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Author: Yihong Du Publisher: World Scientific ISBN: 9812566244 Category : Mathematics Languages : en Pages : 202
Book Description
The maximum principle induces an order structure for partial differential equations, and has become an important tool in nonlinear analysis. This book is the first of two volumes to systematically introduce the applications of order structure in certain nonlinear partial differential equation problems.The maximum principle is revisited through the use of the Krein-Rutman theorem and the principal eigenvalues. Its various versions, such as the moving plane and sliding plane methods, are applied to a variety of important problems of current interest. The upper and lower solution method, especially its weak version, is presented in its most up-to-date form with enough generality to cater for wide applications. Recent progress on the boundary blow-up problems and their applications are discussed, as well as some new symmetry and Liouville type results over half and entire spaces. Some of the results included here are published for the first time.
Author: Yihong Du Publisher: World Scientific ISBN: 9812566244 Category : Mathematics Languages : en Pages : 202
Book Description
The maximum principle induces an order structure for partial differential equations, and has become an important tool in nonlinear analysis. This book is the first of two volumes to systematically introduce the applications of order structure in certain nonlinear partial differential equation problems.The maximum principle is revisited through the use of the Krein-Rutman theorem and the principal eigenvalues. Its various versions, such as the moving plane and sliding plane methods, are applied to a variety of important problems of current interest. The upper and lower solution method, especially its weak version, is presented in its most up-to-date form with enough generality to cater for wide applications. Recent progress on the boundary blow-up problems and their applications are discussed, as well as some new symmetry and Liouville type results over half and entire spaces. Some of the results included here are published for the first time.
Author: Yihong Du Publisher: World Scientific ISBN: 9812774440 Category : Mathematics Languages : en Pages : 202
Book Description
The maximum principle induces an order structure for partial differential equations, and has become an important tool in nonlinear analysis. This book is the first of two volumes to systematically introduce the applications of order structure in certain nonlinear partial differential equation problems. The maximum principle is revisited through the use of the Krein-Rutman theorem and the principal eigenvalues. Its various versions, such as the moving plane and sliding plane methods, are applied to a variety of important problems of current interest. The upper and lower solution method, especially its weak version, is presented in its most up-to-date form with enough generality to cater for wide applications. Recent progress on the boundary blow-up problems and their applications are discussed, as well as some new symmetry and Liouville type results over half and entire spaces. Some of the results included here are published for the first time. Sample Chapter(s). Chapter 1: Krein-Rutman Theorem and the Principal Eigenvalue (128 KB). Contents: KreinOCoRutman Theorem and the Principal Eigenvalue; Maximum Principles Revisited; The Moving Plane Method; The Method of Upper and Lower Solutions; The Logistic Equation; Boundary Blow-Up Problems; Symmetry and Liouville Type Results Over Half and Entire Spaces. Readership: Researchers and postgraduate students in partial differential equations."
Author: Dumitru Motreanu Publisher: Springer Science & Business Media ISBN: 1461493234 Category : Mathematics Languages : en Pages : 459
Book Description
This book focuses on nonlinear boundary value problems and the aspects of nonlinear analysis which are necessary to their study. The authors first give a comprehensive introduction to the many different classical methods from nonlinear analysis, variational principles, and Morse theory. They then provide a rigorous and detailed treatment of the relevant areas of nonlinear analysis with new applications to nonlinear boundary value problems for both ordinary and partial differential equations. Recent results on the existence and multiplicity of critical points for both smooth and nonsmooth functional, developments on the degree theory of monotone type operators, nonlinear maximum and comparison principles for p-Laplacian type operators, and new developments on nonlinear Neumann problems involving non-homogeneous differential operators appear for the first time in book form. The presentation is systematic, and an extensive bibliography and a remarks section at the end of each chapter highlight the text. This work will serve as an invaluable reference for researchers working in nonlinear analysis and partial differential equations as well as a useful tool for all those interested in the topics presented.
Author: Jane Cronin Publisher: American Mathematical Soc. ISBN: 0821815113 Category : Electronic books Languages : en Pages : 198
Book Description
The topological methods based on fixed-point theory and on local topological degree which have been developed by Leray, Schauder, Nirenberg, Cesari and others for the study of nonlinear differential equations are here described in detail, beginning with elementary considerations. The reader is not assumed to have any knowledge of topology beyond the theory of point sets in Euclidean n-space which ordinarily forms part of a course in advanced calculus. The methods are first developed for Euclidean n-space and applied to the study of existence and stability of periodic and almost-periodic solutions of systems of ordinary differential equations, both quasi-linear and with ``large'' nonlinearities. Then, after being extended to infinite-dimensional ``function-spaces'', these methods are applied to integral equations, partial differential equations and further problems concerning periodic solutions of ordinary differential equations.
Author: Dominic G B Edelen Publisher: World Scientific ISBN: 9814505684 Category : Science Languages : en Pages : 341
Book Description
The purpose of the book is to provide research workers in applied mathematics, physics, and engineering with practical geometric methods for solving systems of nonlinear partial differential equations. The first two chapters provide an introduction to the more or less classical results of Lie dealing with symmetries and similarity solutions. The results, however, are presented in the context of contact manifolds rather than the usual jet bundle formulation and provide a number of new conclusions. The remaining three chapters present essentially new methods of solution that are based on recent publications of the authors'. The text contains numerous fully worked examples so that the reader can fully appreciate the power and scope of the new methods. In effect, the problem of solving systems of nonlinear partial differential equations is reduced to the problem of solving families of autonomous ordinary differential equations. This allows the graphs of solutions of the system of partial differential equations to be realized as certain leaves of a foliation of an appropriately defined contact manifold. In fact, it is often possible to obtain families of solutions whose graphs foliate an open subset of the contact manifold. These ideas are extended in the final chapter by developing the theory of transformations that map a foliation of a contact manifold onto a foliation. This analysis gives rise to results of surprising depth and practical significance. In particular, an extended Hamilton-Jacobi method for solving systems of partial differential equations is obtained.
Author: Vicentiu D. Radulescu Publisher: Hindawi Publishing Corporation ISBN: 9774540395 Category : Differential equations, Elliptic Languages : en Pages : 205
Book Description
This book provides a comprehensive introduction to the mathematical theory of nonlinear problems described by elliptic partial differential equations. These equations can be seen as nonlinear versions of the classical Laplace equation, and they appear as mathematical models in different branches of physics, chemistry, biology, genetics, and engineering and are also relevant in differential geometry and relativistic physics. Much of the modern theory of such equations is based on the calculus of variations and functional analysis. Concentrating on single-valued or multivalued elliptic equations with nonlinearities of various types, the aim of this volume is to obtain sharp existence or nonexistence results, as well as decay rates for general classes of solutions. Many technically relevant questions are presented and analyzed in detail. A systematic picture of the most relevant phenomena is obtained for the equations under study, including bifurcation, stability, asymptotic analysis, and optimal regularity of solutions. The method of presentation should appeal to readers with different backgrounds in functional analysis and nonlinear partial differential equations. All chapters include detailed heuristic arguments providing thorough motivation of the study developed later on in the text, in relationship with concrete processes arising in applied sciences. A systematic description of the most relevant singular phenomena described in this volume includes existence (or nonexistence) of solutions, unicity or multiplicity properties, bifurcation and asymptotic analysis, and optimal regularity. The book includes an extensive bibliography and a rich index, thus allowing for quick orientation among the vast collection of literature on the mathematical theory of nonlinear phenomena described by elliptic partial differential equations.
Author: Themistocles M Rassias Publisher: World Scientific ISBN: 9814513652 Category : Mathematics Languages : en Pages : 571
Book Description
Contents: Fixed Point Theory and Nonlinear Problems (Th Rassias)Global Linearization Iterative Methods and Nonlinear Partial Differential Equations III (M Altman)On Generalized Power Series and Generalized Operational Calculus and Its Application (M Al-Bassam)Multiple Solutions to Parametrized Nonlinear Differential Systems from Nielsen Fixed Point Theory (R Brown)The topology of Ind-Affine Sets (P Cherenack)Almost Approximately Polynomial Functions (P Cholewa)Cohomology Classes and Foliated Manifolds (M Craioveanu & M Puta)Bifurcation and Nonlinear Instability in Applied Mathematics (L Debnath)The Stability of Weakly Additive Functional (H Drljevic)Index Theory for G-Bundle Pairs with Applications to Borsuk-Ulam Type Theorems for G-Sphere Bundles (E Fadell & S Husseini)Nonlinear Approximation and Moment Problem (J S Hwang & G D Lin)Periods in Equicontinuous Topological Dynamical Systems (A Iwanik et al.)Continuation Theorems for Semi-Linear Equations in Banach Spaces: A Survey (J Mawhin & K Rybakowski)On Contractifiable Self-Mappings (P Meyers)Normal Structures and Nonexpansive Mappings in Banach Spaces (J Nelson et al.): Survey on Uniqueness and Classification Theorems for Minimal Surfaces (Th Rassias)Contractive Definitions (B Rhoades)On KY Fan's Theorem and Its Applications (S Singh)Fixed Points of Amenable Semigroups of Differentiable Operators (P Soardi)Research Problems on Nonlinear Equations (Th Rassias) Readership: Mathematicians and applied scientists. Keywords:Nonlinear Analysis;Nonlinear Partial Differential Equations III;Polynomial Functions;Cohomology Classes;Foliated Manifolds;Topological Dynamical Systems;Minimal Surfaces;Differentiable Operators;Nonlinear Equations
Author: E.E. Rosinger Publisher: Elsevier ISBN: 9780080872759 Category : Mathematics Languages : en Pages : 379
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: Yihong Du
Author: Yihong Du
Author: Yihong Du
Author: Yihong Du
Author: 
Author: Dumitru Motreanu
Author: Jane Cronin
Author: Dominic G B Edelen
Author: Vicentiu D. Radulescu
Author: Themistocles M Rassias
Author: E.E. Rosinger