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Books > Science & Mathematics > Mathematics > Numerical analysis
Discover a simple, direct approach that highlights the basics you
need within A FIRST COURSE IN THE FINITE ELEMENT METHOD, 6E. This
unique book is written so both undergraduate and graduate students
can easily comprehend the content without the usual prerequisites,
such as structural analysis. The book is written primarily as a
basic learning tool for students, like you, in civil and mechanical
engineering who are primarily interested in stress analysis and
heat transfer. The text offers ideal preparation for utilizing the
finite element method as a tool to solve practical physical
problems.
This well-respected book introduces readers to the theory and
application of modern numerical approximation techniques. Providing
an accessible treatment that only requires a calculus prerequisite,
the authors explain how, why, and when approximation techniques can
be expected to work-and why, in some situations, they fail. A
wealth of examples and exercises develop readers' intuition, and
demonstrate the subject's practical applications to important
everyday problems in math, computing, engineering, and physical
science disciplines. Three decades after it was first published,
Burden, Faires, and Burden's NUMERICAL ANALYSIS remains the
definitive introduction to a vital and practical subject.
Addresses computational methods that have proven efficient for the
solution of a large variety of nonlinear elliptic problems. These
methods can be applied to many problems in science and engineering,
but this book focuses on their application to problems in continuum
mechanics and physics. This book differs from others on the topic
by:* Presenting examples of the power and versatility of
operator-splitting methods.* Providing a detailed introduction to
alternating direction methods of multipliers and their
applicability to the solution of nonlinear (possibly non-smooth)
problems from science and engineering.* Showing that nonlinear
least-squares methods, combined with operator-splitting and
conjugate gradient algorithms, provide efficient tools for the
solution of highly nonlinear problems.
This work is devoted to fixed point theory as well as the theory of
accretive operators in Banach spaces. The goal is to develop, in
self-contained way, the main results in both theories. Special
emphasis is given to the study how both theories can be used to
study the existence and uniqueness of solution of several types of
partial differential equations and integral equations.
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.
Complex analysis is found in many areas of applied mathematics,
from fluid mechanics, thermodynamics, signal processing, control
theory, mechanical and electrical engineering to quantum mechanics,
among others. And of course, it is a fundamental branch of pure
mathematics. The coverage in this text includes advanced topics
that are not always considered in more elementary texts. These
topics include, a detailed treatment of univalent functions,
harmonic functions, subharmonic and superharmonic functions,
Nevanlinna theory, normal families, hyperbolic geometry, iteration
of rational functions, and analytic number theory. As well, the
text includes in depth discussions of the Dirichlet Problem,
Green's function, Riemann Hypothesis, and the Laplace transform.
Some beautiful color illustrations supplement the text of this most
elegant subject.
Numerical Analysis, Second Edition, is a modern and readable text
for the undergraduate audience. This book covers not only the
standard topics but also some more advanced numerical methods being
used by computational scientists and engineers-topics such as
compression, forward and backward error analysis, and iterative
methods of solving equations-all while maintaining a level of
discussion appropriate for undergraduates. Each chapter contains a
Reality Check, which is an extended exploration of relevant
application areas that can launch individual or team projects.
MATLAB(r) is used throughout to demonstrate and implement numerical
methods. The Second Edition features many noteworthy improvements
based on feedback from users, such as new coverage of Cholesky
factorization, GMRES methods, and nonlinear PDEs.
Functions and their properties have been part of the rigorous
precollege curriculum for decades. And functional equations have
been a favorite topic of the leading national and international
mathematical competitions. Yet the subject has not received equal
attention by authors at an introductory level. The majority of the
books on the topic remain unreachable to the curious and
intelligent precollege student. The present book is an attempt to
eliminate this disparity. The book opens with a review chapter on
functions, which collects the relevant foundational information on
functions, plus some material potentially new to the reader. The
next chapter presents a working definition of functional equations
and explains the difficulties in trying to systematize the theory.
With each new chapter, the author presents methods for the solution
of a particular group of equations. Each chapter is complemented
with many solved examples, the majority of which are taken from
mathematical competitions and professional journals. The book ends
with a chapter of unsolved problems and some other auxiliary
material. The book is an invaluable resource for precollege and
college students who want to deepen their knowledge of functions
and their properties, for teachers and instructors who wish to
enrich their curricula, and for any lover of mathematical
problem-solving techniques.
This book includes discussions related to solutions of such tasks
as: probabilistic description of the investment function;
recovering the income function from GDP estimates; development of
models for the economic cycles; selecting the time interval of
pseudo-stationarity of cycles; estimating
characteristics/parameters of cycle models; analysis of accuracy of
model factors. All of the above constitute the general principles
of a theory explaining the phenomenon of economic cycles and
provide mathematical tools for their quantitative description. The
introduced theory is applicable to macroeconomic analyses as well
as econometric estimations of economic cycles.
This book is the second edition of the first complete study and
monograph dedicated to singular traces. The text offers, due to the
contributions of Albrecht Pietsch and Nigel Kalton, a complete
theory of traces and their spectral properties on ideals of compact
operators on a separable Hilbert space. The second edition has been
updated on the fundamental approach provided by Albrecht Pietsch.
For mathematical physicists and other users of Connes'
noncommutative geometry the text offers a complete reference to
traces on weak trace class operators, including Dixmier traces and
associated formulas involving residues of spectral zeta functions
and asymptotics of partition functions.
This book is a description of why and how to do Scientific
Computing for fundamental models of fluid flow. It contains
introduction, motivation, analysis, and algorithms and is closely
tied to freely available MATLAB codes that implement the methods
described. The focus is on finite element approximation methods and
fast iterative solution methods for the consequent linear(ized)
systems arising in important problems that model incompressible
fluid flow. The problems addressed are the Poisson equation,
Convection-Diffusion problem, Stokes problem and Navier-Stokes
problem, including new material on time-dependent problems and
models of multi-physics. The corresponding iterative algebra based
on preconditioned Krylov subspace and multigrid techniques is for
symmetric and positive definite, nonsymmetric positive definite,
symmetric indefinite and nonsymmetric indefinite matrix systems
respectively. For each problem and associated solvers there is a
description of how to compute together with theoretical analysis
that guides the choice of approaches and describes what happens in
practice in the many illustrative numerical results throughout the
book (computed with the freely downloadable IFISS software). All of
the numerical results should be reproducible by readers who have
access to MATLAB and there is considerable scope for
experimentation in the "computational laboratory " provided by the
software. Developments in the field since the first edition was
published have been represented in three new chapters covering
optimization with PDE constraints (Chapter 5); solution of unsteady
Navier-Stokes equations (Chapter 10); solution of models of
buoyancy-driven flow (Chapter 11). Each chapter has many
theoretical problems and practical computer exercises that involve
the use of the IFISS software. This book is suitable as an
introduction to iterative linear solvers or more generally as a
model of Scientific Computing at an advanced undergraduate or
beginning graduate level.
This companion piece to the author's 2018 book, A Software
Repository for Orthogonal Polynomials, focuses on Gaussian
quadrature and the related Christoffel function. The book makes
Gauss quadrature rules of any order easily accessible for a large
variety of weight functions and for arbitrary precision. It also
documents and illustrates known as well as original approximations
for Gauss quadrature weights and Christoffel functions. The
repository contains 60 datasets, each dealing with a particular
weight function. Included are classical, quasi-classical, and, most
of all, nonclassical weight functions and associated orthogonal
polynomials.
Electroencephalography and magnetoencephalography are the two most
efficient techniques to study the functional brain. This book
completely aswers the fundamental mathematical question of
uniqueness of the representations obtained using these techniques,
and also covers many other concrete results for special geometric
models of the brain, presenting the research of the authors and
their groups in the last two decades.
This book presents a novel approach to umbral calculus, which uses
only elementary linear algebra (matrix calculus) based on the
observation that there is an isomorphism between Sheffer
polynomials and Riordan matrices, and that Sheffer polynomials can
be expressed in terms of determinants. Additionally, applications
to linear interpolation and operator approximation theory are
presented in many settings related to various families of
polynomials.
This multi-volume handbook is the most up-to-date and comprehensive
reference work in the field of fractional calculus and its numerous
applications. This third volume collects authoritative chapters
covering several numerical aspects of fractional calculus,
including time and space fractional derivatives, finite differences
and finite elements, and spectral, meshless, and particle methods.
An increasing complexity of models used to predict real-world
systems leads to the need for algorithms to replace complex models
with far simpler ones, while preserving the accuracy of the
predictions. This two-volume handbook covers methods as well as
applications. This first volume focuses on real-time control
theory, data assimilation, real-time visualization,
high-dimensional state spaces and interaction of different
reduction techniques.
The book aims at giving a monographic presentation of the abstract
harmonic analysis of hypergroups, while combining it with applied
topics of spectral analysis, approximation by orthogonal expansions
and stochastic sequences. Hypergroups are locally compact Hausdorff
spaces equipped with a convolution, an involution and a unit
element. Related algebraic structures had already been studied by
Frobenius around 1900. Their axiomatic characterisation in harmonic
analysis was later developed in the 1970s. Hypergoups naturally
emerge in seemingly different application areas as time series
analysis, probability theory and theoretical physics.The book
presents harmonic analysis on commutative and polynomial
hypergroups as well as weakly stationary random fields and
sequences thereon. For polynomial hypergroups also difference
equations and stationary sequences are considered. At greater
extent than in the existing literature, the book compiles a rather
comprehensive list of hypergroups, in particular of polynomial
hypergroups. With an eye on readers at advanced undergraduate and
graduate level, the proofs are generally worked out in careful
detail. The bibliography is extensive.
This monograph studies optimization problems for rigid punches in
elastic media and for high-speed penetration of rigid strikers into
deformed elastoplastic, concrete, and composite media using
variational calculations, tools from functional analysis, and
stochastic and min-max (guaranteed) optimization approaches with
incomplete data. The book presents analytical and numerical results
developed by the authors during the last ten years.
Nonlinear matrix equations arise frequently in applied science and
engineering. This is the first book to provide a unified treatment
of structure-preserving doubling algorithms, which have been
recently studied and proven effective for notoriously challenging
problems, such as fluid queue theory and vibration analysis for
high-speed trains. The authors present recent developments and
results for the theory of doubling algorithms for nonlinear matrix
equations associated with regular matrix pencils, and highlight the
use of these algorithms in achieving robust solutions for
notoriously challenging problems that other methods cannot.
Structure-Preserving Doubling Algorithms for Nonlinear Matrix
Equations is intended for researchers and computational scientists.
Graduate students may also find it of interest.
The book covers fundamentals of the theory of optimal methods for
solving ill-posed problems, as well as ways to obtain accurate and
accurate-by-order error estimates for these methods. The methods
described in the current book are used to solve a number of inverse
problems in mathematical physics. Contents Modulus of continuity of
the inverse operator and methods for solving ill-posed problems
Lavrent'ev methods for constructing approximate solutions of linear
operator equations of the first kind Tikhonov regularization method
Projection-regularization method Inverse heat exchange problems
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