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Modern Mathematical Methods for Scientists and Engineers is a
modern introduction to basic topics in mathematics at the
undergraduate level, with emphasis on explanations and applications
to real-life problems. There is also an 'Application' section at
the end of each chapter, with topics drawn from a variety of areas,
including neural networks, fluid dynamics, and the behavior of
'put' and 'call' options in financial markets. The book presents
several modern important and computationally efficient topics,
including feedforward neural networks, wavelets, generalized
functions, stochastic optimization methods, and numerical methods.A
unique and novel feature of the book is the introduction of a
recently developed method for solving partial differential
equations (PDEs), called the unified transform. PDEs are the
mathematical cornerstone for describing an astonishingly wide range
of phenomena, from quantum mechanics to ocean waves, to the
diffusion of heat in matter and the behavior of financial markets.
Despite the efforts of many famous mathematicians, physicists and
engineers, the solution of partial differential equations remains a
challenge.The unified transform greatly facilitates this task. For
example, two and a half centuries after Jean d'Alembert formulated
the wave equation and presented a solution for solving a simple
problem for this equation, the unified transform derives in a
simple manner a generalization of the d'Alembert solution, valid
for general boundary value problems. Moreover, two centuries after
Joseph Fourier introduced the classical tool of the Fourier series
for solving the heat equation, the unified transform constructs a
new solution to this ubiquitous PDE, with important analytical and
numerical advantages in comparison to the classical solutions. The
authors present the unified transform pedagogically, building all
the necessary background, including functions of real and of
complex variables and the Fourier transform, illustrating the
method with numerous examples.Broad in scope, but pedagogical in
style and content, the book is an introduction to powerful
mathematical concepts and modern tools for students in science and
engineering.
This volume gathers selected contributions from the participants of
the Banff International Research Station (BIRS) workshop Coupled
Mathematical Models for Physical and Biological Nanoscale Systems
and their Applications, who explore various aspects of the
analysis, modeling and applications of nanoscale systems, with a
particular focus on low dimensional nanostructures and coupled
mathematical models for their description. Due to the vastness,
novelty and complexity of the interfaces between mathematical
modeling and nanoscience and nanotechnology, many important areas
in these disciplines remain largely unexplored. In their efforts to
move forward, multidisciplinary research communities have come to a
clear understanding that, along with experimental techniques,
mathematical modeling and analysis have become crucial to the
study, development and application of systems at the nanoscale. The
conference, held at BIRS in autumn 2016, brought together experts
from three different communities working in fields where coupled
mathematical models for nanoscale and biosystems are especially
relevant: mathematicians, physicists (both theorists and
experimentalists), and computational scientists, including those
dealing with biological nanostructures. Its objectives: summarize
the state-of-the-art; identify and prioritize critical problems of
major importance that require solutions; analyze existing
methodologies; and explore promising approaches to addressing the
challenges identified. The contributions offer up-to-date
introductions to a range of topics in nano and biosystems, identify
important challenges, assess current methodologies and explore
promising approaches. As such, this book will benefit researchers
in applied mathematics, as well as physicists and biologists
interested in coupled mathematical models and their analysis for
physical and biological nanoscale systems that concern applications
in biotechnology and medicine, quantum information processing and
optoelectronics.
Modern Mathematical Methods for Scientists and Engineers is a
modern introduction to basic topics in mathematics at the
undergraduate level, with emphasis on explanations and applications
to real-life problems. There is also an 'Application' section at
the end of each chapter, with topics drawn from a variety of areas,
including neural networks, fluid dynamics, and the behavior of
'put' and 'call' options in financial markets. The book presents
several modern important and computationally efficient topics,
including feedforward neural networks, wavelets, generalized
functions, stochastic optimization methods, and numerical methods.A
unique and novel feature of the book is the introduction of a
recently developed method for solving partial differential
equations (PDEs), called the unified transform. PDEs are the
mathematical cornerstone for describing an astonishingly wide range
of phenomena, from quantum mechanics to ocean waves, to the
diffusion of heat in matter and the behavior of financial markets.
Despite the efforts of many famous mathematicians, physicists and
engineers, the solution of partial differential equations remains a
challenge.The unified transform greatly facilitates this task. For
example, two and a half centuries after Jean d'Alembert formulated
the wave equation and presented a solution for solving a simple
problem for this equation, the unified transform derives in a
simple manner a generalization of the d'Alembert solution, valid
for general boundary value problems. Moreover, two centuries after
Joseph Fourier introduced the classical tool of the Fourier series
for solving the heat equation, the unified transform constructs a
new solution to this ubiquitous PDE, with important analytical and
numerical advantages in comparison to the classical solutions. The
authors present the unified transform pedagogically, building all
the necessary background, including functions of real and of
complex variables and the Fourier transform, illustrating the
method with numerous examples.Broad in scope, but pedagogical in
style and content, the book is an introduction to powerful
mathematical concepts and modern tools for students in science and
engineering.
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