Approximation of Additive Integral Operators on Smooth Curves

Part of the Frontiers in Mathematics book series (FM)


We start with polynomial and spline approximation methods for the Cauchy singular integral equation
$$ \begin{gathered} (Au)(t) \equiv a(t)\phi (t) + \frac{{b(t)}} {{\pi i}}\int_\Gamma {\frac{{\phi (\tau ) d\tau }} {{\tau - t}}} + \overline {c(t)\phi (t)} + \overline {\frac{{d(t)}} {{\pi i}}\int_\Gamma {\frac{{\phi (\tau ) d\tau }} {{\tau - t}}} } \hfill \\ + \int\limits_\Gamma {k_0 (t,\tau )} \phi (\tau ) d\tau + \overline {\int\limits_\Gamma {k_1 (t,\tau )} \phi (\tau ) d\tau } = f(t), t \in \Gamma \hfill \\ \end{gathered} $$
where Γ is a simple closed Lyapunov contour. The coefficients a, b, c, d belong to given functional classes, and conditions imposed on the kernels k0(t, τ), k1(t, τ) ensure the compactness of the corresponding integral operators. The study of the stability of different approximation methods for equation (2.1) takes into account the smoothness of the coefficients a, b, c, d. Usually this is easier for equations with continuous coefficients. One encounters two main problems for equation (2.1). The first is connected with non-linearity of the operators considered over the field of complex numbers ℂ. However, this non-linearity can be called ‘weak’, and the corresponding difficulties are not of fundamental importance. Much more important is that the structure of the approximation sequences do not allow us to get a ‘good’ factorization of these sequences, even in the case where the coefficients a, b, c, d are continuous. Therefore, initially, approximation methods for Cauchy integral equations with conjugation were developed in two directions. One way is not to apply approximation methods to equation (2.1) directly but rather to an associated system of singular integral equations without conjugation [40, 118, 119, 120]. This leads to an unnecessary increase in the size of the systems of algebraic equations obtained. Some efforts were made to construct approximation methods working without passing to associated systems of singular integral equations [225, 226], but those methods are applicable to a restricted number of equations, since analytic functions in factorizations of certain combinations of the coefficients of equation (2.1) must satisfy additional metric relations.


Integral Operator Collocation Method Singular Integral Equation Singular Integral Operator Quadrature Method 
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© Birkhäuser Verlag AG 2008

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