Boundary Integral Equations

ISBN-13: 9783030711290 / Angielski / Miękka / 2022 / 804 str.

George C. Hsiao;Wolfgang L. Wendland

Opis

This book is devoted to the mathematical foundation of boundary integral equations. The combination of ?nite element analysis on the boundary with these equations has led to very e?cient computational tools, the boundary element methods (see e.g., the authors [139] and Schanz and Steinbach (eds.) [267]). Although we do not deal with the boundary element discretizations in this book, the material presented here gives the mathematical foundation of these methods. In order to avoid over generalization we have con?ned ourselves to the treatment of elliptic boundary value problems. The central idea of eliminating the ?eld equations in the domain and - ducing boundary value problems to equivalent equations only on the bou- ary requires the knowledge of corresponding fundamental solutions, and this idea has a long history dating back to the work of Green [107] and Gauss [95, 96]. Today the resulting boundary integral equations still serve as a major tool for the analysis and construction of solutions to boundary value problems.

Recenzje

"This second edition also expands the area of applicability of boundary integral equations ... . The authors are outstanding and well known researchers in the mathematical foundations of numerical methods, mainly the boundary element methods, for solving various boundary integral equations. The multitude of their original results is the basis of this impressive volume which, however, requires a fairly high level of mathematical knowledge in order to be useful." (Calin Ioan Gheorghiu, zbMATH 1477.65007, 2022)

Spis treści

Introduction.- Boundary Integral Equations.- Representation Formulae.- Sobolev Spaces.- Variational Formulations.- Electromagnetic Fields.- Introduction to Pseudodifferential Operators.- Pseudodifferential Operators as Integral Operators.- Pseudodifferential and Boundary Integral Operators.- Integral Equations on Recast as Pseudodifferential Equations.- Boundary Integral Equations on Curves in R^2. Remarks on Pseudodifferential Operators for Maxwell Equations.- Appendix A: Local Coordinates.- Appendix B: Vector Field Identities, Integration Formulae.- References.- Index.

O autorze

George C. Hsiao received a bachelor's degree in Civil Engineering from  National Taiwan University, a master's degree from Carnegie Institute of Technology in the same field,  and a doctorate degree in Mathematics  from Carnegie Mellon University. He is now the Carl J. Rees Professor of Mathematics Emeritus at the University of Delaware from which he retired in September 2012 after 43 years on the faculty of the Department of Mathematical Sciences. His primary research interests are integral equations and partial differential equations with their applications in mathematical physics and continuum mechanics.

Wolfgang L. Wendland, now Professor Emeritus at the University Stuttgart was studying mechanical engineering and mathematics at the Technical University Berlin and became Full Professor for Mathematics 1970-1986 at the TU Darmstadt and 1986-2005 at the University Stuttgart. His research interests are in Applied Mathematics with emphasis on partial differential equations and integral equations as well as approximation and numerical methods with applications to continuum mechanics of flow and elasticity problems.

Both authors are well known for their fundamental work on boundary integral equations and related topics.

Blurb

This is the second edition of the book which has two additional new chapters on Maxwell’s equations as well as a section on properties of solution spaces of Maxwell’s equations and their trace spaces. These two new chapters, which summarize the most up-to-date results in the literature for the Maxwell’s equations, are sufficient enough to serve as a self-contained introductory book on the modern mathematical theory of boundary integral equations in electromagnetics.

The book now contains 12 chapters and is divided into two parts. The first six chapters present modern mathematical theory of boundary integral equations that arise in fundamental problems in continuum mechanics and electromagnetics based on the approach of variational formulations of the equations. The second six chapters present an introduction to basic classical theory of the pseudo-differential operators. The aforementioned corresponding boundary integral operators can now be recast as pseudo-differential operators. These serve as concrete examples that illustrate the basic ideas of how one may apply the theory of pseudo-differential operators and their calculus to obtain additional properties for the corresponding boundary integral operators. These two different approaches are complementary to each other. Both serve as the mathematical foundation of the boundary element methods, which have become extremely popular and efficient computational tools for boundary problems in applications.

This book contains a wide spectrum of boundary integral equations arising in fundamental problems in continuum mechanics and electromagnetics. The book is a major scholarly contribution to the modern approaches of boundary integral equations, and should be accessible and useful to a large community of advanced graduate students and researchers in mathematics, physics, and engineering.