Applied Numerical Methods

Greenspan, Donald; Carnahan, Brice; Luther, H.; Wilkes, James O.

doi:10.2307/2004855

Cited by 708 publications

(51 citation statements)

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“…where T (x, t) is the absolute temperature. The physical requirement that T (x, t) ≥ 0 leads to the following relation between the step-sizes [3,4]:…”

Section: Introductionmentioning

confidence: 99%

“…The situation for nonlinear evolution equations is more complex. Standard explicit methods for constructing finite-difference schemes that impose a condition of local stability generally lead to functional relations between the step-sizes that also involve the values of the dependent variable [1,2,3,4]. However, the desired result would be a formulation of explicit schemes such that the functional relations between the time and space step-sizes would depend only on these step-sizes and not also on the value of the dependent variable.…”

Section: Introductionmentioning

confidence: 99%

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Relation between the time and space step‐sizes in nonstandard finite‐difference schemes for the Fisher equation

Mickens¹

1997

Numer. Methods Partial Differential Eq.

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“…where T (x, t) is the absolute temperature. The physical requirement that T (x, t) ≥ 0 leads to the following relation between the step-sizes [3,4]:…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Relation between the time and space step‐sizes in nonstandard finite‐difference schemes for the Fisher equation

Mickens¹

1997

Numer. Methods Partial Differential Eq.

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“…Microsoft Excel software has been employed to process the experimental data. Regression analysis concerning each quality indicator of the cleaner, by means of the least-squares method, has been performed [30]. As is known, this criterion is a technique for fitting the "best" model function f(x) to a data set formed by m points (x i ,ŷ i ), I = 1, .…”

Section: Discussionmentioning

confidence: 99%

A New Spiral Potato Cleaner to Enhance the Removal of Impurities and Soil Clods in Potato Harvesting

et al. 2020

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Sustainability in the agricultural field suggests the conservation and maintenance of a natural environment of soil. Nevertheless, in the potato production chain, the mechanized harvest is carried out with the concurring removal of impurities and fertile soil. The authors have developed a new spiral potato cleaner which is able to capture and efficiently remove soil lumps of various sizes and shapes, as well as various plant residues. Theoretical and experimental studies have been performed on this soil clod cleaner to determine the structural and kinematic parameters that provide efficient capturing, motion and sifting down of the soil, through the gaps between the helices of its cleaning spirals. An analytical description of the motion of the clod of soil has been made and a system of differential equations has been compiled, whose numerical solution made it possible to determine the indicated reasonable operative parameters of the developed spiral potato cleaner. The results of the experimental research confirmed those ones obtained through the numerical solution of the mathematical model, i.e., rational angular speed ω of the rotation of the spirals from 20.0 to 30.0 rad s−1 and the radius R of spirals between 0.12 and 0.15 m.

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“…Engng. 2003; 19:601-614) original method, an important feature of the new method is that no iterations are needed except in the preparatory phase for some cases.Runge-Kutta methods [1][2][3][4][5][6][7][8] or information from several different steps (multi-step methods) such as Adams-Bashforth or Adams-Moulton methods [1, 3-5, 8, 9].Even though there are many special methods for special cases (like stiff problems for example, see [10]) there is still a certain lack in efficiency when solving high-order non-linear ODEs. Runge-Kutta methods for example, especially those higher-order ones, take a considerable number of evaluations of the governing differential equation in each step.In modern times symbolic calculation packages, such as Maple and Mathematica, offer a variety of analytical manipulation capabilities.…”

mentioning

confidence: 99%

“…Runge-Kutta methods [1][2][3][4][5][6][7][8] or information from several different steps (multi-step methods) such as Adams-Bashforth or Adams-Moulton methods [1, 3-5, 8, 9].…”

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confidence: 99%

On the improvement of a numerical method for solving high‐order non‐linear ordinary differential equations

Zhu

Hammel

2006

Commun. Numer. Meth. Engng.

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SUMMARYThere have been many approaches to solve ordinary differential equations numerically. Even though many numerical methods can provide very good approximate solutions they need considerable calculation effort, often through iterations. The advance of symbolic manipulation packages such as Maple gives the opportunity for new approaches to this type of problems. This paper will discuss an improvement to one of these new approaches enabled by the availability of these packages, to obtain a numerical solution of an initial-value problem governed by a high-order non-linear ordinary differential equation. The method we propose here is based on Zhu and Phan's earlier paper (Commun. Numer. Meth. Engng. 2003; 19:601-614) but we shall show an improved order of accuracy for the case of the highest-order derivative being of an even order. In any case it provides the ability to further increase the order of accuracy under certain conditions of differentiability. Like Zhu and Phan's (Commun. Numer. Meth. Engng. 2003; 19:601-614) original method, an important feature of the new method is that no iterations are needed except in the preparatory phase for some cases.

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Applied Numerical Methods

Cited by 708 publications

References 0 publications

Relation between the time and space step‐sizes in nonstandard finite‐difference schemes for the Fisher equation

Relation between the time and space step‐sizes in nonstandard finite‐difference schemes for the Fisher equation

A New Spiral Potato Cleaner to Enhance the Removal of Impurities and Soil Clods in Potato Harvesting

On the improvement of a numerical method for solving high‐order non‐linear ordinary differential equations

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