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Partial Differential Equations - An Introduction to Analytical and Numerical Methods (Hardcover, 1st ed. 2023)
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Partial Differential Equations - An Introduction to Analytical and Numerical Methods (Hardcover, 1st ed. 2023)
Series: Graduate Texts in Mathematics, 294
Expected to ship within 9 - 15 working days
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This textbook introduces the study of partial differential
equations using both analytical and numerical methods. By
intertwining the two complementary approaches, the authors create
an ideal foundation for further study. Motivating examples from the
physical sciences, engineering, and economics complete this
integrated approach. A showcase of models begins the book,
demonstrating how PDEs arise in practical problems that involve
heat, vibration, fluid flow, and financial markets. Several
important characterizing properties are used to classify
mathematical similarities, then elementary methods are used to
solve examples of hyperbolic, elliptic, and parabolic equations.
From here, an accessible introduction to Hilbert spaces and the
spectral theorem lay the foundation for advanced methods. Sobolev
spaces are presented first in dimension one, before being extended
to arbitrary dimension for the study of elliptic equations. An
extensive chapter on numerical methods focuses on finite difference
and finite element methods. Computer-aided calculation with Maple
(TM) completes the book. Throughout, three fundamental examples are
studied with different tools: Poisson's equation, the heat
equation, and the wave equation on Euclidean domains. The
Black-Scholes equation from mathematical finance is one of several
opportunities for extension. Partial Differential Equations offers
an innovative introduction for students new to the area. Analytical
and numerical tools combine with modeling to form a versatile
toolbox for further study in pure or applied mathematics.
Illuminating illustrations and engaging exercises accompany the
text throughout. Courses in real analysis and linear algebra at the
upper-undergraduate level are assumed.
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