De Gruyter Series in Applied and Numerical Mathematics
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Edited by:
Rémi Abgrall
The series is devoted to the publication of high-level monographs and specialized graduate texts which cover the whole spectrum of applied mathematics, including its numerical aspects. The focus of the series is on the interplay between mathematical and numerical analysis, and also on its applications to mathematical models in the physical and life sciences.
The aim of the series is to be an active forum for the dissemination of up-to-date information in the form of authoritative works that will serve the applied mathematics community as the basis for further research.
Editorial Board
Rémi Abgrall, Universität Zürich, Switzerland
José Antonio Carrillo de la Plata, University of Oxford, UK
Jean-Michel Coron, Université Pierre et Marie Curie, Paris, France
Athanassios S. Fokas, Cambridge University, UK
Irene Fonseca, Carnegie Mellon University, Pittsburgh, USA
Topics
Methods for solving problems of mathematical physics can be divided into the following four classes.
Analytical methods (the method of separation of variables, the method of characteristics, the method of Green's functions, etc.) methods have a relatively low degree of universality, i.e. focused on solving rather narrow classes of problems.
Approximate analytical methods (projection, variational methods, small parameter method, operational methods, various iterative methods) are more versatile than analytical ones.
Numerical methods (finite difference method, direct method, control volume method, finite element method, etc.) are very universal methods.
Probabilistic methods (Monte Carlo methods) are highly versatile. Can be used to calculate discontinuous solutions. However, they require large amounts of calculations and, as a rule, they lose with the computational complexity of the above methods when solving such problems to which these methods are applicable.
Comparing methods for solving problems of mathematical physics, it is impossible to give unconditional primacy to any of them. Any of them may be the best for solving problems of a certain class.
The proposed method of moving nodes for boundary value problems of differential equations combines a combination of numerical and analytical methods. In this case, we can obtain, on the one hand, an approximately analytical solution of the problem, which is not related to the methods listed above. On the other hand, this method allows one to obtain compact discrete approximations of the original problem. Note that obtaining an approximately analytical solution of differential equations is based on numerical methods. The nature of numerical methods also allows obtaining an approximate analytical expression for solving differential equations
This book delves into the recent findings and research methods in the existence and regularity theory for Non-Newtonian Fluids with Variable Power-Law. The aim of this book is not only to introduce recent results and research methods in the existence and regularity theory, such as higher integrability, higher differentiability, and Holder continuity for flows of non-Newtonian fluids with variable power-laws, but also to summarize much of the existing literature concerning these topics. While this book mainly focuses on steady-state flows of non-Newtonian fluids, the methods and ideas presented in this book can be applied to unsteady flows (as discussed in Chapter 7) and other related problems such as complex non-Newtonian fluids, plasticity, elasticity, p(x)-Laplacian type systems, and so on.
The book is intended for researchers and graduate students in the field of mathematical fluid mechanics and partial differential equations with variable exponents. It is expected to contribute to the advancement of mathematics and its applications.
Metamaterials are advanced composite materials which have exotic and powerful properties. Their complicated microstructures make metamaterials challenging to model, requiring the use of sophisticated mathematical techniques. This book uses a from-first-principles approach (based on boundary integral methods and asymptotic analysis) to study a class of high-contrast metamaterials. These mathematical techniques are applied to the problem of designing graded metamaterials that replicate the function of the cochlea.
Nonlinear evolution equations are widely used to describe nonlinear phenomena in natural and social sciences. However, they are usually quite difficult to solve in most instances. This book introduces the finite difference methods for solving nonlinear evolution equations. The main numerical analysis tool is the energy method. This book covers the difference methods for the initial-boundary value problems of twelve nonlinear partial differential equations. They are Fisher equation, Burgers' equation, regularized long-wave equation, Korteweg-de Vries equation, Camassa-Holm equation, Schrödinger equation, Kuramoto-Tsuzuki equation, Zakharov equation, Ginzburg-Landau equation, Cahn-Hilliard equation, epitaxial growth model and phase field crystal model. This book is a monograph for the graduate students and science researchers majoring in computational mathematics and applied mathematics. It will be also useful to all researchers in related disciplines.
This book on finite element-based computational methods for solving incompressible viscous fluid flow problems shows readers how to apply operator splitting techniques to decouple complicated computational fluid dynamics problems into a sequence of relatively simpler sub-problems at each time step, such as hemispherical cavity flow, cavity flow of an Oldroyd-B viscoelastic flow, and particle interaction in an Oldroyd-B type viscoelastic fluid. Efficient and robust numerical methods for solving those resulting simpler sub-problems are introduced and discussed. Interesting computational results are presented to show the capability of methodologies addressed in the book.
This book uses numerical analysis as the main tool to investigate methods in machine learning and neural networks. The efficiency of neural network representations for general functions and for polynomial functions is studied in detail, together with an original description of the Latin hypercube method and of the ADAM algorithm for training. Furthermore, unique features include the use of Tensorflow for implementation session, and the description of on going research about the construction of new optimized numerical schemes.
This two-volume monograph is a comprehensive and up-to-date presentation of the theory and applications of kinetic equations. The second volume covers discrete velocity models of the Boltzmann equation, results on the Landau equation, and numerical (deterministic and stochastic) methods for the solution of kinetic equations.
This two-volume monograph is a comprehensive and up-to-date presentation of the theory and applications of kinetic equations. The first volume covers many-particle dynamics, Maxwell models of the Boltzmann equation (including their exact and self-similar solutions), and hydrodynamic limits beyond the Navier-Stokes level.
This book studies the existence and uniqueness of solutions to parabolic-type equations with irregular coefficients and/or initial conditions. It elaborates on the DiPerna-Lions theory of renormalized solutions to linear transport equations and related equations, and also examines the connection between the results on the partial differential equation and the well-posedness of the underlying stochastic/ordinary differential equation.
This book contains a first systematic study of compressible fluid flows subject to stochastic forcing. The bulk is the existence of dissipative martingale solutions to the stochastic compressible Navier-Stokes equations. These solutions are weak in the probabilistic sense as well as in the analytical sense. Moreover, the evolution of the energy can be controlled in terms of the initial energy. We analyze the behavior of solutions in short-time (where unique smooth solutions exists) as well as in the long term (existence of stationary solutions). Finally, we investigate the asymptotics with respect to several parameters of the model based on the energy inequality.
Contents
Part I: Preliminary results
Elements of functional analysis
Elements of stochastic analysis
Part II: Existence theory
Modeling fluid motion subject to random effects
Global existence
Local well-posedness
Relative energy inequality and weak–strong uniqueness
Part III: Applications
Stationary solutions
Singular limits
Scientists and engineers are mainly using Richardson extrapolation as a computational tool for increasing the accuracy of various numerical algorithms for the treatment of systems of ordinary and partial differential equations and for improving the computational efficiency of the solution process by the automatic variation of the time-stepsizes. A third issue, the stability of the computations, is very often the most important one and, therefore, it is the major topic studied in all chapters of this book.
Clear explanations and many examples make this text an easy-to-follow handbook for applied mathematicians, physicists and engineers working with scientific models based on differential equations.
Contents
The basic properties of Richardson extrapolation
Richardson extrapolation for explicit Runge-Kutta methods
Linear multistep and predictor-corrector methods
Richardson extrapolation for some implicit methods
Richardson extrapolation for splitting techniques
Richardson extrapolation for advection problems
Richardson extrapolation for some other problems
General conclusions
This monograph is concerned with free-boundary problems of partial differential equations arising in the physical sciences and in engineering. The existence and uniqueness of solutions to the Hele-Shaw problem are derived and techniques to deal with the Muskat problem are discussed. Based on these, mathematical models for the dynamics of cracks in underground rocks and in-situ leaching are developed.
Contents
Introduction
The Hele–Shaw problem
A joint motion of two immiscible viscous fluids
Mathematical models of in-situ leaching
Dynamics of cracks in rocks
Elements of continuum mechanics
