Unified Transform for Boundary Value Problems: Applications and Advances
Unified Transform for Boundary Value Problems: Applications and Advances
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DOI：
10.1137/1.9781611973822
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发表时间：
2014-12
期刊：
影响因子：
--
通讯作者：
A. Fokas;B. Pelloni
中科院分区：
文献类型：
--
作者：
A. Fokas;B. Pelloni
这本书描述了最先进的发展和应用的统一变换及其与边界元方法的关系。作者从几个不同的角度给出了边值问题的解，特别是由偏微分方程组(PDE)模拟的问题的类型。讨论了统一变换在线性和可积非线性偏微分方程组边值问题分析和数值模拟中的最新应用，以及求解线性椭圆型偏微分方程组的一种成熟的数值方法--密切相关的边界元方法。正文分为三个部分。第一部分介绍了关于线性、非线性发展问题和椭圆问题的新的理论结果。构造了几类边值问题的新的显式解表示，并进行了严格的分析。第二部分是椭圆型问题的变分公式的详细概述。它将统一变换方法与边界元方法放在一个经典的背景下，强调了它的新颖性。第三部分介绍了基于边界元方法和统一变换的最新数值应用。读者：本书面向致力于解决物理和工程中边值问题的应用和数值研究的数学家和科学家。
This book describes state-of-the-art advances and applications of the unified transform and its relation to the boundary element method. The authors present the solution of boundary value problems from several different perspectives, in particular the type of problems modeled by partial differential equations (PDEs). They discuss recent applications of the unified transform to the analysis and numerical modeling of boundary value problems for linear and integrable nonlinear PDEs and the closely related boundary element method, a well-established numerical approach for solving linear elliptic PDEs. The text is divided into three parts. Part I contains new theoretical results on linear and nonlinear evolutionary and elliptic problems. New explicit solution representations for several classes of boundary value problems are constructed and rigorously analyzed. Part II is a detailed overview of variational formulations for elliptic problems. It places the unified transform approach in a classic context alongside the boundary element method and stresses its novelty. Part III presents recent numerical applications based on the boundary element method and on the unified transform. Audience: This book is intended for applied and numerical research mathematicians and scientists working on the solution of boundary value problems in physics and engineering.