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Advanced engineering analysis : the calculus of variations and functional analysis with applications in mechanics
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Description
Rating
Title
Advanced
engineering
analysis
: the
calculus
of
variations
and
functional
analysis
with
applications
in
mechanics
Creator
Lebedev, L. P.
Contributors
Cloud, Michael J.
Eremeyev, Victor A.
World Scientific (Firm)
DescriptionAbstract
Advanced
Engineering
Analysis
is
a
textbook
on
modern
engineering
analysis
,
covering
the
calculus
of
variations
,
functional
analysis
,
control
theory
, as
well
as
applications
of these
disciplines
to
mechanics
. The
book
offers
a
brief
and
concise
,
yet
complete
explanation
of
essential
theory
and
applications
.
It
contains
exercises
with
hints
and
solutions
,
ideal
for
selfstudy
.
DescriptionTable Of Contents
1
.
Basic
calculus
of
variations
.
1.1
.
Introduction
.
1.2
.
Euler's
equation
for the
simplest
problem
.
1.3
.
Properties
of
extremals
of the
simplest
functional
.
1.4
.
Ritz's
method
.
1.5
.
Natural
boundary
conditions
.
1.6
.
Extensions
to
more
general
functionals
.
1.7
.
Functionals
depending
on
functions
in
many
variables
.
1.8
. A
functional
with
integrand
depending
on
partial
derivatives
of
higher
order
.
1.9
. The
first
variation
.
1.10
.
Isoperimetric
problems
.
1.11
.
General
form
of the
first
variation
.
1.12
.
Movable
ends
of
extremals
.
1.13
.
Broken
extremals
:
WeierstrassErdmann
conditions
and
related
problems
.
1.14
.
Sufficient
conditions
for
minimum
.
1.15
.
Exercises

2
.
Applications
of the
calculus
of
variations
in
mechanics
.
2.1
.
Elementary
problems
for
elastic
structures
.
2.2
.
Some
extremal
principles
of
mechanics
.
2.3
.
Conservation
laws
.
2.4
.
Conservation
laws
and
Noether's
theorem
.
2.5
.
Functionals
depending
on
higher
derivatives
of
y
.
2.6
.
Noether's
theorem
,
general
case
.
2.7
.
Generalizations
.
2.8
.
Exercises

3
.
Elements
of
optimal
control
theory
.
3.1
. A
variational
problem
as an
optimal
control
problem
.
3.2
.
General
problem
of
optimal
control
.
3.3
.
Simplest
problem
of
optimal
control
.
3.4
.
Fundamental
solution
of a
linear
ordinary
differential
equation
.
3.5
. The
simplest
problem
,
continued
.
3.6
.
Pontryagin's
maximum
principle
for the
simplest
problem
.
3.7
.
Some
mathematical
preliminaries
.
3.8
.
General
terminal
control
problem
.
3.9
.
Pontryagin's
maximum
principle
for the
terminal
optimal
problem
.
3.10
.
Generalization
of the
terminal
control
problem
.
3.11
.
Small
variations
of
control
function
for
terminal
control
problem
.
3.12
. A
discrete
version
of
small
variations
of
control
function
for
generalized
terminal
control
problem
.
3.13
.
Optimal
time
control
problems
.
3.14
.
Final
remarks
on
control
problems
.
3.15
.
Exercises
. ;
8
4
.
Functional
analysis
.
4.1
. A
normed
space
as a
metric
space
.
4.2
.
Dimension
of a
linear
space
and
separability
.
4.3
.
Cauchy
sequences
and
Banach
spaces
.
4.4
. The
completion
theorem
.
4.5
.
L[symbol]
spaces
and the
Lebesgue
integral
.
4.6
.
Sobolev
spaces
.
4.7
.
Compactness
.
4.8
.
Inner
product
spaces
,
Hilbert
spaces
.
4.9
.
Operators
and
functionals
.
4.10
.
Contraction
mapping
principle
.
4.11
.
Some
approximation
theory
.
4.12
Orthogonal
decomposition
of a
Hilbert
space
and the
Riesz
representation
theorem
.
4.13
.
Basis
,
GramSchmidt
procedure
, and
Fourier
series
in
Hilbert
space
.
4.14
.
Weak
convergence
.
4.15
.
Adjoint
and
selfadjoint
operators
.
4.16
.
Compact
operators
.
4.17
.
Closed
operators
.
4.18
. On the
Sobolev
imbedding
theorem
.
4.19
.
Some
energy
spaces
in
mechanics
.
4.20
.
Introduction
to
spectral
concepts
.
4.21
. The
Fredholm
theory
in
Hilbert
spaces
.
4.22
.
Exercises

5
.
Applications
of
functional
analysis
in
mechanics
.
5.1
.
Some
mechanics
problems
from the
standpoint
of the
calculus
of
variations
; the
virtual
work
principle
.
5.2
.
Generalized
solution
of the
equilibrium
problem
for a
clamped
rod
with
springs
.
5.3
.
Equilibrium
problem
for a
clamped
membrane
and its
generalized
solution
.
5.4
.
Equilibrium
of a
free
membrane
.
5.5
.
Some
other
equilibrium
problems
of
linear
mechanics
.
5.6
. The
Ritz
and
BubnovGalerkin
methods
.
5.7
. The
HamiltonOstrogradski
principle
and
generalized
setup
of
dynamical
problems
in
classical
mechanics
.
5.8
.
Generalized
setup
of
dynamic
problem
for
membrane
.
5.9
.
Other
dynamic
problems
of
linear
mechanics
.
5.10
. The
Fourier
method
.
5.11
. An
eigenfrequency
boundary
value
problem
arising
in
linear
mechanics
.
5.12
. The
spectral
theorem
.
5.13
. The
Fourier
method
,
continued
.
5.14
.
Equilibrium
of a
von
Karman
plate
.
5.15
. A
unilateral
problem
.
5.16
.
Exercises
.
Publisher
World Scientific Pub. Co.
Subject
Engineering mathematics.
Mechanics, Applied  Mathematics.
Functional analysis.
Identifier
9789814390484
(electronic
bk.)
;
981439047X
;
9789814390477
;
http://www.worldscientific.com/worldscibooks/10.1142/8401#t=toc
Language
eng
Type
Electronic books.
FormatExtent
x, 489 p. : ill.
Date
c2012
.
OCLC number
874498321
CONTENTdm number
441
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