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This monograph deals with some of the latest results in nonlinear
mechanics, obtained recently by the use of a modernized version of
Bogoljubov's method of successive changes of variables which
ensures rapid convergence. This method visualised as early as 1934
by Krylov and Bogoljubov provides an effective tool for solving
many interesting problems of nonlinear mechanics. It led, in
particular, to the solution of the problem of the existence of a
quasi periodic regime, with the restriction that approximate
solutions obtained in the general case involved divergent series.
Recently, making use of the research of Kolmogorov and Arno'ld,
Bogoljubov has modernised the method of successive substitutions in
such a way that the convergence of the corresponding expansions is
ensured. This book consists of a short Introduction and seven
chapters. The first chapter presents the results obtained by
BogoIjubov in 1963 on the extension of the method of successive
substitutions and the study of quasi periodic solutions applied to
non-conservative systems (inter alia making explicit the dependence
of these solutions on the parameter, indicating methods of
obtaining asymptotic and convergent series for them, etc.)."
Systems of conservation laws arise naturally in physics and chemistry. Continuing where the previous volume left off, the author considers the maximum principle from the viewpoints of both viscous approximation and numerical schemes. Convergence is studied through compensated compactness. The author applies this tool to the description of large amplitude wave propagation. Small waves are studied through geometrical optics. Special structures are presented in chapters on rich and Temple systems. Finally, Serre explains why the initial-boundary value problem is far from trivial, with descriptions of the Kreiss-Lopatinski condition for well-posedness, with applications to shock wave stability, and certain problems in boundary layer theory. Throughout the presentation is reasonably self-contained, with large numbers of exercises and full discussion of all the ideas. This will make it ideal as a text for graduate courses in the area of partial differential equations.
Systems of conservation laws arise naturally in physics and
chemistry. To understand them and their consequences (shock waves,
finite velocity wave propagation) properly in mathematical terms
requires, however, knowledge of a broad range of topics. This book
sets up the foundations of the modern theory of conservation laws,
describing the physical models and mathematical methods, leading to
the Glimm scheme. Building on this the author then takes the reader
to the current state of knowledge in the subject. The maximum
principle is considered from the viewpoint of numerical schemes and
also in terms of viscous approximation. Small waves are studied
using geometrical optics methods. Finally, the initial-boundary
problem is considered in depth. Throughout, the presentation is
reasonably self-contained, with large numbers of exercises and full
discussion of all the ideas. This will make it ideal as a text for
graduate courses in the area of partial differential equations.
These 6 volumes - the result of a 10 year collaboration between the
authors, two of France's leading scientists and both distinguished
international figures - compile the mathematical knowledge required
by researchers in mechanics, physics, engineering, chemistry and
other branches of application of mathematics for the theoretical
and numerical resolution of physical models on computers. Since the
publication in 1924 of the "Methoden der mathematischen Physik" by
Courant and Hilbert, there has been no other comprehensive and
up-to-date publication presenting the mathematical tools needed in
applications of mathematics in directly implementable form. The
advent of large computers has in the meantime revolutionised
methods of computation and made this gap in the literature
intolerable: the objective of the present work is to fill just this
gap. Many phenomena in physical mathematics may be modeled by a
system of partial differential equations in distributed systems: a
model here means a set of equations, which together with given
boundary data and, if the phenomenon is evolving in time, initial
data, defines the system. The advent of high-speed computers has
made it possible for the first time to calculate values from models
accurately and rapidly. Researchers and engineers thus have a
crucial means of using numerical results to modify and adapt
arguments and experiments along the way. Every facet of technical
and industrial activity has been affected by these developments.
Modeling by distributed systems now also supports work in many
areas of physics (plasmas, new materials, astrophysics,
geophysics), chemistry and mechanics and is finding increasing use
in the life sciences.
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