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This book is designed as an advanced undergraduate or a first-year graduate course for students from various disciplines like applied mathematics, physics, engineering. The main purpose is on the one hand to train the students to appreciate the interplay between theory and modelling in problems arising in the applied sciences; on the other hand to give them a solid theoretical background for numerical methods, such as finite elements. Accordingly, this textbook is divided into two parts. The first one has a rather elementary character with the goal of developing and studying basic problems from the macro-areas of diffusion, propagation and transport, waves and vibrations. Ideas and connections with concrete aspects are emphasized whenever possible, in order to provide intuition and feeling for the subject. For this part, a knowledge of advanced calculus and ordinary differential equations is required. Also, the repeated use of the method of separation of variables assumes some basic results from the theory of Fourier series, which are summarized in an appendix. The main topic of the second part is the development of Hilbert space methods for the variational formulation and analysis of linear boundary and initial-boundary value problemsemph{. }% Given the abstract nature of these chapters, an effort has been made to provide intuition and motivation for the various concepts and results. The understanding of these topics requires some basic knowledge of Lebesgue measure and integration, summarized in another appendix. At the end of each chapter, a number of exercises at different level of complexity is included. The most demanding problems are supplied with answers or hints. The exposition if flexible enough to allow substantial changes without compromising the comprehension and to facilitate a selection of topics for a one or two semester course.

Differential equations, Partial. --- Electronic books. -- local. --- Laplacian operator. --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Operator, Laplacian --- Mathematics. --- Partial differential equations. --- Partial Differential Equations. --- Partial differential equations --- Math --- Science --- Differential equations, Partial --- Differential equations, partial.

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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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Cet ouvrage est consacré à une introduction aux problèmes inverses elliptiques et paraboliques. L’ objectif est de présenter quelques méthodes récentes pour établir des résultats d’unicité et de stabilité. Seront traités quelques problèmes inverses elliptiques devenus maintenant classiques, tels que la conductivité inverse, la détection de corrosion ou de fissures et les problèmes spectraux inverses. Parmi les problèmes inverses paraboliques considérés figurent le problème classique de retrouver une distribution initiale de la chaleur et la localisation de sources, de chaleur ou de pollution par exemple. Les problèmes d’identification de non linéarités seront aussi étudiés. Cet ouvrage s’adresse à tous ceux qui souhaitent s’ intéresser à l’analyse mathématique des problèmes inverses. This volume is devoted to an introduction of elliptic and parabolic inverse problems. The goal is to present some recent methods for establishing uniqueness and stability results. A number of classical elliptic inverse problems are studied, e.g. the inverse conductivity problem, the detection of corrosion or cracks and inverse spectral problems. Among the parabolic inverse problems, the classic problem of finding an initial distribution of heat and the location of sources is considered. This volume will be of interest to all those who want to learn the mathematical analysis of inverse problems.

Differential equations. --- Mathematical analysis. --- Mathematics. --- Inverse problems (Differential equations) --- Differential equations, Elliptic --- Differential equations, Parabolic --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Differential equations, Elliptic. --- Differential equations, Parabolic. --- Elliptic differential equations --- Elliptic partial differential equations --- Linear elliptic differential equations --- Parabolic differential equations --- Parabolic partial differential equations --- Partial differential equations. --- Partial Differential Equations. --- Differential equations, Partial --- Differential equations, Linear --- Differential equations, partial. --- Partial differential equations --- Inverse problems (Differential equations).

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This volume contains lecture notes on some topics in geometric analysis, a growing mathematical subject which uses analytical techniques, mostly of partial differential equations, to treat problems in differential geometry and mathematical physics. The presentation of the material should be rather accessible to non-experts in the field, since the presentation is didactic in nature. The reader will be provided with a survey containing some of the most exciting topics in the field, with a series of techniques used to treat such problems.

Geometric analysis --- Differential equations, Partial --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Geometric analysis PDEs (Geometric partial differential equations) --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Partial differential equations. --- Physics. --- Analysis. --- Partial Differential Equations. --- Mathematical Methods in Physics. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Partial differential equations --- 517.1 Mathematical analysis --- Mathematical analysis --- Math --- Science --- Geometry --- Global analysis (Mathematics). --- Differential equations, partial. --- Mathematical physics. --- Physical mathematics --- Physics --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Cetraro <2007>

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This volume offers researchers the opportunity to catch up with important developments in the field of numerical analysis and scientific computing and to get in touch with state-of-the-art numerical techniques. The book has three parts. The first one is devoted to the use of wavelets to derive some new approaches in the numerical solution of PDEs, showing in particular how the possibility of writing equivalent norms for the scale of Besov spaces allows to develop some new methods. The second part provides an overview of the modern finite-volume and finite-difference shock-capturing schemes for systems of conservation and balance laws, with emphasis on providing a unified view of such schemes by identifying the essential aspects of their construction. In the last part a general introduction is given to the discontinuous Galerkin methods for solving some classes of PDEs, discussing cell entropy inequalities, nonlinear stability and error estimates.

Differential equations, Partial -- Numerical solutions -- Congresses. --- Electronic books. -- local. --- Galerkin methods -- Congresses. --- Wavelets (Mathematics) -- Congresses. --- Calculus --- Applied Mathematics --- Mathematics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Differential equations, Partial --- Wavelets (Mathematics) --- Galerkin methods --- Numerical solutions --- Sinc-Galerkin methods --- Sinc methods --- Mathematics. --- Partial differential equations. --- Numerical analysis. --- Numerical Analysis. --- Partial Differential Equations. --- Numerical analysis --- Differential equations, partial. --- Partial differential equations --- Mathematical analysis