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Covers simultaneously rigorous mathematics, general physical
principles and engineering applications with practical interest
Provides interpretation of results with the help of illustrations
Includes detailed proofs of all results
Covers simultaneously rigorous mathematics, general physical
principles and engineering applications with practical interest
Provides interpretation of results with the help of illustrations
Includes detailed proofs of all results
This book is part of the series "Mathematics and Physics Applied to
Science and Technology." It combines rigorous mathematics with
general physical principles to model practical engineering systems
with a detailed derivation and interpretation of results. The book
presents the mathematical theory of partial differential equations
and methods of solution satisfying initial and boundary conditions.
It includes applications to acoustic, elastic, water,
electromagnetic and other waves, to the diffusion of heat, mass and
electricity, and to their interactions. The author covers
simultaneously rigorous mathematics, general physical principles
and engineering applications with practical interest. The book
provides interpretation of results with the help of illustrations
throughout and discusses similar phenomena, such as the diffusion
of heat, electricity and mass. The book is intended for graduate
students and engineers working with mathematical models and can be
applied to problems in mechanical, aerospace, electrical and other
branches of engineering.
Building on the author's previous book in the series, Complex
Analysis with Applications to Flows and Fields (CRC Press, 2010),
Transcendental Representations with Applications to Solids and
Fluids focuses on four infinite representations: series expansions,
series of fractions for meromorphic functions, infinite products
for functions with infinitely many zeros, and continued fractions
as alternative representations. This book also continues the
application of complex functions to more classes of fields,
including incompressible rotational flows, compressible
irrotational flows, unsteady flows, rotating flows, surface tension
and capillarity, deflection of membranes under load, torsion of
rods by torques, plane elasticity, and plane viscous flows. The two
books together offer a complete treatment of complex analysis,
showing how the elementary transcendental functions and other
complex functions are applied to fluid and solid media and force
fields mainly in two dimensions. The mathematical developments
appear in odd-numbered chapters while the physical and engineering
applications can be found in even-numbered chapters. The last
chapter presents a set of detailed examples. Each chapter begins
with an introduction and concludes with related topics. Written by
one of the foremost authorities in aeronautical/aerospace
engineering, this self-contained book gives the necessary
mathematical background and physical principles to build models for
technological and scientific purposes. It shows how to formulate
problems, justify the solutions, and interpret the results.
Linear Differential Equations and Oscillators is the first book
within Ordinary Differential Equations with Applications to
Trajectories and Vibrations, Six-volume Set. As a set, they are the
fourth volume in the series Mathematics and Physics Applied to
Science and Technology. This first book consists of chapters 1 and
2 of the fourth volume. The first chapter covers linear
differential equations of any order whose unforced solution can be
obtained from the roots of a characteristic polynomial, namely
those: (i) with constant coefficients; (ii) with homogeneous power
coefficients with the exponent equal to the order of derivation.
The method of characteristic polynomials is also applied to (iii)
linear finite difference equations of any order with constant
coefficients. The unforced and forced solutions of (i,ii,iii) are
examples of some general properties of ordinary differential
equations. The second chapter applies the theory of the first
chapter to linear second-order oscillators with one
degree-of-freedom, such as the mechanical mass-damper-spring-force
system and the electrical self-resistor-capacitor-battery circuit.
In both cases are treated free undamped, damped, and amplified
oscillations; also forced oscillations including beats, resonance,
discrete and continuous spectra, and impulsive inputs. Describes
general properties of differential and finite difference equations,
with focus on linear equations and constant and some power
coefficients Presents particular and general solutions for all
cases of differential and finite difference equations Provides
complete solutions for many cases of forcing including resonant
cases Discusses applications to linear second-order mechanical and
electrical oscillators with damping Provides solutions with forcing
including resonance using the characteristic polynomial, Green' s
functions, trigonometrical series, Fourier integrals and Laplace
transforms
Non-Linear Differential Equations and Dynamical Systems is the
second book within Ordinary Differential Equations with
Applications to Trajectories and Vibrations, Six-volume Set. As a
set, they are the fourth volume in the series Mathematics and
Physics Applied to Science and Technology. This second book
consists of two chapters (chapters 3 and 4 of the set). The first
chapter considers non-linear differential equations of first order,
including variable coefficients. A first-order differential
equation is equivalent to a first-order differential in two
variables. The differentials of order higher than the first and
with more than two variables are also considered. The applications
include the representation of vector fields by potentials. The
second chapter in the book starts with linear oscillators with
coefficients varying with time, including parametric resonance. It
proceeds to non-linear oscillators including non-linear resonance,
amplitude jumps, and hysteresis. The non-linear restoring and
friction forces also apply to electromechanical dynamos. These are
examples of dynamical systems with bifurcations that may lead to
chaotic motions. Presents general first-order differential
equations including non-linear like the Ricatti equation Discusses
differentials of the first or higher order in two or more variables
Includes discretization of differential equations as finite
difference equations Describes parametric resonance of linear time
dependent oscillators specified by the Mathieu functions and other
methods Examines non-linear oscillations and damping of dynamical
systems including bifurcations and chaotic motions
Higher-Order Differential Equations and Elasticity is the third
book within Ordinary Differential Equations with Applications to
Trajectories and Vibrations, Six-volume Set. As a set, they are the
fourth volume in the series Mathematics and Physics Applied to
Science and Technology. This third book consists of two chapters
(chapters 5 and 6 of the set). The first chapter in this book
concerns non-linear differential equations of the second and higher
orders. It also considers special differential equations with
solutions like envelopes not included in the general integral. The
methods presented include special differential equations, whose
solutions include the general integral and special integrals not
included in the general integral for myriad constants of
integration. The methods presented include dual variables and
differentials, related by Legendre transforms, that have
application in thermodynamics. The second chapter concerns
deformations of one (two) dimensional elastic bodies that are
specified by differential equations of: (i) the second-order for
non-stiff bodies like elastic strings (membranes); (ii)
fourth-order for stiff bodies like bars and beams (plates). The
differential equations are linear for small deformations and
gradients and non-linear otherwise. The deformations for beams
include bending by transverse loads and buckling by axial loads.
Buckling and bending couple non-linearly for plates. The
deformations depend on material properties, for example isotropic
or anisotropic elastic plates, with intermediate cases such as
orthotropic or pseudo-isotropic. Discusses differential equations
having special integrals not contained in the general integral,
like the envelope of a family of integral curves Presents
differential equations of the second and higher order, including
non-linear and with variable coefficients Compares relation of
differentials with the principles of thermodynamics Describes
deformations of non-stiff elastic bodies like strings and membranes
and buckling of stiff elastic bodies like bars, beams, and plates
Presents linear and non-linear waves in elastic strings, membranes,
bars, beams, and plates
Singular Differential Equations and Special Functions is the fifth
book within Ordinary Differential Equations with Applications to
Trajectories and Vibrations, Six-volume Set. As a set they are the
fourth volume in the series Mathematics and Physics Applied to
Science and Technology. This fifth book consists of one chapter
(chapter 9 of the set). The chapter starts with general classes of
differential equations and simultaneous systems for which the
properties of the solutions can be established 'a priori', such as
existence and unicity of solution, robustness and uniformity with
regard to changes in boundary conditions and parameters, and
stability and asymptotic behavior. The book proceeds to consider
the most important class of linear differential equations with
variable coefficients, that can be analytic functions or have
regular or irregular singularities. The solution of singular
differential equations by means of (i) power series; (ii)
parametric integral transforms; and (iii) continued fractions lead
to more than 20 special functions; among these is given greater
attention to generalized circular, hyperbolic, Airy, Bessel and
hypergeometric differential equations, and the special functions
that specify their solutions. Includes existence, unicity,
robustness, uniformity, and other theorems for non-linear
differential equations Discusses properties of dynamical systems
derived from the differential equations describing them, using
methods such as Liapunov functions Includes linear differential
equations with periodic coefficients, including Floquet theory,
Hill infinite determinants and multiple parametric resonance
Details theory of the generalized Bessel differential equation, and
of the generalized, Gaussian, confluent and extended hypergeometric
functions and relations with other 20 special functions Examines
Linear Differential Equations with analytic coefficients or regular
or irregular singularities, and solutions via power series,
parametric integral transforms, and continued fractions
Simultaneous Differential Equations and Multi-Dimensional
Vibrations is the fourth book within Ordinary Differential
Equations with Applications to Trajectories and Vibrations,
Six-volume Set. As a set, they are the fourth volume in the series
Mathematics and Physics Applied to Science and Technology. This
fourth book consists of two chapters (chapters 7 and 8 of the set).
The first chapter concerns simultaneous systems of ordinary
differential equations and focuses mostly on the cases that have a
matrix of characteristic polynomials, namely linear systems with
constant or homogeneous power coefficients. The method of the
matrix of characteristic polynomials also applies to simultaneous
systems of linear finite difference equations with constant
coefficients. The second chapter considers linear multi-dimensional
oscillators with any number of degrees of freedom including
damping, forcing, and multiple resonance. The discrete oscillators
may be extended from a finite number of degrees-of-freedom to
infinite chains. The continuous oscillators correspond to waves in
homogeneous or inhomogeneous media, including elastic, acoustic,
electromagnetic, and water surface waves. The combination of
propagation and dissipation leads to the equations of mathematical
physics. Presents simultaneous systems of ordinary differential
equations and their elimination for a single ordinary differential
equation Includes cases with a matrix of characteristic
polynomials, including simultaneous systems of linear differential
and finite difference equations with constant coefficients Covers
multi-dimensional oscillators with damping and forcing, including
modal decomposition, natural frequencies and coordinates, and
multiple resonance Discusses waves in inhomogeneous media, such as
elastic, electromagnetic, acoustic, and water waves Includes
solutions of partial differential equations of mathematical physics
by separation of variables leading to ordinary differential
equations
Volume IV of the series "Mathematics and Physics Applied to Science
and Technology," this comprehensive six-book set covers: Linear
Differential Equations and Oscillators Non-linear Differential
Equations and Dynamical Systems Higher-order Differential Equations
and Elasticity Simultaneous Systems of Differential Equations and
Multi-dimensional Oscillators Singular Differential Equations and
Special Functions Classification and Examples of Differential
Equations and their Applications
Complex Analysis with Applications to Flows and Fields presents the
theory of functions of a complex variable, from the complex plane
to the calculus of residues to power series to conformal mapping.
The book explores numerous physical and engineering applications
concerning potential flows, the gravity field, electro- and
magnetostatics, steady heat conduction, and other problems. It
provides the mathematical results to sufficiently justify the
solution of these problems, eliminating the need to consult
external references. The book is conveniently divided into four
parts. In each part, the mathematical theory appears in
odd-numbered chapters while the physical and engineering
applications can be found in even-numbered chapters. Each chapter
begins with an introduction or summary and concludes with related
topics. The last chapter in each section offers a collection of
many detailed examples. This self-contained book gives the
necessary mathematical background and physical principles to build
models for technological and scientific purposes. It shows how to
formulate problems, justify the solutions, and interpret the
results.
Classification and Examples of Differential Equations and their
Applications is the sixth book within Ordinary Differential
Equations with Applications to Trajectories and Vibrations,
Six-volume Set. As a set, they are the fourth volume in the series
Mathematics and Physics Applied to Science and Technology. This
sixth book consists of one chapter (chapter 10 of the set). It
contains 20 examples related to the preceding five books and
chapters 1 to 9 of the set. It includes two recollections: the
first with a classification of differential equations into 500
standards and the second with a list of 500 applications. The
ordinary differential equations are classified in 500 standards
concerning methods of solution and related properties, including:
(i) linear differential equations with constant or homogeneous
coefficients and finite difference equations; (ii) linear and
non-linear single differential equations and simultaneous systems;
(iii) existence, unicity and other properties; (iv) derivation of
general, particular, special, analytic, regular, irregular, and
normal integrals; (v) linear differential equations with variable
coefficients including known and new special functions. The theory
of differential equations is applied to the detailed solution of
500 physical and engineering problems including: (i) one- and
multidimensional oscillators, with damping or amplification, with
non-resonant or resonant forcing; (ii) single, non-linear, and
parametric resonance; (iii) bifurcations and chaotic dynamical
systems; (iv) longitudinal and transversal deformations and
buckling of bars, beams, and plates; (v) trajectories of particles;
(vi) oscillations and waves in non-uniform media, ducts, and wave
guides. Provides detailed solution of examples of differential
equations of the types covered in tomes l-5 of the set (Ordinary
Differential Equations with Applications to Trajectories and
Vibrations, Six -volume Set) Includes physical and engineering
problems that extend those presented in the tomes 1-6 (Ordinary
Differential Equations with Applications to Trajectories and
Vibrations, Six-volume Set) Includes a classification of ordinary
differential equations and their properties into 500 standards that
can serve as a look-up table of methods of solution Covers a
recollection of 500 physical and engineering problems and sub-cases
that involve the solution of differential equations Presents the
problems used as examples including formulation, solution, and
interpretation of results
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