Titlebook: Integral Equations; Theory and Numerical Wolfgang Hackbusch Book 1995 Birkhäuser Verlag 1995 Approximation.Integral equation.Interpolation

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Integral Equations978-3-0348-9215-5Series ISSN 0373-3149 Series E-ISSN 2296-6072

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Introduction,We begin by recalling the following example from the analysis of ordinary differential equations. Consider the initial value problem . Integration from . to . reduces this to the integral equation .. One reason why the reformulation (2) is of interest is because it is more suitable than (1) for proving existence and uniqueness of a solutions.

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Theory of Fredholm Integral Equations of the Second Kind,In the very end of the last century, Erik Ivar Fredholm (Stockholm) investigated those equations, which are now named in honour of him. Together with results of Hilbert, his theory led to the development of functional analysis, which took shape in the beginning of this century.

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Numerical Treatment of Fredholm Integral Equations of the Second Kind,In §§4.4.2-S and §4.4.7, several possibilities for constructing the discrete analogue . of an operator . will be described. The general properties of such approximations will be investigated in the following section.

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,Abel’s Integral Equation,The following Volterra integral equation of the first kind is due to Abel (1823): .. Since the denominator . has a zero at y=x, the integral in (1) is to be understood in the improper sense (cf. §6.1.3) and Abel’s integral equation is an example of a weakly singular equation.

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Singular Integral Equations,Let the function . be defined on .=[.] and, possibly, be singular at an interior point .∈(.). Recall that the improper integral was defined by.if both limits exist (cf. §6.1.3). By Remark 6.1.2a, the improper integral exists for . (.): = |.|. with .>-1. For.(i.e, s=-1) one obtains