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The book is intended as an advanced undergraduate or first-year graduate course for students from various disciplines, including applied mathematics, physics and engineering. It has evolved from courses offered on partial differential equations (PDEs) over the last several years at the Politecnico di Milano. These courses had a twofold purpose: on the one hand, to teach students to appreciate the interplay between theory and modeling in problems arising in the applied sciences, and on the other to provide them with a solid theoretical background in numerical methods, such as finite elements. Accordingly, this textbook is divided into two parts. The first part, chapters 2 to 5, is more elementary in nature and focuses on developing and studying basic problems from the macro-areas of diffusion, propagation and transport, waves and vibrations. In turn the second part, chapters 6 to 11, concentrates on the development of Hilbert spaces methods for the variational formulation and the analysis of (mainly) linear boundary and initial-boundary value problems.

Mathematics. --- Mathematical Modeling and Industrial Mathematics. --- Partial Differential Equations. --- Mathematical Applications in the Physical Sciences. --- Appl.Mathematics/Computational Methods of Engineering. --- Differential equations, partial. --- Engineering mathematics. --- Mathématiques --- Mathématiques de l'ingénieur --- Mathematics --- Differential equations, Partial --- Differential equations, Elliptic --- Boundary value problems --- Geometry, Differential --- Functions --- Diffusion --- Engineering & Applied Sciences --- Applied Mathematics --- Numerical solutions --- Gases --- Liquids --- Analysis (Mathematics) --- Differential geometry --- Boundary conditions (Differential equations) --- Elliptic differential equations --- Elliptic partial differential equations --- Linear elliptic differential equations --- Partial differential equations --- Math --- Partial differential equations. --- Mathematical physics. --- Mathematical models. --- Applied mathematics. --- Mathematical and Computational Engineering. --- Engineering --- Engineering analysis --- Mathematical analysis --- Physical mathematics --- Physics --- Models, Mathematical --- Simulation methods --- Geometry, Differential. --- Functions. --- Diffusion. --- Differential equations, Partial. --- Differential equations, Elliptic. --- Boundary value problems. --- Numerical solutions.

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Il testo costituisce una introduzione alla teoria delle equazioni a derivate parziali, strutturata in modo da abituare il lettore ad una sinergia tra modellistica e aspetti teorici. La prima parte riguarda le più note equazioni della fisica-matematica, idealmente raggruppate nelle tre macro-aree diffusione, propagazione e trasporto, onde e vibrazioni. Nella seconda parte si presenta la formulazione variazionale dei principali problemi iniziali e/o al bordo e la loro analisi con i metodi dell'Analisi Funzionale negli spazi di Hilbert.

Mathematics. --- Partial differential equations. --- Partial Differential Equations. --- Partial differential equations --- Math --- Differential equations, partial.

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The book is intended as an advanced undergraduate or first-year graduate course for students from various disciplines, including applied mathematics, physics and engineering. It has evolved from courses offered on partial differential equations (PDEs) over the last several years at the Politecnico di Milano. These courses had a twofold purpose: on the one hand, to teach students to appreciate the interplay between theory and modeling in problems arising in the applied sciences, and on the other to provide them with a solid theoretical background in numerical methods, such as finite elements. Accordingly, this textbook is divided into two parts. The first part, chapters 2 to 5, is more elementary in nature and focuses on developing and studying basic problems from the macro-areas of diffusion, propagation and transport, waves and vibrations. In turn the second part, chapters 6 to 11, concentrates on the development of Hilbert spaces methods for the variational formulation and the analysis of (mainly) linear boundary and initial-boundary value problems.

Mathematics. --- Partial differential equations. --- Mathematical physics. --- Mathematical models. --- Applied mathematics. --- Engineering mathematics. --- Mathematical Modeling and Industrial Mathematics. --- Partial Differential Equations. --- Mathematical Applications in the Physical Sciences. --- Appl.Mathematics/Computational Methods of Engineering. --- Engineering --- Engineering analysis --- Models, Mathematical --- Physical mathematics --- Physics --- Partial differential equations --- Math --- Mathematics --- Mathematical analysis --- Simulation methods --- Science --- Differential equations, partial. --- Mathematical and Computational Engineering. --- Differential equations, Partial.

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This textbook presents problems and exercises at various levels of difficulty in the following areas: Classical Methods in PDEs (diffusion, waves, transport, potential equations); Basic Functional Analysis and Distribution Theory; Variational Formulation of Elliptic Problems; and Weak Formulation for Parabolic Problems and for the Wave Equation. Thanks to the broad variety of exercises with complete solutions, it can be used in all basic and advanced PDE courses.

Mathematics. --- Functional Analysis. --- Partial Differential Equations. --- Mathematical Applications in the Physical Sciences. --- Mathematical Physics. --- Functional analysis. --- Differential equations, partial. --- Mathématiques --- Analyse fonctionnelle --- Differential equations, Partial --- Differential equations, Elliptic --- Boundary value problems --- Geometry, Differential --- Functions --- Diffusion --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Numerical solutions --- Gases --- Liquids --- Analysis (Mathematics) --- Differential geometry --- Boundary conditions (Differential equations) --- Elliptic differential equations --- Elliptic partial differential equations --- Linear elliptic differential equations --- Partial differential equations --- Partial differential equations. --- Mathematical physics. --- Physical mathematics --- Physics --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Math --- Science --- Boundary value problems. --- Differential equations, Elliptic. --- Differential equations, Partial. --- Diffusion. --- Geometry, Differential. --- Functions. --- Numerical solutions. --- Numerical analysis --- Differential equations --- Mathematical analysis --- Numbers, Complex --- Set theory --- Separation (Technology) --- Solution (Chemistry) --- Solutions, Solid --- Matter --- Packed towers --- Semiconductor doping --- Differential equations, Linear --- Functions of complex variables --- Mathematical physics --- Initial value problems --- Properties

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This book presents a concise, clear, and consistent account of the methodology of phase synchronization, an extension of modal analysis to decouple any linear system in real space. It expounds on the novel theory of phase synchronization and presents recent advances, while also providing relevant background on classical decoupling theories that are used in structural analysis. The theory is illustrated with a broad range of examples. The theoretical development is also supplemented by applications to engineering problems. In addition, the methodology is implemented in a MATLAB algorithm which can be used to solve many of the illustrative examples in the book. This book is suited for researchers, practicing engineers, and graduate students in various fields of engineering, mathematics, and physical science. Provides a concise review of classical decoupling techniques; Explains the physical intuition behind the method of phase synchronization; Illustrates the relationship between phase synchronization and classical decoupling theories and demonstrates the utility of phase synchronization; Reinforces concepts by illustrative examples; Provides a MATLAB algorithm for simulations.

Vibration. --- Dynamical systems. --- Dynamics. --- Differential equations. --- Partial differential equations. --- Phase transitions (Statistical physics). --- Vibration, Dynamical Systems, Control. --- Ordinary Differential Equations. --- Partial Differential Equations. --- Phase Transitions and Multiphase Systems. --- Phase changes (Statistical physics) --- Phase transitions (Statistical physics) --- Phase rule and equilibrium --- Statistical physics --- Partial differential equations --- 517.91 Differential equations --- Differential equations --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Cycles --- Sound