Abel’s Integral Equation and Singular Integral Equations

Abdul-Majid Wazwaz ()
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Abdul-Majid Wazwaz: Saint Xavier University

Chapter Chapter 7 in Linear and Nonlinear Integral Equations, 2011, pp 237-260 from Springer

Abstract: 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 7.1 $$f\left( x \right) = \lambda \int_{g\left( x \right)}^{h\left( x \right)} {K\left( {x,t} \right)u\left( t \right)dt,} $$ or of the second kind 7.2 $$u\left( x \right) = f\left( x \right) = \lambda \int_{g\left( x \right)}^{h\left( x \right)} {K\left( {x,t} \right)u\left( t \right)dt,} $$ 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.

Keywords: Integral Equation; Recurrence Relation; Singular Integral Equation; Noise Term; Volterra Integral Equation (search for similar items in EconPapers)
Date: 2011
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DOI: 10.1007/978-3-642-21449-3_7

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