Nonlinear Singular Integral Equations

Abel’s integral equation, linear or nonlinear, occurs in many branches of scientific fields [1], such as microscopy, seismology, radio astronomy, electron emission, atomic scattering, radar ranging, plasma diagnostics, X-ray radiography, and optical fiber evaluation. Linear Abel’s integral equation is the earliest example of an integral equation. In Chapter 2, Abel’s integral equation was defined as a singular integral equation. Volterra integral equations of the first kind 17.1 $$f\left( x \right) = \lambda \int_{g\left( x \right)}^{h\left( x \right)} {K\left( {x,t} \right)u\left( t \right)dt,} $$ or of the second kind 17.2 $$u\left( x \right) = f\left( x \right) + \int_{g\left( x \right)}^{h\left( x \right)} {K\left( {x,t} \right)u\left( t \right)dt,} $$ are called singular [2–8] if: 1. one of the limits of integration g(x), h(x) or both are infinite, or 2. if the kernel K(x, t) becomes infinite at one or more points at the range of integration.