Page 5 - Handbook Of Integral Equations

P. 5

FOREWORD

Integral equations are encountered in various ﬁelds of science and numerous applications (in
elasticity, plasticity, heat and mass transfer, oscillation theory, ﬂuid dynamics, ﬁltration theory,
electrostatics, electrodynamics, biomechanics, game theory, control, queuing theory, electrical en-
gineering, economics, medicine, etc.).
Exact (closed-form) solutions of integral equations play an important role in the proper un-
derstanding of qualitative features of many phenomena and processes in various areas of natural
science. Lots of equations of physics, chemistry and biology contain functions or parameters which
are obtained from experiments and hence are not strictly ﬁxed. Therefore, it is expedient to choose
the structure of these functions so that it would be easier to analyze and solve the equation. As a
possible selection criterion, one may adopt the requirement that the model integral equation admit a
solution in a closed form. Exact solutions can be used to verify the consistency and estimate errors
of various numerical, asymptotic, and approximate methods.
More than 2100 integral equations and their solutions are given in the ﬁrst part of the book
(Chapters 1–6). A lot of new exact solutions to linear and nonlinear equations are included. Special
attention is paid to equations of general form, which depend on arbitrary functions. The other
equations contain one or more free parameters (the book actually deals with families of integral
equations); it is the reader’s option to ﬁx these parameters. Totally, the number of equations
described in this handbook is an order of magnitude greater than in any other book currently
available.
The second part of the book (Chapters 7–14) presents exact, approximate analytical, and numer-
ical methods for solving linear and nonlinear integral equations. Apart from the classical methods,
some new methods are also described. When selecting the material, the authors have given a
pronounced preference to practical aspects of the matter; that is, to methods that allow effectively
“constructing” the solution. For the reader’s better understanding of the methods, each section is
supplied with examples of speciﬁc equations. Some sections may be used by lecturers of colleges
and universities as a basis for courses on integral equations and mathematical physics equations for
graduate and postgraduate students.
For the convenience of a wide audience with different mathematical backgrounds, the authors
tried to do their best, wherever possible, to avoid special terminology. Therefore, some of the methods
are outlined in a schematic and somewhat simpliﬁed manner, with necessary references made to
books where these methods are considered in more detail. For some nonlinear equations, only
solutions of the simplest form are given. The book does not cover two-, three- and multidimensional
integral equations.
The handbook consists of chapters, sections and subsections. Equations and formulas are
numbered separately in each section. The equations within a section are arranged in increasing
order of complexity. The extensive table of contents provides rapid access to the desired equations.
For the reader’s convenience, the main material is followed by a number of supplements, where
some properties of elementary and special functions are described, tables of indeﬁnite and deﬁnite
integrals are given, as well as tables of Laplace, Mellin, and other transforms, which are used in the
book.
The ﬁrst and second parts of the book, just as many sections, were written so that they could be
read independently from each other. This allows the reader to quickly get to the heart of the matter.

1 2 3 4 5 6 7 8 9 10