Abel’s Integral Equation and Singular Integral Equations

Abstract

Abel’s integral equation 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. Abel’s integral equation is the earliest example of an integral equation [2]. In Chapter 2, Abel’s integral equation was defined as a singular integral equation. Volterra integral equations of the first kind

$$f\left( x \right) = \lambda \int_{g\left( x \right)}^{h\left( x \right)} {K\left( {x,t} \right)u\left( t \right)dt,} $$

(7.1)

or of the second kind

$$u\left( x \right) = f\left( x \right) = \lambda \int_{g\left( x \right)}^{h\left( x \right)} {K\left( {x,t} \right)u\left( t \right)dt,} $$

(7.2)

are called singular [3–4] 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.