Fourier Series And Numerical Methods For Partial Differential Equations


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Fourier Series and Numerical Methods for Partial Differential Equations


Fourier Series and Numerical Methods for Partial Differential Equations

Author: Richard Bernatz

language: en

Publisher: John Wiley & Sons

Release Date: 2010-07-30


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The importance of partial differential equations (PDEs) in modeling phenomena in engineering as well as in the physical, natural, and social sciences is well known by students and practitioners in these fields. Striking a balance between theory and applications, Fourier Series and Numerical Methods for Partial Differential Equations presents an introduction to the analytical and numerical methods that are essential for working with partial differential equations. Combining methodologies from calculus, introductory linear algebra, and ordinary differential equations (ODEs), the book strengthens and extends readers' knowledge of the power of linear spaces and linear transformations for purposes of understanding and solving a wide range of PDEs. The book begins with an introduction to the general terminology and topics related to PDEs, including the notion of initial and boundary value problems and also various solution techniques. Subsequent chapters explore: The solution process for Sturm-Liouville boundary value ODE problems and a Fourier series representation of the solution of initial boundary value problems in PDEs The concept of completeness, which introduces readers to Hilbert spaces The application of Laplace transforms and Duhamel's theorem to solve time-dependent boundary conditions The finite element method, using finite dimensional subspaces The finite analytic method with applications of the Fourier series methodology to linear version of non-linear PDEs Throughout the book, the author incorporates his own class-tested material, ensuring an accessible and easy-to-follow presentation that helps readers connect presented objectives with relevant applications to their own work. Maple is used throughout to solve many exercises, and a related Web site features Maple worksheets for readers to use when working with the book's one- and multi-dimensional problems. Fourier Series and Numerical Methods for Partial Differential Equations is an ideal book for courses on applied mathematics and partial differential equations at the upper-undergraduate and graduate levels. It is also a reliable resource for researchers and practitioners in the fields of mathematics, science, and engineering who work with mathematical modeling of physical phenomena, including diffusion and wave aspects.

Partial Differential Equations


Partial Differential Equations

Author: Mark S. Gockenbach

language: en

Publisher: SIAM

Release Date: 2010-12-02


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A fresh, forward-looking undergraduate textbook that treats the finite element method and classical Fourier series method with equal emphasis.

Analytic Methods for Partial Differential Equations


Analytic Methods for Partial Differential Equations

Author: G. Evans

language: en

Publisher: Springer Science & Business Media

Release Date: 2012-12-06


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The subject of partial differential equations holds an exciting and special position in mathematics. Partial differential equations were not consciously created as a subject but emerged in the 18th century as ordinary differential equations failed to describe the physical principles being studied. The subject was originally developed by the major names of mathematics, in particular, Leonard Euler and Joseph-Louis Lagrange who studied waves on strings; Daniel Bernoulli and Euler who considered potential theory, with later developments by Adrien-Marie Legendre and Pierre-Simon Laplace; and Joseph Fourier's famous work on series expansions for the heat equation. Many of the greatest advances in modern science have been based on discovering the underlying partial differential equation for the process in question. J ames Clerk Maxwell, for example, put electricity and magnetism into a unified theory by estab lishing Maxwell's equations for electromagnetic theory, which gave solutions for problems in radio wave propagation, the diffraction of light and X-ray developments. Schrodinger's equation for quantum mechankal processes at the atomic level leads to experimentally verifiable results which have changed the face of atomic physics and chemistry in the 20th century. In fluid mechanics, the Navier-Stokes' equations form a basis for huge number-crunching activities associated with such widely disparate topics as weather forcasting and the design of supersonic aircraft. Inevitably the study of partial differential equations is a large undertaking, and falls into several areas of mathematics.