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Abstrak :
This book presents a new approach to analyzing initial-boundary value problems for integrable partial differential equations (PDEs) in two dimensions, a method that the author first introduced in 1997 and which is based on ideas of the inverse scattering transform. This method is unique in also yielding novel integral representations for the explicit solution of linear boundary value problems, which include such classical problems as the heat equation on a finite interval and the Helmholtz equation in the interior of an equilateral triangle.

Abstrak :
This book is the most comprehensive, up-to-date account of the popular numerical methods for solving boundary value problems in ordinary differential equations. It aims at a thorough understanding of the field by giving an in-depth analysis of the numerical methods by using decoupling principles. Numerous exercises and real-world examples are used throughout to demonstrate the methods and the theory. Although first published in 1988, this republication remains the most comprehensive theoretical coverage of the subject matter, not available elsewhere in one volume. Many problems, arising in a wide variety of application areas, give rise to mathematical models which form boundary value problems for ordinary differential equations. These problems rarely have a closed form solution, and computer simulation is typically used to obtain their approximate solution. This book discusses methods to carry out such computer simulations in a robust, efficient, and reliable manner.

Abstrak :
For more than 30 years, this two-volume set has helped prepare graduate students to use partial differential equations and integral equations to handle significant problems arising in applied mathematics, engineering, and the physical sciences. Originally published in 1967, this graduate-level introduction is devoted to the mathematics needed for the modern approach to boundary value problems using Green's functions and using eigenvalue expansions. Now a part of SIAM's Classics series, these volumes contain a large number of concrete, interesting examples of boundary value problems for partial differential equations that cover a variety of applications that are still relevant today. For example, there is substantial treatment of the Helmholtz equation and scattering theory--subjects that play a central role in contemporary inverse problems in acoustics and electromagnetic theory.

Abstrak :
This is the first book published in English devoted solely to the boundary function method, which is one of the asymptotic methods. This method provides an effective and simple way to obtain asymptotic approximations for the solutions of certain ordinary and partial differential equations containing small parameters in front of the highest derivatives. These equations, called singularly perturbed equations, are often used in modeling. In addition to numerous examples, the book includes discussions on singularly perturbed problems from chemical kinetics and heat conduction, semiconductor device modeling, and mathematical biology. The book also contains a variety of original ideas and explicit calculations previously available only in journal literature, as well as many concrete applied problems illustrating the boundary function method algorithms. Quite general asymptotic results described in the book are rigorous in the sense that, along with the asymptotic algorithms, in most cases the theorems on estimation of the remainder terms are presented. A survey of results of Russian mathematicians on the subject is provided; many of these results are not well known in the West.

Abstrak :
Tobias Nau addresses initial boundary value problems in cylindrical space domains with the aid of modern techniques from functional analysis and operator theory. In particular, the author uses concepts from Fourier analysis of functions with values in Banach spaces and the operator-valued functional calculus of sectorial operators. He applies abstract results to concrete problems in cylindrical space domains such as the heat equation subject to numerous boundary conditions and equations arising from fluid dynamics.

Abstrak :
Lectures on a unified theory of and practical procedures for the numerical solution of very general classes of linear and nonlinear two point boundary-value problems.