general and type independent grounds, to equations that are not analytic.
Author: E.E. Rosinger
Publisher: Elsevier
ISBN: 0080872573
Category : Science
Languages : en
Pages : 429
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Book Description
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.
Author: E.E. Rosinger
Publisher: Elsevier
ISBN: 0080872751
Category : Mathematics
Languages : en
Pages : 403
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Book Description
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.
Author: Boling Guo
Publisher: World Scientific
ISBN: 9814544264
Category :
Languages : en
Pages : 267
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Book Description
Contents: Direct and Inverse Diffraction by Periodic Structures (G Bao)Weak Flow of H-Systems (Y-M Chen)Strongly Compact Attractor for Dissipative Zakharov Equations (B-L Guo et al.)C∞-Solutions of Generalized Porous Medium Equations (M Ôtani & Y Sugiyama)Cauchy Problem for Generalized IMBq Equation (G-W Chen & S-B Wang)Inertial Manifolds for a Nonlocal Kuramoto–Sivashinsky Equation (J-Q Duan et al.)Weak Solutions of the Generalized Magnetic Flow Equations (S-H He & Z-D Dai)The Solution of Hammerstein Integral Equation Without Coercive Conditions (Y-L Shu)Global Behaviour of the Solution of Nonlinear Forest Evolution Equation (D-J Wang)Uniqueness of Generalized Solutions for Semiconductor Equations (J-S Xing & Y Hu)On the Vectorial Hamilton–Jacobi System (B-S Yan)An Integrable Hamiltonian System Associated with cKdV Hierarchy (J-S Zhang et al.)and other papers Readership: Mathematicians. Keywords:Diffraction;Weak Flow;Zakharov Equations;Porous Medium Equations;Cauchy Problem;IMBq Equation;Kuramoto-Sivashinsky Equation;Magnetic Flow Equations;Hammerstein Integral Equation;Nonlinear Forest Evolution Equation;Uniqueness;Generalized Solutions;Semiconductor Equations;HamiltonâJacobi System;Hamiltonian System;cKdV Hierarchy
Author: Inna Shingareva
Publisher: Springer Science & Business Media
ISBN: 370910517X
Category : Mathematics
Languages : en
Pages : 372
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Book Description
The emphasis of the book is given in how to construct different types of solutions (exact, approximate analytical, numerical, graphical) of numerous nonlinear PDEs correctly, easily, and quickly. The reader can learn a wide variety of techniques and solve numerous nonlinear PDEs included and many other differential equations, simplifying and transforming the equations and solutions, arbitrary functions and parameters, presented in the book). Numerous comparisons and relationships between various types of solutions, different methods and approaches are provided, the results obtained in Maple and Mathematica, facilitates a deeper understanding of the subject. Among a big number of CAS, we choose the two systems, Maple and Mathematica, that are used worldwide by students, research mathematicians, scientists, and engineers. As in the our previous books, we propose the idea to use in parallel both systems, Maple and Mathematica, since in many research problems frequently it is required to compare independent results obtained by using different computer algebra systems, Maple and/or Mathematica, at all stages of the solution process. One of the main points (related to CAS) is based on the implementation of a whole solution method (e.g. starting from an analytical derivation of exact governing equations, constructing discretizations and analytical formulas of a numerical method, performing numerical procedure, obtaining various visualizations, and comparing the numerical solution obtained with other types of solutions considered in the book, e.g. with asymptotic solution).
Author: Tran Duc Van
Publisher: CRC Press
ISBN: 9781584880165
Category : Mathematics
Languages : en
Pages : 256
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Book Description
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.
Author: J.M. Ball
Publisher: Springer Science & Business Media
ISBN: 9400971893
Category : Mathematics
Languages : en
Pages : 476
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Book Description
This volume contains the proceedings of a NATO/London Mathematical Society Advanced Study Institute held in Oxford from 25 July - 7 August 1982. The institute concerned the theory and applications of systems of nonlinear partial differential equations, with emphasis on techniques appropriate to systems of more than one equation. Most of the lecturers and participants were analysts specializing in partial differential equations, but also present were a number of numerical analysts, workers in mechanics, and other applied mathematicians. The organizing committee for the institute was J.M. Ball (Heriot-Watt), T.B. Benjamin (Oxford), J. Carr (Heriot-Watt), C.M. Dafermos (Brown), S. Hildebrandt (Bonn) and J.S. pym (Sheffield) . The programme of the institute consisted of a number of courses of expository lectures, together with special sessions on different topics. It is a pleasure to thank all the lecturers for the care they took in the preparation of their talks, and S.S. Antman, A.J. Chorin, J.K. Hale and J.E. Marsden for the organization of their special sessions. The institute was made possible by financial support from NATO, the London Mathematical Society, the u.S. Army Research Office, the u.S. Army European Research Office, and the u.S. National Science Foundation. The lectures were held in the Mathematical Institute of the University of Oxford, and residential accommodation was provided at Hertford College.
Author: Elemer E. Rosinger
Publisher:
ISBN: 9780444703101
Category : Differential equations, Nonlinear
Languages : en
Pages : 409
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Book Description
Author: Andrei D. Polyanin
Publisher: CRC Press
ISBN: 1135440816
Category : Mathematics
Languages : en
Pages : 835
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Book Description
The Handbook of Nonlinear Partial Differential Equations is the latest in a series of acclaimed handbooks by these authors and presents exact solutions of more than 1600 nonlinear equations encountered in science and engineering--many more than any other book available. The equations include those of parabolic, hyperbolic, elliptic and other types, and the authors pay special attention to equations of general form that involve arbitrary functions. A supplement at the end of the book discusses the classical and new methods for constructing exact solutions to nonlinear equations. To accommodate different mathematical backgrounds, the authors avoid wherever possible the use of special terminology, outline some of the methods in a schematic, simplified manner, and arrange the equations in increasing order of complexity. Highlights of the Handbook:
Author: W. F. Ames
Publisher: Academic Press
ISBN: 1483221504
Category : Mathematics
Languages : en
Pages : 335
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Book Description
Nonlinear Partial Differential Equations: A Symposium on Methods of Solution is a collection of papers presented at the seminar on methods of solution for nonlinear partial differential equations, held at the University of Delaware, Newark, Delaware on December 27-29, 1965. The sessions are divided into four Symposia: Analytic Methods, Approximate Methods, Numerical Methods, and Applications. Separating 19 lectures into chapters, this book starts with a presentation of the methods of similarity analysis, particularly considering the merits, advantages and disadvantages of the methods. The subsequent chapters describe the fundamental ideas behind the methods for the solution of partial differential equation derived from the theory of dynamic programming and from finite systems of ordinary differential equations. These topics are followed by reviews of the principles to the lubrication approximation and compressible boundary-layer flow computation. The discussion then shifts to several applications of nonlinear partial differential equations, including in electrical problems, two-phase flow, hydrodynamics, and heat transfer. The remaining chapters cover other solution methods for partial differential equations, such as the synergetic approach. This book will prove useful to applied mathematicians, physicists, and engineers.
