A Unified Approach To Boundary Value Problems
A Unified Approach To Boundary Value Problems
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DOI：
10.1137/1.9780898717068
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发表时间：
2008
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影响因子：
--
通讯作者：
A. Fokas
中科院分区：
文献类型：
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作者：
A. Fokas
这本书提出了一种分析二维可积偏微分方程组(PDE)初边值问题的新方法，这种方法是作者在1997年首次提出的，它是基于逆散射变换的思想。这种方法的独特之处还在于给出了线性边值问题显式解的新的积分表示，其中包括有限区间上的热方程和等边三角形内部的Helmholtz方程等经典问题。作者S的透彻介绍让感兴趣的读者迅速吸收了本书的基本结果，避免了许多计算细节。书中讨论了一些新的发展，包括半直线上和有限区间上线性发展方程的一种新的变换方法；某些积分的解析逆，如运动边界的衰减Radon变换和Dirichlet-to-Neumann映射；凸多边形中椭圆型偏微分方程组的解析和数值方法；以及可积的非线性偏微分方程组。结语提供了作者S新方法所用问题的清单，提出了有待解决的问题，并对如何将该方法应用于三维问题进行了一瞥。听众：边值问题的统一方法适用于高等本科生和研究生一年级的边值问题课程。应用数学家、工程师、理论物理学家、数学生物学家和其他使用PDE的学者也会发现这本书很有价值。内容：前言；导论；第一章：半直线上的发展方程；第二章：有限区间上的发展方程；第三章：渐近性和一种新的数值技巧；第四章：从偏微分方程组到经典变换；第五章：Riemann Hilbert和d-Bar问题；第六章：傅立叶变换及其变分；第七章：衰减Radon变换的逆与医学成像；第八章：运动边界上的狄利克雷到诺依曼映射；第九章：散度公式、整体关系和Lax对；第十章：半直线和有限区间上积分表示的重新推导；第11章：多边形域上的基本椭圆型偏微分方程解；第12章：简单多边形域上椭圆型偏微分方程组的新变换方法；第13章：Riemann Hilbert问题的表述；第14章：傅立叶平面上的配置法；第15章：从线性到可积的非线性偏微分方程组；第16章：半直线上的非线性可积偏微分方程组；第17章：可线性化的边界条件；第18章：广义Dirichlet到Neumann映射；第19章：振动Riemann Hilbert问题的渐近性；引言；书目；指数。
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. The author s thorough introduction allows the interested reader to quickly assimilate the essential results of the book, avoiding many computational details. Several new developments are addressed in the book, including a new transform method for linear evolution equations on the half-line and on the finite interval; analytical inversion of certain integrals such as the attenuated radon transform and the Dirichlet-to-Neumann map for a moving boundary; analytical and numerical methods for elliptic PDEs in a convex polygon; and integrable nonlinear PDEs. An epilogue provides a list of problems on which the author s new approach has been used, offers open problems, and gives a glimpse into how the method might be applied to problems in three dimensions. Audience: A Unified Approach to Boundary Value Problems is appropriate for courses in boundary value problems at the advanced undergraduate and first-year graduate levels. Applied mathematicians, engineers, theoretical physicists, mathematical biologists, and other scholars who use PDEs will also find the book valuable. Contents: Preface; Introduction; Chapter 1: Evolution Equations on the Half-Line; Chapter 2: Evolution Equations on the Finite Interval; Chapter 3: Asymptotics and a Novel Numerical Technique; Chapter 4: From PDEs to Classical Transforms; Chapter 5: Riemann Hilbert and d-Bar Problems; Chapter 6: The Fourier Transform and Its Variations; Chapter 7: The Inversion of the Attenuated Radon Transform and Medical Imaging; Chapter 8: The Dirichlet to Neumann Map for a Moving Boundary; Chapter 9: Divergence Formulation, the Global Relation, and Lax Pairs; Chapter 10: Rederivation of the Integral Representations on the Half-Line and the Finite Interval; Chapter 11: The Basic Elliptic PDEs in a Polygonal Domain; Chapter 12: The New Transform Method for Elliptic PDEs in Simple Polygonal Domains; Chapter 13: Formulation of Riemann Hilbert Problems; Chapter 14: A Collocation Method in the Fourier Plane; Chapter 15: From Linear to Integrable Nonlinear PDEs; Chapter 16: Nonlinear Integrable PDEs on the Half-Line; Chapter 17: Linearizable Boundary Conditions; Chapter 18: The Generalized Dirichlet to Neumann Map; Chapter 19: Asymptotics of Oscillatory Riemann Hilbert Problems; Epilogue; Bibliography; Index.