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This textbook gives an introduction to Partial Differential
Equations (PDEs), for any reader wishing to learn and understand
the basic concepts, theory, and solution techniques of elementary
PDEs. The only prerequisite is an undergraduate course in Ordinary
Differential Equations. This work contains a comprehensive
treatment of the standard second-order linear PDEs, the heat
equation, wave equation, and Laplace's equation. First-order and
some common nonlinear PDEs arising in the physical and life
sciences, with their solutions, are also covered.This textbook
includes an introduction to Fourier series and their properties, an
introduction to regular Sturm-Liouville boundary value problems,
special functions of mathematical physics, a treatment of
nonhomogeneous equations and boundary conditions using methods such
as Duhamel's principle, and an introduction to the finite
difference technique for the numerical approximation of solutions.
All results have been rigorously justified or precise references to
justifications in more advanced sources have been cited. Appendices
providing a background in complex analysis and linear algebra are
also included for readers with limited prior exposure to those
subjects.The textbook includes material from which instructors
could create a one- or two-semester course in PDEs. Students may
also study this material in preparation for a graduate school
(masters or doctoral) course in PDEs.
This textbook gives an introduction to Partial Differential
Equations (PDEs), for any reader wishing to learn and understand
the basic concepts, theory, and solution techniques of elementary
PDEs. The only prerequisite is an undergraduate course in Ordinary
Differential Equations. This work contains a comprehensive
treatment of the standard second-order linear PDEs, the heat
equation, wave equation, and Laplace's equation. First-order and
some common nonlinear PDEs arising in the physical and life
sciences, with their solutions, are also covered.This textbook
includes an introduction to Fourier series and their properties, an
introduction to regular Sturm-Liouville boundary value problems,
special functions of mathematical physics, a treatment of
nonhomogeneous equations and boundary conditions using methods such
as Duhamel's principle, and an introduction to the finite
difference technique for the numerical approximation of solutions.
All results have been rigorously justified or precise references to
justifications in more advanced sources have been cited. Appendices
providing a background in complex analysis and linear algebra are
also included for readers with limited prior exposure to those
subjects.The textbook includes material from which instructors
could create a one- or two-semester course in PDEs. Students may
also study this material in preparation for a graduate school
(masters or doctoral) course in PDEs.
This textbook provides an introduction to financial mathematics and
financial engineering for undergraduate students who have completed
a three- or four-semester sequence of calculus courses. It
introduces the theory of interest, discrete and continuous random
variables and probability, stochastic processes, linear
programming, the Fundamental Theorem of Finance, option pricing,
hedging, and portfolio optimization. This third edition expands on
the second by including a new chapter on the extensions of the
Black-Scholes model of option pricing and a greater number of
exercises at the end of each chapter. More background material and
exercises added, with solutions provided to the other chapters,
allowing the textbook to better stand alone as an introduction to
financial mathematics. The reader progresses from a solid grounding
in multivariable calculus through a derivation of the Black-Scholes
equation, its solution, properties, and applications. The text
attempts to be as self-contained as possible without relying on
advanced mathematical and statistical topics. The material
presented in this book will adequately prepare the reader for
graduate-level study in mathematical finance.
Anyone with an interest in learning about the mathematical modeling
of prices of financial derivatives such as bonds, futures, and
options can start with this book, whereby the only mathematical
prerequisite is multivariable calculus. The necessary theory of
interest, statistical, stochastic, and differential equations are
developed in their respective chapters, with the goal of making
this introductory text as self-contained as possible.In this
edition, the chapters on hedging portfolios and extensions of the
Black-Scholes model have been expanded. The chapter on optimizing
portfolios has been completely re-written to focus on the
development of the Capital Asset Pricing Model. The binomial model
due to Cox-Ross-Rubinstein has been enlarged into a standalone
chapter illustrating the wide-ranging utility of the binomial model
for numerically estimating option prices. There is a completely new
chapter on the pricing of exotic options. The appendix now features
linear algebra with sufficient background material to support a
more rigorous development of the Arbitrage Theorem.The new edition
has more than doubled the number of exercises compared to the
previous edition and now contains over 700 exercises. Thus,
students completing the book will gain a deeper understanding of
the development of modern financial mathematics.
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