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Positive quadratures for volterra equations

Positive Quadraturen für Volterra-Gleichungen

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Abstract

The present paper deals with discretizations to linear Volterra equations which preserve the possible positivity of the Volterra operator. It is shown that the method must be implicit and e.g. that the repeated trapezoidal rule has this property. It is then shown how this property can be used in studying the asymptotic behaviour of the solutionsx n of the discretized equations, asn→∞ with a fixed steph.

Zusammenfassung

Die vorgelegte Arbeit beschäftigt sich mit der Diskretisierung von linearen Volterra-Gleichungen, welche die mögliche Positivität des Volterra-Operators erhalten. Es wird gezeigt, daß die Methode implizit sein muß und daß z. B. die Trapezregel diese Eigenschaft hat. Es wird weiters gezeigt, daß diese Eigenschaft benützt werden kann, um die asymptotischen Eigenschaften der Lösungenx n der diskretisierten Gleichung zu studieren, wennn→∞ bei einer festen Schrittweiteh.

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References

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Nevanlinna, O. Positive quadratures for volterra equations. Computing 16, 349–357 (1976). https://doi.org/10.1007/BF02252083

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Keywords

  • Asymptotic Behaviour
  • Computational Mathematic
  • Trapezoidal Rule
  • Discretized Equation
  • Present Paper Deal