Ce livre constitue une introduction au Calcul Scientifique. Son objectif est de présenter des méthodes numériques permettant de résoudre avec un ordinateur certains problèmes mathématiques qui ne peuvent être traités simplement avec un papier et un crayon. La présentation de ces méthodes est rendue vivante par le recours systématique à Matlab et Octave dont les principales commandes sont introduites progressivement.
Ce livre présente les bases théoriques et méthodologiques de l'analyse numérique, en se concentrant sur la stabilité, la précision et la complexité des algorithmes. Il couvre des thèmes variés tels que la résolution de systèmes, l'optimisation et l'intégration numérique, avec de nombreux exemples et des programmes MATLAB pour une application concrète.
Le livre a pour but de prA(c)senter les fondements thA(c)oriques et mA(c)thodologiques de l'analyse numA(c)rique. Une attention toute particuliA]re est portA(c)e sur les concepts de stabilitA(c), prA(c)cision et complexitA(c) des algorithmes.Les mA(c)thodes modernes relatives aux thA]mes suivants sont prA(c)sentA(c)es et analysA(c)es en dA(c) rA(c)solution des systA]mes linA(c)aires et non linA(c)aires, approximation polynAmiale, optimisation, intA(c)gration numA(c)rique, polynAme orthogonaux, transformations rapides, A(c)quations diffA(c)rentielles ordinaires.Les techniques prA(c)sentA(c)es sont illustrA(c)es par de nombreux tableaux et figures. Beaucoup d'exemples et de contre-exemples sont proposA(c)s pour permettre au lecteur de dA(c)velopper son sens critique.Une caractA(c)ristique principale du livre rA(c)side dans l'abondance des programmes MATLAB qui accompagnent toutes les mA(c)thodes numA(c)riques proposA(c)es et qui les illustrent par des applications concrA]tes. Le lecteur dA(c)tient ainsi tous les outils pour acquA(c)rir de solides connaissances thA(c)oriques et les appliquer directement sur ordinateur.Cet ouvrage s'adresse aux A(c)tudiants du second cycle des universitA(c)s, aux A(c)lA]ves des A(c)coles d'ingA(c)nieurs, et plus gA(c)nA(c)ralement, A toutes les personnes qui pratiquent le calcul scientifique.
This book delves into the numerical approximation of partial differential equations (PDEs), aiming to illustrate various numerical methods, particularly those derived from the variational formulation of PDEs. It covers stability and convergence analysis, error bounds, and algorithmic implementation aspects, balancing theoretical analysis with practical applications. The text addresses a variety of problems, including linear and nonlinear, steady and time-dependent scenarios, with both smooth and non-smooth solutions. It also explores model equations and several (initial-) boundary value problems relevant to multiple application fields. Part I focuses on general numerical methods for discretizing PDEs, developing a comprehensive theory around Galerkin methods and their variants (such as Petrov Galerkin and generalized Galerkin), alongside collocation methods for spatial discretization. This theoretical framework is then applied to two significant numerical subspace realizations: the finite element method (including conforming, non-conforming, mixed, and hybrid types) and the spectral method (utilizing Legendre and Chebyshev expansions).
"One of the purposes of this book is to provide the mathematical foundations of numerical methods, to analyze their basic theoretical properties (stability, accuracy, and computational complexity), and to demonstrate their performances on examples and counterexamples, which outline their pros and cons. This is done using the MATLAB software environment, which is user-friendly and widely adopted. Within any specific class of problems, the most appropriate scientific computing algorithms are reviewed, their theoretical analyses are carried out, and the expected results are verified on a MATLAB computer implementation. Every chapter is supplied with examples, exercises, and applications of the discussed theory to the solution of real-life problems.". "This book is addressed to senior undergraduate and graduate students with particular focus on degree courses in engineering, mathematics, physics, and computer science. The attention paid to the applications and the related development of software makes it valuable also to researchers and users of scientific computing in a large variety of professional fields."--BOOK JACKET.
Focusing on the numerical modeling of partial differential equations, this book covers essential concepts and emphasizes algorithmic and computer implementation. It includes straightforward programs designed for practical use, making it accessible for readers interested in applying these techniques in computational settings.
Simulating the behavior of a human heart, predicting tomorrow's weather, optimizing the aerodynamics of a sailboat, finding the ideal cooking time for a hamburger: to solve these problems, cardiologists, meteorologists, sportsmen, and engineers can count on math help.
When Big Data and Mathematical Models Meet
Covid-19 has shown us the importance of mathematical and statistical models to interpret reality, provide forecasts, and explore future scenarios. Algorithms, artificial neural networks, and machine learning help us discover the opportunities and pitfalls of a world governed by mathematics and artificial intelligence.
Numerische Mathematik ist ein zentrales Gebiet der Mathematik, das für vielfältige Anwendungen die Grundlage bildet und das alle Studierenden der Mathematik, Ingenieurwissenschaften, Informatik und Physik kennenlernen. Das vorliegende Lehrbuch ist eine didaktisch exzellente, besonders sorgfältig ausgearbeitete Einführung für Anfänger. Eines der Ziele dieses Buches ist es, die mathematischen Grundlagen der numerischen Methoden zu liefern, ihre grundlegenden theoretischen Eigenschaften (Stabilität, Genauigkeit, Komplexität)zu analysieren, und ihre Leistungsfähigkeit an Beispielen und Gegenbeispielen mittels MATLAB zu demonstrieren. Die besondere Sorgfalt, die den Anwendungen und betreffenden Softwareentwicklungen gewidmet wurde, macht das vorliegende Werk auch für Studenten mit abgeschlossenem Studium, Wissenschaftler und Anwender des wissenschaftlichen Rechnens in vielen Berufsfeldern zu einem unverzichtbaren Arbeitsmittel.
In questo testo si introducono i concetti elementari di modellistica numerica di problemi differenziali alle derivate parziali. Si considerano le classiche equazioni lineari ellittiche, paraboliche ed iperboliche, ma anche altre equazioni, quali quelle di diffusione e trasporto, di Navier-Stokes, e le leggi di conservazione, e si forniscono numerosi esempi fisici che stanno alla base di tali equazioni. Quindi si analizzano metodi di risoluzione numerica basati su elementi finiti, differenze finite e metodi spettrali. Il volume h adatto agli studenti dei corsi di laurea di indirizzo scientifico (Ingegneria, Fisica, Matematica, Chimica, Scienza dell'Informazione) e consigliato ai ricercatori del mondo accademico ed extra-accademico che vogliano avvicinarsi a questo interessante ramo della matematica applicata.