• Honam Mathematical Journal
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• v.36 no.2
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• pp.291-303
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• 2014

In this paper, through a direct computation with subintervals partitioning [0, 1], we compute better a posteriori bounds for the average case error of the difference between the true value of $I(f)=\int_{0}^{1}f(x)dx$ with $f{\in}C^r$[0, 1] minus the composite trapezoidal rule and the composite trapezoidal rule minus the basic trapezoidal rule for $r{\geq}3$ by using zero mean-Gaussian.

• Honam Mathematical Journal
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• v.29 no.2
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• pp.245-257
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• 2007

We showed in [2] that if $r\leq2$, then the average error between simple Trapezoidal rule and the composite Trapezoidal rule on two consecutive subintervals is proportional to $h^{2r+3}$ using zero mean Gaussian distribution under the assumption that we have subintervals (for simplicity equal length) partitioning and that each subinterval has the length. In this paper, if $r\geq3$, we show that zero mean Gaussian distribution of average error between simple Trapezoidal rule and the composite Trapezoidal rule on two consecutive subintervals is bounded by $Ch^8$.

• Honam Mathematical Journal
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• v.30 no.3
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• pp.581-587
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• 2008

In this paper, to have a better a posteriori error bound of the average case error between the true value of I(f) and the Trapezoidal rule on subintervals using zero mean-Gaussian, we prove that a new average error between the difference of the true value of I(f) from the composite Trapezoidal rule and that of the composite Trapezoidal rule from the simple Trapezoidal rule is bounded by $c_rH^{2r+3}$ through direct computation of constants $c_r$ for r ${\leq}$ 2 under the assumption that we have subintervals (for simplicity equal length h) partitioning [0, 1].

• Journal of applied mathematics & informatics
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• v.8 no.1
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• pp.231-242
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• 2001

We showed in [2] that if r≤2, zero mean Gaussian of average error of the Trapezoidal rule is proportional to h/sub i//sup 2r+3/ on the interval [0,1]. In this paper, if r≥3, we show that zero mean Gaussian of average error of the Trapezoidal rule is bounded by Ch⁴/sub i/h⁴/sub j/.

• Journal of applied mathematics & informatics
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• v.13 no.1_2
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• pp.365-372
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• 2003

We show that if ${\gamma}$ $\leq$ 2, the average error of the composite Trapezoidal rule on two consecutive intervals is proportional to h$\^$2h+3/ where h is the length of each subinterval of the interval [0, 1]. As a result, we show that the Trapezoidal rule with equally spaced points is optimal in the average case setting when ${\gamma}$ $\leq$ 2.

• Communications of the Korean Mathematical Society
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• v.13 no.4
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• pp.903-911
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• 1998

We show that if r $\leq$ 2, the average error of the Trapezoidal rule is proportional to $n^{-min{r＋l, 3}}$ where n is the number of mesh points on the interval [D, 1]. As a result, we show that the Trapezoidal rule with equally spaced points is optimal in the average case setting when r $\leq$ 2.

• Journal of the Korean Mathematical Society
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• v.41 no.6
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• pp.957-976
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• 2004

It is well known that the application of the nonlinear coordinate transformations is useful for efficient numerical evaluation of weakly singular integrals. In this paper, we consider the trapezoidal rule combined with a nonlinear transformation $\Omega$$_{m}$(b;$\chi$), containing a parameter b, proposed first by Yun [14]. It is shown that the trapezoidal rule with the transformation $\Omega$$_{m}$(b;$\chi$), like the case of the Gauss-Legendre quadrature rule, can improve the asymptotic truncation error by using a moderately large b. By several examples, we compare the numerical results of the present method with those of some existing methods. This shows the superiority of the transformation $\Omega$$_{m}$(b;$\chi$).TEX>).

• Journal of Electrical Engineering and Technology
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• v.8 no.4
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• pp.872-878
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• 2013

In industrial control systems, flow measurement is a very important issue. It is frequently needed to calculate how much total fluid or gas flows through a cross-section. Flow volume measurement tools use simple sampling or rectangle methods. Actually, flow volume measurement process is an integration process. For this reason, measurement systems using instantaneous sampling technique cause considerably high errors. In order to make more accurate flow measurement, numerical integration methods should be used. Literally, for numerical integration method, Rectangular, Trapezoidal, Simpson, Romberg and Gaussian Quadrature methods are suggested. Among these methods, trapezoidal rule method is quite easy to calculate and is notably more accurate and contains no restrictive conditions. Therefore, it is especially convenient for the portable flow volume measurement systems. In this study, the volume measurement of air which is flowing through a cross-section is achieved by using PLC ladder diagram. The measurements are done using two different approaches. Trapezoidal rule method is proposed to measure the flow sensor signal to minimize measurement errors due to the classical sampling method as a different approach. It is concluded that the trapezoidal rule method is more effective than the classical sampling.

• The KIPS Transactions:PartA
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• v.12A no.5 s.95
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• pp.391-394
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• 2005

In this paper, we study the average case error of the Trapezoidal rule using zero mean-Gaussian. Assume that we have n subintervals (for simplicity equal length) partitioning [0,1] and that each subinterval has the length h. Then, for $r{\leq}2$, we show that the average error between simple Trapezoidal rule and the composite Trapezoidal rule on two consecutive subintervals is bounded by $h^{2r+3}$ through direct computation of constants $c_r$.

• Journal of applied mathematics & informatics
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• v.13 no.1_2
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• pp.353-363
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• 2003

In [2], if ${\gamma}$ $\leq$ 2, the average error of the composite Trapezoidal rule on two consecutive intervals turned out to be proportional to h$\^$2r+3/ where ${\gamma}$ is the number of differentiablity and h is the length of each uniform subinterval of the interval [0, 1] In this paper, we show that if ${\gamma}$ $\geq$ 3, the average error of the composite Trapezoidal rule on two consecutive intervals is bounded by Ch$\^$8/.