Many physical problems that are usually solved by differential equation techniques can be
solved more effectively by integral equation methods. This work focuses exclusively on singular
integral equations and on the distributional solutions of these equations. A large number of
beautiful mathematical concepts are required to find such solutions which in tum can be
applied to a wide variety of scientific fields - potential theory me chanics fluid dynamics
scattering of acoustic electromagnetic and earth quake waves statistics and population
dynamics to cite just several. An integral equation is said to be singular if the kernel is
singular within the range of integration or if one or both limits of integration are infinite.
The singular integral equations that we have studied extensively in this book are of the
following type. In these equations f (x) is a given function and g(y) is the unknown function.
1. The Abel equation x x) = l g (y) d 0 a 1. ( Ct y ( ) a X - Y 2. The Cauchy type integral
equation b g (y) g(x)= (x)+).. l--dy a y-x where).. is a parameter. x Preface 3. The extension
b g (y) a (x) g (x) = J (x) +).. l--dy a y-x of the Cauchy equation. This is called the Carle
man equation.
