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Since the publication of "Spectral Methods in Fluid Dynamics", spectral methods, particularly in their multidomain version, have become firmly established as a mainstream tool for scientific and engineering computation. While retaining the tight integration between the theoretical and practical aspects of spectral methods that was the hallmark of the earlier book, Canuto et al. now incorporate the many improvements in the algorithms and the theory of spectral methods that have been made since 1988. The initial treatment Fundamentals in Single Domains discusses the fundamentals of the approximation of solutions to ordinary and partial differential equations on single domains by expansions in smooth, global basis functions. The first half of the book provides the algorithmic details of orthogonal expansions, transform methods, spectral discretization of differential equations plus their boundary conditions, and solution of the discretized equations by direct and iterative methods. The second half furnishes a comprehensive discussion of the mathematical theory of spectral methods on single domains, including approximation theory, stability and convergence, and illustrative applications of the theory to model boundary-value problems. Both the algorithmic and theoretical discussions cover spectral methods on tensor-product domains, triangles and tetrahedra. All chapters are enhanced with material on the Galerkin with numerical integration version of spectral methods. The discussion of direct and iterative solution methods is greatly expanded as are the set of numerical examples that illustrate the key properties of the various types of spectral approximations and the solution algorithms. A companion book "Evolution to Complex Geometries and Applications to Fluid Dynamics" contains an extensive survey of the essential algorithmic and theoretical aspects of spectral methods for complex geometries and provides detailed discussions of spectral algorithms for fluid dynamics in simple and complex geometries. .

Spectral theory (Mathematics) --- Numerical analysis. --- Differential equations, Partial --- Numerical solutions. --- Numerical analysis --- Mathematical analysis --- Functional analysis --- Hilbert space --- Measure theory --- Transformations (Mathematics) --- Computer science --- Mathematical physics. --- Hydraulic engineering. --- Classical and Continuum Physics. --- Numerical and Computational Physics, Simulation. --- Computational Mathematics and Numerical Analysis. --- Mathematical Methods in Physics. --- Engineering Fluid Dynamics. --- Fluid- and Aerodynamics. --- Mathematics. --- Engineering, Hydraulic --- Engineering --- Fluid mechanics --- Hydraulics --- Shore protection --- Physical mathematics --- Physics --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Mathematics --- Continuum physics. --- Physics. --- Computer mathematics. --- Fluid mechanics. --- Fluids. --- Mechanics --- Hydrostatics --- Permeability --- Hydromechanics --- Continuum mechanics --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Classical field theory --- Continuum physics

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The purpose of the volume is to provide a support for a first course in Mathematical Analysis, along the lines of the recent Programme Specifications for mathematical teaching in European universities. The contents are organised to appeal especially to Engineering, Physics and Computer Science students, all areas in which mathematical tools play a crucial role. Basic notions and methods of differential and integral calculus for functions of one real variable are presented in a manner that elicits critical reading and prompts a hands-on approach to concrete applications. The layout has a specifically-designed modular nature, allowing the instructor to make flexible didactical choices when planning an introductory lecture course. The book may in fact be employed at three levels of depth. At the elementary level the student is supposed to grasp the very essential ideas and familiarise with the corresponding key techniques. Proofs to the main results befit the intermediate level, together with several remarks and complementary notes enhancing the treatise. The last, and farthest-reaching, level consists of links to online material, which enable the strongly motivated reader to explore further into the subject. Definitions and properties are furnished with substantial examples to stimulate the learning process. Over 350 solved exercises complete the text, at least half of which guide the reader to the solution.

Mathematics. --- Analysis. --- Functional Analysis. --- Integral Equations. --- Integral Transforms, Operational Calculus. --- Ordinary Differential Equations. --- Partial Differential Equations. --- Global analysis (Mathematics). --- Functional analysis. --- Integral equations. --- Integral Transforms. --- Differential Equations. --- Differential equations, partial. --- Mathématiques --- Analyse globale (Mathématiques) --- Analyse fonctionnelle --- Equations intégrales --- Electronic books. -- local. --- Mathematical analysis. --- Applied Mathematics --- Engineering & Applied Sciences --- Math --- 517.1 Mathematical analysis --- Mathematical analysis --- Analysis (Mathematics). --- Integral transforms. --- Operational calculus. --- Differential equations. --- Partial differential equations. --- Science --- Partial differential equations --- 517.91 Differential equations --- Differential equations --- Operational calculus --- Electric circuits --- Integral equations --- Transform calculus --- Transformations (Mathematics) --- Equations, Integral --- Functional equations --- Functional analysis --- Functional calculus --- Calculus of variations --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Analyse mathématique. --- Advanced calculus --- Analysis (Mathematics) --- Algebra