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Among the theoretical methods for solving many problems of applied
mathematics, physics, and technology, asymptotic methods often
provide results that lead to obtaining more effective algorithms of
numerical evaluation. Presenting the mathematical methods of
perturbation theory, Introduction to Asymptotic Methods reviews the
most important methods of singular perturbations within the scope
of application of differential equations. The authors take a
challenging and original approach based on the integrated
mathematical-analytical treatment of various objects taken from
interdisciplinary fields of mechanics, physics, and applied
mathematics. This new hybrid approach will lead to results that
cannot be obtained by standard theories in the field. Emphasizing
fundamental elements of the mathematical modeling process, the book
provides comprehensive coverage of asymptotic approaches, regular
and singular perturbations, one-dimensional non-stationary
non-linear waves, Pade approximations, oscillators with negative
Duffing type stiffness, and differential equations with
discontinuous nonlinearities. The book also offers a method of
construction for canonical variables transformation in parametric
form along with a number of examples and applications. The book is
applications oriented and features results and literature citations
that have not been seen in the Western Scientific Community. The
authors emphasize the dynamics of the development of perturbation
methods and present the development of ideas associated with this
wide field of research.
This book is devoted to researchers and teachers, as well as
graduate students, undergraduates and bachelors in engineering
mechanics, nano-mechanics, nanomaterials, nanostructures and
applied mathematics. It presents a collection of the latest
developments in the field of nonlinear (chaotic) dynamics of mass
distributed-parameter nanomechanical structures, providing a
rigorous and comprehensive study of modeling nonlinear phenomena.
It is written in a unique pedagogical style particularly suitable
for independent study and self-education. In addition, the book
achieves a good balance between Western and Eastern extensive
studies of the mathematical problems of nonlinear vibrations of
structural members.
From the reviews: "A unique feature of this book is the nice blend
of engineering vividness and mathematical rigour. [...] The authors
are to be congratulated for their valuable contribution to the
literature in the area of theoretical thermoelasticity and
vibration of plates." Journal of Sound and Vibration
This book is devoted to researchers and teachers, as well as
graduate students, undergraduates and bachelors in engineering
mechanics, nano-mechanics, nanomaterials, nanostructures and
applied mathematics. It presents a collection of the latest
developments in the field of nonlinear (chaotic) dynamics of mass
distributed-parameter nanomechanical structures, providing a
rigorous and comprehensive study of modeling nonlinear phenomena.
It is written in a unique pedagogical style particularly suitable
for independent study and self-education. In addition, the book
achieves a good balance between Western and Eastern extensive
studies of the mathematical problems of nonlinear vibrations of
structural members.
From the reviews: "A unique feature of this book is the nice blend
of engineering vividness and mathematical rigour. [...] The authors
are to be congratulated for their valuable contribution to the
literature in the area of theoretical thermoelasticity and
vibration of plates." Journal of Sound and Vibration
This book offers a valuable methodological approach to the
state-of-the-art of the classical plate/shell mathematical models,
exemplifying the vast range of mathematical models of nonlinear
dynamics and statics of continuous mechanical structural members.
The main objective highlights the need for further study of the
classical problem of shell dynamics consisting of mathematical
modeling, derivation of nonlinear PDEs, and of finding their
solutions based on the development of new and effective numerical
techniques. The book is designed for a broad readership of graduate
students in mechanical and civil engineering, applied mathematics,
and physics, as well as to researchers and professionals interested
in a rigorous and comprehensive study of modeling non-linear
phenomena governed by PDEs.
This monograph, addressing researchers as well as engineers, is devoted to nonclassical thermoelastic modelling of the nonlinear dynamics of shells. Differential equations of different dimensionality and different type have to be combined and nonlinearities of different geometrical, physical or elasto-plastic categories are addressed. Special emphasis is given to the Bubnov--Galerkin method. It can be applied to many problems in the theory of plates and shells, even those with very complex geometries, holes and various boundary conditions. The authors made every effort to keep the text intelligible for both practitioners and graduate students, although they offer a rigorous treatment of both purely mathematical and numerical approaches presented so that the reader can understand, analyse and track the nonlinear dynamics of spatial systems (shells) with thermomechanical behaviours.
This book focuses on the computational analysis of nonlinear
vibrations of structural members (beams, plates, panels, shells),
where the studied dynamical problems can be reduced to the
consideration of one spatial variable and time. The reduction is
carried out based on a formal mathematical approach aimed at
reducing the problems with infinite dimension to finite ones. The
process also includes a transition from governing nonlinear partial
differential equations to a set of finite number of ordinary
differential equations.Beginning with an overview of the recent
results devoted to the analysis and control of nonlinear dynamics
of structural members, placing emphasis on stability, buckling,
bifurcation and deterministic chaos, simple chaotic systems are
briefly discussed. Next, bifurcation and chaotic dynamics of the
Euler-Bernoulli and Timoshenko beams including the geometric and
physical nonlinearity as well as the elastic-plastic deformations
are illustrated. Despite the employed classical numerical analysis
of nonlinear phenomena, the various wavelet transforms and the four
Lyapunov exponents are used to detect, monitor and possibly control
chaos, hyper-chaos, hyper-hyper-chaos and deep chaos exhibited by
rectangular plate-strips and cylindrical panels.The book is
intended for post-graduate and doctoral students, applied
mathematicians, physicists, teachers and lecturers of universities
and companies dealing with a nonlinear dynamical system, as well as
theoretically inclined engineers of mechanical and civil
engineering.
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