Solution of the volterra equation of renewal theory with the galerkin technique using cubic splines

Deligönül, Z. Șeyda; Bilgen, Semih

doi:10.1080/00949658408810751

Cited by 28 publications

(8 citation statements)

References 11 publications

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“…In [25], Tortorella has proposed two new methods for computing renewal functions (the ''trapezoidal'' and the ''Simpson'' methods) which he has tested with success in [26] against [2,10,11,23]. We here compare our results to those by Tortorella for two examples from [26].…”

Section: Comparison With Tortorella's Methods [2526]supporting

confidence: 50%

“…The results given by the ''trapezoidal'' and the Simpson'' methods from [26] are respectively referred to as M T (t) and M S (t). For the first example (previously in [11]), the distribution of U 1 is given by F U 1 ðtÞ ¼ 0:75ð1 À e À2t Þ þ 0:25ð1 À e À4t Þ and the exact renewal function (no delay case) is R U 1 ðtÞ ¼ t þ 0:75ð1 À e À0:8t Þ (see [24]). All the results by Tortorella are coherent with ours (see Fig.…”

Section: Comparison With Tortorella's Methods [2526]mentioning

confidence: 99%

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Discrete random bounds for general random variables and applications to reliability

Mercier¹

2007

European Journal of Operational Research

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Section: Comparison With Tortorella's Methods [2526]supporting

confidence: 50%

Section: Comparison With Tortorella's Methods [2526]mentioning

confidence: 99%

Discrete random bounds for general random variables and applications to reliability

Mercier¹

2007

European Journal of Operational Research

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“…The method proposed here extends the earlier one (Delig6niil, Bilgen, 1984), modifying the quadrature integration scheme to take care of the singularity of the Weibull distribution function. For cases where c~> 1, no singularity is involved, hance the method can be successfully used with pure Gauss-Legendre quadrature.…”

Section: Resultsmentioning

confidence: 99%

Weibull renewal equation solution

Bilgen

Deligönül

1987

Computing

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ZusammenfassungWeibull Renewal Equation Solution. 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.Liisung der Weibullschen Erneuerungsgleichung. Die bekannten Methoden zur Ltsung der Weibullschen Erneuerungsgleichung sind ffir einige Parameter der Weibulldichte wegen der Singularitfit im Ursprung ungenau. Die bier vorgeschlagene Ltsungsmethode gestattet Rechnungen mit jeder gewtinschten Genauigkeit und insbesondere auch fiir Parameterwerte im singul/iren Bereich. Wir beniitzen kubische Spline-Approximation als unbekannte Erneuerungsfunktion, die Methoden yon Galerkin zur Ltsung der Integralgleichung und fiir die Integrale Gauss-bzw. Gauss-Jacobi-Quadratur. Die Ergebnisse vergleichen wir mit Resultaten, die durch Simulation erhalten wurden.

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“…The one numerical integration method is done through cubic spline method that is proposed by Cleroux and McConalogue [24], and then it is developed by Baxter et al [25] for Weibull, Gamma, Lognormal, truncated Normal, and inverse Gaussian BFD. Deligönül and Bilgen [26] propose a method using cubic spline and Galerkin technique to obtain RNFs. The well-known numerical integration method is Riemann-Stieljies (RS) method that is proposed by Min Xie [8].…”

Section: Introductionmentioning

confidence: 99%

The Estimation of Renewal Functions Using the Mean Value Theorem for Integrals (MeVTI) Method

Sasongko¹,

Mahatma²

2016

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In the analysis of warranty, renewal functions are important in acquiring the expected number of failures of a nonrepairable component in a time interval. It is very difficult and complicated -if at all possible- to obtain a renewal function analytically. This paper proposes a numerical integration method for estimating renewal functions in the terms of renewal integral equations. The estimation is done through the Mean Value Theorem for Integrals (MeVTI) method after modifying the variable of the renewal integral equations. The accuracy of the estimation is measured by its comparison against the existing analytical approach of renewal functions, those are for Exponential, Erlang, Gamma, and Normal baseline failure distributions. The estimation of the renewal function for a Weibull baseline failure distribution as the results of the method is compared to that of the well-known numerical integration approaches, the Riemann-Stieljies and cubic spline methods. Keywords : Mean Value Theorem for Integrals, Renewal Functions, Renewal Integral Equations.

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Solution of the volterra equation of renewal theory with the galerkin technique using cubic splines

Cited by 28 publications

References 11 publications

Discrete random bounds for general random variables and applications to reliability

Discrete random bounds for general random variables and applications to reliability

Weibull renewal equation solution

The Estimation of Renewal Functions Using the Mean Value Theorem for Integrals (MeVTI) Method

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