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Author: M.B. Oberguggenberger
Publisher: Elsevier
ISBN: 0080872921
Category : Mathematics
Languages : en
Pages : 449
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Book Description
This work inaugurates a new and general solution method for arbitrary continuous nonlinear PDEs. The solution method is based on Dedekind order completion of usual spaces of smooth functions defined on domains in Euclidean spaces. However, the nonlinear PDEs dealt with need not satisfy any kind of monotonicity properties. Moreover, the solution method is completely type independent. In other words, it does not assume anything about the nonlinear PDEs, except for the continuity of their left hand term, which includes the unkown function. Furthermore the right hand term of such nonlinear PDEs can in fact be given any discontinuous and measurable function.
Author: M.B. Oberguggenberger
Publisher: Elsevier
ISBN: 0080872921
Category : Mathematics
Languages : en
Pages : 449
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Book Description
This work inaugurates a new and general solution method for arbitrary continuous nonlinear PDEs. The solution method is based on Dedekind order completion of usual spaces of smooth functions defined on domains in Euclidean spaces. However, the nonlinear PDEs dealt with need not satisfy any kind of monotonicity properties. Moreover, the solution method is completely type independent. In other words, it does not assume anything about the nonlinear PDEs, except for the continuity of their left hand term, which includes the unkown function. Furthermore the right hand term of such nonlinear PDEs can in fact be given any discontinuous and measurable function.
Author: Elemer Rosinger
Publisher: CreateSpace
ISBN: 9781503334809
Category :
Languages : en
Pages : 202
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Book Description
Contrary to widespread perception, there has ever since 1994 been a unified, general, that is, type independent theory for the existence and regularity of solutions for very large classes of nonlinear systems of PDEs, with possibly associated initial and/or boundary value problems, see [21,22], and for further developments [1-3,47-56,58,64-66]. This solution method is based on the Dedekind order completion of suitable spaces of piece-wise smooth functions on the Euclidean domains of definition of the respective PDEs. All the solutions obtained have a blanket, universal, minimal regularity property, namely, they can be assimilated with usual measurable functions or even with Hausdorff continuous functions on the respective Euclidean domains. It is important to note that the use of the order completion method does not require any monotonicity conditions on the nonlinear systems of PDEs involved. One of the major advantages of the order completion method is that it eliminates the algebra based dichotomy "linear versus nonlinear" PDEs, treating both cases equally. Furthermore, the order completion method does not introduce the dichotomy "monotonous versus non-monotonous" PDEs. None of the known functional analytic methods can exhibit such a powerful general performance, since in addition to topology, such methods are significantly based on algebra, and vector spaces do inevitably differentiate between linear and nonlinear entities. The power of the order completion method is also shown in its ability to solve equations far more general than PDEs, and give in fact necessary and sufficient conditions for the existence of their solutions, as well as explicit expressions for the solutions obtained.In the case of PDEs, another advantage of the order completion method is that in treating initial and/or boundary value problems it avoids the considerable additional difficulties which the usual functional analytic methods encounter.Nevertheless, there are certain basic connections and similarities between the usual functional analytic methods in solving PDEs, and on the other hand, the order completion method. And in fact, the ancient equation x^2 = 2 which had two and a half millennia ago created such a terrible conundrum for Pythagoras, has ultimately been solved by a simple version of the order completion method. Indeed, its irrational solution was obtained in the set R of real numbers, while that set itself was obtained by the Dedekind order completion of the set Q of rational numbers, when that latter set is considered with its natural order relation.
Author: Ivan Lirkov
Publisher: Springer Science & Business Media
ISBN: 3642125344
Category : Computers
Languages : en
Pages : 855
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Book Description
The 7th International Conference on Large-Scale Scienti?c Computations (LSSC 2009) was held in Sozopol, Bulgaria, June 4–8, 2009. The conference was organized and sponsored by the Institute for Parallel Processing at the B- garian Academy of Sciences. The conference was devoted to the 70th birthday anniversary of Professor Zahari Zlatev. The Bulgarian Academy of Sciences awarded him the Marin Drinov medal on ribbon for his outstanding results in environmental mat- matics and for his contributions to the Bulgarian mathematical society and the Academy of Sciences. The plenary invited speakers and lectures were: – P. Arbenz, “?Finite Element Analysis of Human Bone Structures” – Y. Efendiev, “Mixed Multiscale Finite Element Methods Using Limited Global Information” – U. Langer, “Fast Solvers for Non-Linear Time-Harmonic Problems” – T. Manteu?el, “First-Order System Least-Squares Approach to Resistive Magnetohydrodynamic Equations” – K. Sabelfeld, “Stochastic Simulation for Solving Random Boundary Value Problems and Some Applications” – F. Tro ¨ltzsch,“OnFinite ElementErrorEstimatesforOptimalControlPr- lems with Elliptic PDEs” – Z. Zlatev, “On Some Stability Properties of the Richardson Extrapolation Applied Together with the ?-method” The success of the conference and the present volume in particular are an outcome of the joint e?orts of many partnersfrom various institutions and or- nizations. Firstwe wouldlike to thank allthe membersofthe Scienti?c Comm- tee for their valuable contribution forming the scienti?c face of the conference, as well as for their help in reviewing contributed papers. We especially thank the organizers of the special sessions.
Author: Elemer E. Rosinger
Publisher: Springer Science & Business Media
ISBN: 9401590761
Category : Mathematics
Languages : en
Pages : 247
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Book Description
This book presents global actions of arbitrary Lie groups on large classes of generalised functions by using a novel parametric approach. This new method extends and completes earlier results of the author and collaborators, in which global Lie group actions on generalised functions were only defined in the case of projectable or fibre-preserving Lie group actions. The parametric method opens the possibility of dealing with vastly larger classes of Lie semigroup actions which still transform solutions into solutions. These Lie semigroups can contain arbitrary noninvertible smooth mappings. Thus, they cannot be subsemigroups of Lie groups. Audience: This volume is addressed to graduate students and researchers involved in solving linear and nonlinear partial differential equations, and in particular, in dealing with the Lie group symmetries of their classical or generalised solutions.
Author: Michael Oberguggenberger
Publisher: Routledge
ISBN: 1351428039
Category : Mathematics
Languages : en
Pages : 400
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Book Description
Questions regarding the interplay of nonlinearity and the creation and propagation of singularities arise in a variety of fields-including nonlinear partial differential equations, noise-driven stochastic partial differential equations, general relativity, and geometry with singularities. A workshop held at the Erwin-Schrödinger International Institute for Mathematical Physics in Vienna investigated these questions and culminated in this volume of invited papers from experts in the fields of nonlinear partial differential equations, structure theory of generalized functions, geometry and general relativity, stochastic partial differential equations, and nonstandard analysis. The authors provide the latest research relevant to work in partial differential equations, mathematical physics, and nonlinear analysis. With a focus on applications, this books provides a compilation of recent approaches to the problem of singularities in nonlinear models. The theory of differential algebras of generalized functions serves as the central theme of the project, along with its interrelations with classical methods.
Author: M. V. Karasev
Publisher: American Mathematical Soc.
ISBN: 9780821833360
Category : Asymptotic symmetry (Physics)
Languages : en
Pages : 298
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Book Description
The collection consists of four papers in different areas of mathematical physics united by the intrinsic coherence of the asymptotic methods used. The papers describe both the known results and most recent achievements, as well as new concepts and ideas in mathematical analysis of quantum and wave problems. In the introductory paper ``Quantization and Intrinsic Dynamics'' a relationship between quantization of symplectic manifolds and nonlinear wave equations is described and discussed from the viewpoint of the weak asymptotics method (asymptotics in distributions) and the semiclassical approximation method. It also explains a hidden dynamic geometry that arises when using these methods. Three other papers discuss applications of asymptotic methods to the construction of wave-type solutions of nonlinear PDE's, to the theory of semiclassical approximation (in particular, the Whitham method) for nonlinear second-order ordinary differential equations, and to the study of the Schrodinger type equations whose potential wells are sufficiently shallow that the discrete spectrum contains precisely one point. All the papers contain detailed references and are oriented not only to specialists in asymptotic methods, but also to a wider audience of researchers and graduate students working in partial differential equations and mathematical physics.
Author: Robert Hermann
Publisher: Math-Sci Press
ISBN: 9780915692453
Category : Mathematics
Languages : en
Pages : 286
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Book Description
Author: Joshua Allensworth Leslie
Publisher: American Mathematical Soc.
ISBN: 0821829645
Category : Mathematics
Languages : en
Pages : 226
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Book Description
This volume contains papers based on some of the talks given at the NSF-CBMS conference on ``The Geometrical Study of Differential Equations'' held at Howard University (Washington, DC). The collected papers present important recent developments in this area, including the treatment of nontransversal group actions in the theory of group invariant solutions of PDEs, a method for obtaining discrete symmetries of differential equations, the establishment of a group-invariant version of the variational complex based on a general moving frame construction, the introduction of a new variational complex for the calculus of difference equations and an original structural investigation of Lie-Backlund transformations. The book opens with a modern and illuminating overview of Lie's line-sphere correspondence and concludes with several interesting open problems arising from symmetry analysis of PDEs. It offers a rich source of inspiration for new or established researchers in the field. This book can serve nicely as a companion volume to a forthcoming book written by the principle speaker at the conference, Professor Niky Kamran, to be published in the AMS series, CBMS Regional Conference Series in Mathematics.
Author: Nigel J Cutland
Publisher: CRC Press
ISBN: 1000716821
Category : Mathematics
Languages : en
Pages : 273
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Book Description
This book contains expository papers and articles reporting on recent research by leading world experts in nonstandard mathematics, arising from the International Colloquium on Nonstandard Mathematics held at the University of Aveiro, Portugal in July 1994. Nonstandard mathematics originated with Abraham Robinson, and the body of ideas that have developed from this theory of nonstandard analysis now vastly extends Robinson's work with infinitesimals. The range of applications includes measure and probability theory, stochastic analysis, differential equations, generalised functions, mathematical physics and differential geometry, moreover, the theory has implicaitons for the teaching of calculus and analysis. This volume contains papers touching on all of the abovbe topics, as well as a biographical note about Abraham Robinson based on the opening address given by W.A>J> Luxemburg - who knew Robinson - to the Aveiro conference which marked the 20th anniversary of Robinson's death. This book will be of particular interest to students and researchers in nonstandard analysis, measure theory, generalised functions and mathematical physics.
Author: Todor Boyanov
Publisher: Springer Science & Business Media
ISBN: 3540709401
Category : Computers
Languages : en
Pages : 741
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Book Description
This book constitutes the thoroughly refereed post-proceedings of the 6th International Conference on Numerical Methods and Applications, NMA 2006, held in Borovets, Bulgaria, in August 2006. The 84 revised full papers presented together with 3 invited papers were carefully reviewed and selected from 111 submissions. The papers are organized in topical sections on numerical methods for hyperbolic problems, robust preconditioning solution methods, Monte Carlo and quasi-Monte Carlo for diverse applications, metaheuristics for optimization problems, uncertain/control systems and reliable numerics, interpolation and quadrature processes, large-scale computations in environmental modelling, and contributed talks.
