On the evaluation of one-dimensional Cauchy principal value integrals by rules based on cubic spline interpolation

AbstractIn this note we consider the numerical evaluation of one dimensional Cauchy principal value integrals of the form $$\rlap{--} \smallint _a^b \frac{{k(x)f(x)}}{{x - \lambda }}dx, a< \lambda< b,$$ by rules obtained by “subtracting out” the singularity and then applying product quadratures based on cubic spline interpolation at equally spaced nodes. Convergence results are established for Hölder continuous functions of order, μ, 0<μ≤1, and asymptotic rates are obtained for functionsf≠Ck[a, b],k=1, 2, 3 or 4. Some comparisons with other methods and numerical examples are also given.ZusammenfassungIn dieser Arbeit betrachten wir die numerische Bestimmung von eindimensionalen Integralen vom Cauchyschen Hauptwert vom Typ $$\rlap{--} \smallint _a^b \frac{{k(x)f(x)}}{{x - \lambda }}dx, a< \lambda< b,$$ mittels Regeln, die gewonnen wurden durch “Subtraktion” der Singularität und nachfolgender Applikation von Produkt-Quadraturen auf der Basis kubischer Spline-Interpolation auf äquidistanten Knoten. Übereinstimmende Resultate werden durch Hölder-kontinuierliche Funktionen der Ordnung μ, 0<μ≤1, erstellt, und asymptotische Konvergenzraten erhält man für Funktionenf≠Ck[a,b],k=1, 2, 3 oder 4. Vergleiche mit anderen Methoden und numerische Beispiele sind ebenfalls angegeben.

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