Linear and Nonlinear Integral Equations pp 237-260 | Cite as
Abel’s Integral Equation and Singular Integral Equations
Chapter
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 or of the second kind are called singular [3–4] if:
$$f\left( x \right) = \lambda \int_{g\left( x \right)}^{h\left( x \right)} {K\left( {x,t} \right)u\left( t \right)dt,} $$
(7.1)
$$u\left( x \right) = f\left( x \right) = \lambda \int_{g\left( x \right)}^{h\left( x \right)} {K\left( {x,t} \right)u\left( t \right)dt,} $$
(7.2)
- 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.
Keywords
Integral Equation Recurrence Relation Singular Integral Equation Noise Term Volterra Integral Equation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References
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