Praise for the First Edition". . . fills a considerable gap in the numerical analysis literature by providing a self-contained treatment . . . this is an important work written in a clear style . . . warmly recommended to any graduate student or researcher in the field of the numerical solution of partial differential equations."—SIAM ReviewTime-Dependent Problems and Difference Methods, Second Edition continues to provide ...

This book explores the most significant computational methods and the history of their development. It begins with the earliest mathematical / numerical achievements made by the Babylonians and the Greeks, followed by the period beginning in the 16th century. For several centuries the main scientific challenge concerned the mechanics of planetary dynamics, and the book describes the basic numerical methods of that time.In turn, at the end of the Second World War scientific computin...

The book of nature is written in the language of mathematics -- Galileo GalileiHow is it possible to predict weather patterns for tomorrow, with access solely to today’s weather data? And how is it possible to predict the aerodynamic behavior of an aircraft that has yet to be built?The answer is computer simulations based on mathematical models – sets of equations – that describe the underlying physical properties. However, these equation...

The first volume of the proceedings of the 7th conference on "Finite Volumes for Complex Applications" (Berlin, June 2014) covers topics that include convergence and stability analysis, as well as investigations of these methods from the point of view of compatibility with physical principles. It collects together the focused invited papers, as well as the reviewed contributions from internationally leading researchers in the field of analysis of finite volume and related methods. Altogeth...

This book places thermodynamics on a system-theoretic foundation so as to harmonize it with classical mechanics. Using the highest standards of exposition and rigor, the authors develop a novel formulation of thermodynamics that can be viewed as a moderate-sized system theory as compared to statistical thermodynamics. This middle-ground theory involves deterministic large-scale dynamical system models that bridge the gap between classical and statistical thermodynamics.The authors'...

Since the 1980s research on quantum computation has dramatically changed the theoretical perspectives of computer science. Quantum computers could enable unprecedented computational power and revolutionize our cryptographic systems, even our entire electronic communication. This textbook gives an introduction to the theory of quantum computation. The author has chosen an elementary and lean theoretical approach, presupposing mathematical and physical knowledge which is standard in undergra...

The finite element method is a technique for solving problems in applied science and engineering. The essence of this book is the application of the finite element method to the solution of boundary and initial-value problems posed in terms of partial differential equations. The method is developed for the solution of Poisson's equation, in a weighted-residual context, and then proceeds to time-dependent and nonlinear problems. The relationship with the variational approach is also explain...

Problem solving in computing is referred to as computational thinking. The theory behind this concept is challenging in its technicalities, yet simple in its ideas. This book introduces the theory of computation from its inception to current form of complexity; from explanations of how the field of computer science was formed using classical ideas in mathematics by Gödel, to conceptualization of the Turing Machine, to its more recent innovations in quantum computation, hypercomputation, va...

Honoring Andrei Agrachev's 60th birthday, this volume presents recent advances in the interaction between Geometric Control Theory and sub-Riemannian geometry. On the one hand, Geometric Control Theory used the differential geometric and Lie algebraic language for studying controllability, motion planning, stabilizability and optimality for control systems. The geometric approach turned out to be fruitful in applications to robotics, vision modeling, mathematical physics etc. On the other ...

This volume, with a foreword by Sir Roger Penrose, discusses the foundations of computation in relation to nature.It focuses on two main questions:The contributors are world-renowned experts who have helped shape a cutting-edge computational understanding of the universe. They discuss computation in the world from a variety of perspectives, ranging from foundational concepts to pragmatic models to ontological conceptions and philosophical implications.The volume provides a state-of-the-art...

This book is a landmark title in the continuous move from integer to non-integer in mathematics: from integer numbers to real numbers, from factorials to the gamma function, from integer-order models to models of an arbitrary order. For historical reasons, the word 'fractional' is used instead of the word 'arbitrary'.This book is written for readers who are new to the fields of fractional derivatives and fractional-order mathematical models, and feel that they need them for developing more...

Lyapunov exponents lie at the heart of chaos theory, and are widely used in studies of complex dynamics. Utilising a pragmatic, physical approach, this self-contained book provides a comprehensive description of the concept. Beginning with the basic properties and numerical methods, it then guides readers through to the most recent advances in applications to complex systems. Practical algorithms are thoroughly reviewed and their performance is discussed, while a broad set of examples illu...