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.
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.
This introduction to Scientific Computing illustrates several numerical methods for the computer solution of certain classes of mathematical problems. The authors show how to compute the zeros or the integrals of continuous functions, solve linear systems, approximate functions by polynomials and construct accurate approximations for the solution of differential equations. To make the presentation concrete, the programming environment Matlab is adopted as a faithful companion.
"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.
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).