<i>Numerical Integration</i>

Davis, Philip J.; Rabinowitz, Philip D.; Romain, Jacques E.

doi:10.1063/1.3034795

Cited by 168 publications

(263 citation statements)

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“…with h ≤ 0.125) are very small (on the order of 1 percent or smaller) at any time point. Let us note that, for the validation of our numerical method, we have used the exact solution obtained for the simplest boundary and initial conditions as given by (4), (7) and (10). In the case that a different, more complicated boundary condition and/or initial condition are used in the diffusion problem, the exact solution may not always be available in an explicit form; in particular, it is rarely available in the 2D case (considered in sections 5 and 6 below).…”

Section: D Case: Validation Of the Methods And Numerical Testsmentioning

confidence: 99%

“…The accuracy of this integration rule on a uniform grid with the grid step size h is O(h 2 ) (e.g. Davis 1975). …”

Section: D Case: Validation Of the Methods And Numerical Testsmentioning

confidence: 99%

“…The trapezoidal rule of integration can be readily applied for integration over a two-dimensional domain (10). We then compute the integral…”

Section: Discretization Of the 2d Problemmentioning

confidence: 99%

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Some analytical and numerical approaches to understanding trap counts resulting from pest insect immigration

Bearup

Petrovskaya

Petrovskii

2015

Mathematical Biosciences

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Monitoring of pest insects is an important part of the integrated pest management. It aims to provide information about pest insect abundance at a given location. This includes data collection, usually using traps, and their subsequent analysis and/or interpretation. However, interpretation of trap count (number of insects caught over a fixed time) remains a challenging problem. Firstly, an increase in either the population density or insects activity can result in a similar increase in the number of insects trapped (the so called "activity-density" problem). Secondly, a genuine increase of the local population density can be attributed to qualitatively different ecological mechanisms such as multiplication or immigration. Identification of the true factor causing an increase in trap count is important as different mechanisms require different control strategies. In this paper, we consider a mean-field mathematical model of insect trapping based on the diffusion equation. Although the diffusion equation is a well-studied model, its analytical solution in closed form is actually available only for a few special cases, whilst in a more general case the problem has to be solved numerically. We choose finite differences as the baseline numerical method and show that numerical solution of the problem, especially in the realistic 2D case, is not at all straightforward as it requires a sufficiently accurate approximation of the diffusion fluxes. Once the numerical method is justified and tested, we apply it to the corresponding boundary problem where different types of boundary forcing describe different scenarios of pest insect immigration and reveal the corresponding patterns in the trap count growth.

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Section: D Case: Validation Of the Methods And Numerical Testsmentioning

confidence: 99%

“…The accuracy of this integration rule on a uniform grid with the grid step size h is O(h 2 ) (e.g. Davis 1975). …”

Section: D Case: Validation Of the Methods And Numerical Testsmentioning

confidence: 99%

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Some analytical and numerical approaches to understanding trap counts resulting from pest insect immigration

Bearup

Petrovskaya

Petrovskii

2015

Mathematical Biosciences

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“…(5b) and (5c) have no analytical solution, but can be solved numerically, e.g. by adjusted Simpson integration (Davis & Rabinowitz, 1975).…”

Section: Principlementioning

confidence: 99%

The optical rotator

Tandrup

Gundersen

Jensen

1997

Journal of Microscopy

113

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SummaryThe optical rotator is an unbiased, local stereological principle for estimation of cell volume and cell surface area in thick, transparent slabs. The underlying principle was first described in 1993 by Kiêu & Jensen ( J. Microsc. 170,[45][46][47][48][49][50][51] who also derived an estimator of length. In this study we further discuss the methods derived from this principle and present two new local volume estimators.The optical rotator benefits from information obtained in all three dimensions in thick sections but avoids over-/ underprojection problems at the extremes of the cell. Using computer-assisted microscopes the extra measurements demand minimal extra effort and make this estimator even more efficient when it comes to estimation of individual cell size than many of the previous local estimators. We demonstrate the principle of the optical rotator in an example (the cells in the dorsal root ganglion of the rat), evaluate its efficiency and compare it with other unbiased, local stereological principles available for estimation of cell volume and surface area.

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“…three suitable components for magnetic properties: alfa fiber (<110>//RD), cubic fiber (<100>//ND) and RD' fiber ({hkl}<001>), and an unsuitable component: gamma fiber (<111>//ND) 8 . The evaluation of the area under the "fiber curves" by mathematical numeric integration, according to the Simpson's rule 10 , can be seen in Figure 4. The results can hence be interpreted as indicators of the incidence of each type of fiber.…”

Section: Resultsmentioning

confidence: 99%

Effect of Aluminium Addition on the Magnetic Properties of a Semi-Processed Electrical Steel

Marra

Buono

2016

Mat. Res.

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Aluminum is often added to electrical steels since this element causes improvement of the magnetic properties. However the addition of aluminum is usually limited to 0.30 wt% 1 . Therefore, this work aimed at evaluating the influence of higher levels of aluminum in the magnetic properties of semiprocessed steels. To accomplish this, four types of steel were produced in laboratory with different aluminum contents (0.10, 0.30, 0.60 and 1.0 wt%). Carbon, manganese and silicon concentrations were kept constant. It was observed that an increased aluminum content caused reduction of the magnetic core loss, especially of the eddy current component. However, the magnetic permeability virtually wasn't influenced by aluminum concentration.

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Numerical Integration

Cited by 168 publications

References 0 publications

Some analytical and numerical approaches to understanding trap counts resulting from pest insect immigration

Some analytical and numerical approaches to understanding trap counts resulting from pest insect immigration

The optical rotator

Effect of Aluminium Addition on the Magnetic Properties of a Semi-Processed Electrical Steel

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