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The material collected in this volume discusses the present as well as expected future directions of development of the field with particular emphasis on applications. The seven survey articles present different topics in Evolutionary PDE's, written by leading experts.- Review of new results in the area - Continuation of previous volumes in the handbook series covering evolutionary PDEs - New content coverage of DE applications
Differential equations. --- Nonlinear optics. --- Optics, Nonlinear --- Optics --- Lasers --- 517.91 Differential equations --- Differential equations
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Differential equations --- Equations différentielles --- Periodicals. --- Périodiques --- Differential equations. --- Mathematical Sciences --- Applied Mathematics --- Ejournals --- UML. --- differential equations --- Equations, Differential --- Bessel functions --- Calculus --- 517.91 Differential equations --- Mathematics
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Mathematical models --- Differential equations --- Differential equations. --- Mathematical models. --- Models, Mathematical --- Equations, Differential --- 517.91 Differential equations --- Simulation methods --- Bessel functions --- Calculus
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A mathematical theory is introduced in this book to unify a large class of nonlinear partial differential equation (PDE) models for better understanding and analysis of the physical and biological phenomena they represent. The so-called mean field approximation approach is adopted to describe the macroscopic phenomena from certain microscopic principles for this unified mathematical formulation. Two key ingredients for this approach are the notions of “duality” according to the PDE weak solutions and “hierarchy” for revealing the details of the otherwise hidden secrets, such as physical mystery hidden between particle density and field concentration, quantized blow up biological mechanism sealed in chemotaxis systems, as well as multi-scale mathematical explanations of the Smoluchowski–Poisson model in non-equilibrium thermodynamics, two-dimensional turbulence theory, self-dual gauge theory, and so forth. This book shows how and why many different nonlinear problems are inter-connected in terms of the properties of duality and scaling, and the way to analyze them mathematically.
Differential equations, Partial. --- Mathematics. --- Mean field theory. --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Differential equations. --- Partial differential equations. --- Ordinary Differential Equations. --- Partial Differential Equations. --- Partial differential equations --- 517.91 Differential equations --- Differential equations --- Math --- Science --- Differential Equations. --- Differential equations, partial.
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This concise book covers the classical tools of PDE theory used in today's science and engineering: characteristics, the wave propagation, the Fourier method, distributions, Sobolev spaces, fundamental solutions, and Green's functions. The approach is problem-oriented, giving the reader an opportunity to master solution techniques. The theoretical part is rigorous and with important details presented with care. Hints are provided to help the reader restore the arguments to their full rigor. Many examples from physics are intended to keep the book intuitive and to illustrate the applied nature of the subject. The book is useful for a higher-level undergraduate course and for self-study.
Differential equations, Partial. --- Differential equations. --- Electronic books. -- local. --- Differential equations, Partial --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- 517.91 Differential equations --- Differential equations --- Partial differential equations --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Partial differential equations. --- Analysis. --- Partial Differential Equations. --- 517.1 Mathematical analysis --- Mathematical analysis --- Math --- Science --- Global analysis (Mathematics). --- Differential equations, partial. --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic
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"Approximate and Renormgroup Symmetries" deals with approximate transformation groups, symmetries of integro-differential equations and renormgroup symmetries. It includes a concise and self-contained introduction to basic concepts and methods of Lie group analysis, and provides an easy-to-follow introduction to the theory of approximate transformation groups and symmetries of integro-differential equations. The book is designed for specialists in nonlinear physics - mathematicians and non-mathematicians - interested in methods of applied group analysis for investigating nonlinear problems in physical science and engineering. Dr. N.H. Ibragimov is a professor at the Department of Mathematics and Science, Research Centre ALGA, Sweden. He is widely regarded as one of the world's foremost experts in the field of symmetry analysis of differential equations; Dr. V. F. Kovalev is a leading scientist at the Institute for Mathematical Modeling, Russian Academy of Science, Moscow.
Differential equations. --- Lie groups. --- Lie-Gruppe -- Renormierungsgruppe -- Integrodifferentialgleichung -- Symmetrie. --- Renormalization group. --- Symmetry (Mathematics). --- Differential equations --- Lie groups --- Symmetry (Mathematics) --- Renormalization group --- Mathematics --- Calculus --- Physical Sciences & Mathematics --- Invariance (Mathematics) --- Groups, Lie --- 517.91 Differential equations --- Mathematics. --- Applied mathematics. --- Engineering mathematics. --- Applications of Mathematics. --- Engineering --- Engineering analysis --- Mathematical analysis --- Math --- Science --- Group theory --- Automorphisms --- Lie algebras --- Symmetric spaces --- Topological groups
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The name of C. G. J. Jacobi is familiar to every student of mathematics, thanks to the Jacobion determinant, the Hamilton-Jacobi equations in dynamics, and the Jacobi identity for vector fields. Best known for his contributions to the theory of elliptic and abelian functions, Jacobi is also known for his innovative teaching methods and for running the first research seminar in pure mathematics. A record of his lectures on Dynamics given in 1842-43 at Konigsberg, edited by A. Clebsch, has been available in the original German. This is an English translation. It is not just a historical document; the modern reader can learn much about the subject directly from one of its great masters.
Mathematics. --- Mathematics, general. --- Dynamics. --- Hamilton-Jacobi equations. --- Differential equations. --- Math --- Science --- 517.91 Differential equations --- Differential equations --- Equations, Hamilton-Jacobi --- Equations, Jacobi-Hamilton --- Jacobi-Hamilton equations --- Calculus of variations --- Differential equations, Partial --- Hamiltonian systems --- Mechanics --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Physics --- Statics
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Integration of differential equations is a central problem in mathematics and several approaches have been developed by studying analytic, algebraic, and algorithmic aspects of the subject. One of these is Differential Galois Theory, developed by Kolchin and his school, and another originates from the Soliton Theory and Inverse Spectral Transform method, which was born in the works of Kruskal, Zabusky, Gardner, Green and Miura. Many other approaches have also been developed, but there has so far been no intersection between them. This unique introduction to the subject finally brings them together, with the aim of initiating interaction and collaboration between these various mathematical communities. The collection includes a LMS Invited Lecture Course by Michael F. Singer, together with some shorter lecture courses and review articles, all based upon a mini-programme held at the International Centre for Mathematical Sciences (ICMS) in Edinburgh.
Differential equations --- Algebraic number theory --- Differential calculus --- Differential algebra --- Equations différentielles --- Théorie des nombres algébriques --- Calcul différentiel --- Algèbre différentielle --- Congresses --- Congrès --- Algebraic theory --- Calculus, Differential --- Calculus --- 517.91 Differential equations --- Algebra, Differential --- Differential fields --- Algebraic fields
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Problems in Real Analysis: Advanced Calculus on the Real Axis features a comprehensive collection of challenging problems in mathematical analysis that aim to promote creative, non-standard techniques for solving problems. This self-contained text offers a host of new mathematical tools and strategies which develop a connection between analysis and other mathematical disciplines, such as physics and engineering. A broad view of mathematics is presented throughout; the text is excellent for the classroom or self-study. It is intended for undergraduate and graduate students in mathematics, as well as for researchers engaged in the interplay between applied analysis, mathematical physics, and numerical analysis. Key features: *Uses competition-inspired problems as a platform for training typical inventive skills; *Develops basic valuable techniques for solving problems in mathematical analysis on the real axis and provides solid preparation for deeper study of real analysis; *Includes numerous examples and interesting, valuable historical accounts of ideas and methods in analysis; *Offers a systematic path to organizing a natural transition that bridges elementary problem-solving activity to independent exploration of new results and properties.
Measure theory. Mathematical integration --- Mathematical analysis --- Electronic books. -- local. --- Functions of real variables. --- Mathematical analysis. --- Applied Mathematics --- Engineering & Applied Sciences --- Real variables --- 517.1 Mathematical analysis --- Mathematics. --- Analysis (Mathematics). --- Differential equations. --- Applied mathematics. --- Engineering mathematics. --- Numerical analysis. --- Analysis. --- Real Functions. --- Numerical Analysis. --- Ordinary Differential Equations. --- Applications of Mathematics. --- Functions of complex variables --- Global analysis (Mathematics). --- Differential Equations. --- 517.91 Differential equations --- Differential equations --- Math --- Science --- Analysis, Global (Mathematics) --- Differential topology --- Geometry, Algebraic --- Engineering --- Engineering analysis --- Mathematics
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The construction of strict Lyapunov functions is a challenging problem that is of significant ongoing research interest. Although converse Lyapunov function theory guarantees the existence of strict Lyapunov functions in many situations, the Lyapunov functions that converse theory provides are often abstract and nonexplicit, and therefore may not lend themselves to engineering applications. Often, even when a system is known to be stable, one still needs explicit Lyapunov functions; however, once an appropriate strict Lyapunov function has been constructed, many robustness and stabilization problems can be solved almost immediately through standard feedback designs or robustness arguments. By contrast, non-strict Lyapunov functions are often readily constructed, e.g., from passivity, backstepping, or forwarding (especially in the time varying context), or by using the Hamiltonian in Euler–Lagrange systems. Constructions of Strict Lyapunov Functions contains a broad repertoire of Lyapunov constructions for nonlinear systems, focusing on methods for transforming non-strict Lyapunov functions into strict ones. Many important classes of dynamics are covered: Jurdjevic–Quinn systems; time-varying systems satisfying LaSalle or Matrosov conditions; slowly and rapidly time-varying systems; adaptively controlled dynamics; and hybrid systems. The explicitness and simplicity of the constructions make them suitable for feedback design, and for quantifying the effects of uncertainty. Readers will benefit from the authors’ mathematical rigor and unifying, design-oriented approach, as well as the numerous worked examples, covering several applications that are of compelling interest including the adaptive control of chemostats and the stabilization of underactuated ships. Researchers from applied-mathematical and engineering backgrounds working in nonlinear and dynamical systems will find this monograph to be most valuable and for graduate students of control theory it will also be an authoritative source of information on a very important subject.
Lyapunov functions. --- Differential equations. --- 517.91 Differential equations --- Differential equations --- Functions, Liapunov --- Liapunov functions --- Systems theory. --- Vibration. --- Control and Systems Theory. --- Systems Theory, Control. --- Vibration, Dynamical Systems, Control. --- Control, Robotics, Mechatronics. --- Cycles --- Mechanics --- Sound --- Control engineering. --- System theory. --- Dynamical systems. --- Dynamics. --- Robotics. --- Mechatronics. --- Mechanical engineering --- Microelectronics --- Microelectromechanical systems --- Automation --- Machine theory --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Physics --- Statics --- Systems, Theory of --- Systems science --- Science --- Control engineering --- Control equipment --- Control theory --- Engineering instruments --- Programmable controllers --- Philosophy --- Lyapunov functions --- Lyapunov stability
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