by the author to the English edition The book aims to present a powerful new tool of computational mechanics, complex variable boundary integral equations (CV-BIE). The book is conceived as a continuation of the classical monograph by N. I. Muskhelishvili into the computer era. Two years have passed since the Russian edition of the present book. We have seen growing interest in numerical simulation of media with internal structure, and have evidence of the potential of the new methods. The evidence was especially clear in problems relating to multiple grains, blocks, cracks, inclusions and voids. This prompted me, when preparing the English edition, to place more emphasis on such topics. The other change was inspired by Professor Graham Gladwell. It was he who urged me to abridge the chain of formulae and to increase the number of examples. Now the reader will find more examples showing the potential and advantages of the analysis. The first chapter of the book contains a simple exposition of the theory of real variable potentials, including the hypersingular potential and the hypersingular equations. This makes up for the absence of such exposition in current textbooks, and reveals important links between the real variable BIE and the complex variable counterparts. The chapter may also help readers who are learning or lecturing on the boundary element method.

DIVHigh-level treatment of one-dimensional singular integral equations covers Holder Condition, Hilbert and Riemann-Hilbert problems, Dirichlet problem, more. 1953 edition. /div

This 2000 book provided the first detailed exposition of the mathematical theory of boundary integral equations of the first kind on non-smooth domains.

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.

The boundary element method is more appropriate than the finite element method to tackle linear, wave propagation, infinite domain, mobile boundaries and unknown boundaries problems. In some engineering applications, both methods are combined. This book presents the mathematical basis of this method and its computer implementation. Numerous applications to fluid mechanics, mechanics of solids, acoustics and electromagnetism are developed.

This book focuses on surface layers fracture of cyclical contacting bodies (machine parts). Calculation models and calculating procedures of stress-strain states of cyclically contacting solids with cracks, are included. Recommendations for the optimization of operating parameters of joints (contact stresses magnitude, friction/lubrication conditions, materials crack resistance etc) for elements of rolling pairs (wheel–rail systems, backup roll – working roll of rolling mills etc.) and some fretting pairs are formulated.

The author's ambition for this publication was to make BEM accessible to the student as well as to the professional engineer. For this reason, his maintask was to organize and present the material in such a way so that the book becomes "user-friendly" and easy to comprehend, taking into account only the mathematics and mechanics to which students have been exposed during their undergraduate studies. This effort led to an innovative, in many aspects, way of presentingBEM, including the derivation of fundamental solutions, the integral representation of the solutions and the boundary integral equations for various governing differentialequations in a simple way minimizing a recourse to mathematics with which the student is not familiar. The indicial and tensorial notations, though they facilitate the author's work and allow to borrow ready to use expressions from the literature, have been avoided in the present book. Nevertheless, all the necessary preliminary mathematical concepts have been included in order to make the book complete and self-sufficient.Throughout the book, every concept is followed by example problems, which have been worked out in detail and with all the necessary clarifications. Furthermore, each chapter of the book is enriched with problems-to-solve. These problems serve a threefold purpose. Some of them are simple and aim at applying and better understanding the presented theory, some others are more difficult and aim at extending the theory to special cases requiring a deeper understanding of the concepts, and others are small projects which serve the purpose of familiarizing the student with BEM programming and the programs contained in the CD-ROM.The latter class of problems is very important as it helps students to comprehend the usefulness and effectiveness of the method by solving real-life engineering problems. Through these problems students realize that the BEM is a powerful computational tool and not an alternative theoretical approach for dealing with physical problems. My experience in teaching BEM shows that this is the students' most favorite type of problems. They are delighted to solve them, since they integrate their knowledge and make them feel confident in mastering BEM.The CD-ROM which accompanies the book contains the source codes of all the computer programs developed in the book, so that the student or the engineer can use them for the solution of a broad class of problems. Among them are general potential problems, problems of torsion, thermal conductivity,deflection of membranes and plates, flow of incompressible fluids, flow through porous media, in isotropic or anisotropic, homogeneous or composite bodies, as well as plane elastostatic problems in simply or multiply connected domains. As one can readily find out from the variety of the applications, the book is useful for engineers of all disciplines. The author is hopeful that the present book will introduce the reader to BEM in an easy, smooth and pleasant way and also contribute to itsdissemination as a modern robust computational tool for solving engineering problems.

This second edition of Linear Integral Equations continues the emphasis that the first edition placed on applications. Indeed, many more examples have been added throughout the text. Significant new material has been added in Chapters 6 and 8. For instance, in Chapter 8 we have included the solutions of the Cauchy type integral equations on the real line. Also, there is a section on integral equations with a logarithmic kernel. The bibliography at the end of the book has been exteded and brought up to date. I wish to thank Professor B.K. Sachdeva who has checked the revised man uscript and has suggested many improvements. Last but not least, I am grateful to the editor and staff of Birkhauser for inviting me to prepare this new edition and for their support in preparing it for publication. RamP Kanwal CHAYfERl Introduction 1.1. Definition An integral equation is an equation in which an unknown function appears under one or more integral signs Naturally, in such an equation there can occur other terms as well. For example, for a ~ s ~ b; a :( t :( b, the equations (1.1.1) f(s) = ib K(s, t)g(t)dt, g(s) = f(s) + ib K(s, t)g(t)dt, (1.1.2) g(s) = ib K(s, t)[g(t)fdt, (1.1.3) where the function g(s) is the unknown function and all the other functions are known, are integral equations. These functions may be complex-valued functions of the real variables s and t.

This book compiles solutions of linear theory of elasticity problems for isotropic and anisotropic bodies with sharp and rounded notches. It contains an overview of established and recent achievements, and presents the authors’ original solutions in the field considered with extensive discussion. The volume demonstrates through numerous, useful examples the effectiveness of singular integral equations for obtaining exact solutions of boundary problems of the theory of elasticity for bodies with cracks and notches. Incorporating analytical and numerical solutions of the problems of stress concentrations in solid bodies with crack-like defects, this volume is ideal for scientists and PhD students dealing with the problems of theory of elasticity and fracture mechanics.