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Weibull renewal equation solution

Lösung der Weibullschen Erneuerungsgleichung

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Abstract

The existing solution methods for the Weibull Renewal Equation suffer from a lack of sufficient accuracy due to the singularity at the origin for some parameter values of the weibull density. The proposed method of solution provides accuracy to any desired degree of precision for all parameter values particularly in the singular range. The method utilizes a cubic spline approximation of the unknown renewal function and applies the Galerkin technique of integral equation solution. Gaussian quadratures are used to evaluate integrals. The singular nature of the integrand is handled by the Gauss-Jacobi quadrature. Results are compared with those obtained by simulation.

Zusammenfassung

Die bekannten Methoden zur Lösung der Weibullschen Erneuerungsgleichung sind für einige Parameter der Weibulldichte wegen der Singularität im Ursprung ungenau. Die hier vorgeschlagene Lösungsmethode gestattet Rechnungen mit jeder gewünschten Genauigkeit und insbesondere auch für Parameterwerte im singulären Bereich. Wir benützen kubische Spline-Approximation als unbekannte Erneuerungsfunktion, die Methoden von Galerkin zur Lösung der Integralgleichung und für die Integrale Gauss- bzw. Gauss-Jacobi-Quadratur. Die Ergebnisse vergleichen wir mit Resultaten, die durch Simulation erhalten wurden.

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Bilgen, S., Deligönül, Z.S. Weibull renewal equation solution. Computing 39, 71–76 (1987). https://doi.org/10.1007/BF02307714

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  • DOI: https://doi.org/10.1007/BF02307714

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