Handbook of Linear Partial Differential Equations for Engineers and Scientists

Polyanin, Andrei D.; Назайкинский, В. Е.

doi:10.1201/9781420035322

Cited by 469 publications

(504 citation statements)

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“…Equation (B 10) when the left boundary is at the origin has been derived in the past in Masoliver et al [41] and can also be found in Polyanin [31]. The simplified expression when s 0 ¼ 0 and L s ¼ 0 has been reported in equation (2.1) of the main text.…”

Section: A2 Circular Territoriesmentioning

confidence: 99%

“…The subscript 0 for W in equation (2.4) indicates that the initial position of the animal is the origin and the Q n,s parameters are given by 2 Dt/r 2 ) and equation (2.7) becomes the probability distribution of a Brownian walker confined in a circular domain of radius r reported in Polyanin [31]. These limiting procedures, in both Cartesian and polar geometries, are also valid in the more general case, reported in appendix B in equations (B 10) and (B 22), of off-centre initial conditions and off-centre location of the territory centroid.…”

Section: The Animalmentioning

confidence: 99%

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Predicting oscillatory dynamics in the movement of territorial animals

Giuggioli

Potts

2012

J. R. Soc. Interface.

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Understanding ecological processes relies upon the knowledge of the dynamics of each individual component. In the context of animal population ecology, the way animals move and interact is of fundamental importance in explaining a variety of observed patterns. Here, we present a theoretical investigation on the movement dynamics of interacting scent-marking animals. We study how the movement statistics of territorial animals is responsible for the appearance of damped oscillations in the mean square displacement (MSD) of the animals. This non-monotonicity is shown to depend on one dimensionless parameter, given by the ratio of the correlation distance between successive steps to the size of the territory. As that parameter increases, the time dependence of the animal's MSD displays a transition from monotonic, characteristic of Brownian walks, to non-monotonic, characteristic of highly correlated walks. The results presented here represent a novel way of determining the degree of persistence in animal movement processes within confined regions.

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Section: A2 Circular Territoriesmentioning

confidence: 99%

Section: The Animalmentioning

confidence: 99%

Predicting oscillatory dynamics in the movement of territorial animals

Giuggioli

Potts

2012

J. R. Soc. Interface.

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“…where F y v (t) is the amount of AI coming from a cross section of the yarn at y = y v at time t. The solution of this Cauchy problem for a vertical yarn is [7] C v (x, y,t) = F y v (0)…”

Section: The Macro-levelmentioning

confidence: 99%

Diffusion of Active Ingredients in Textiles

Goessens

Malengier

et al. 2012

J Math Model Algor

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Most practical textile models are based on a two scale approach: a one-dimensional fiber model and a fabric model, see [1]. No meso-level is used in between, i.e. the yarn scale is neglected in this setup. For dense textile substrates this seems appropriate as the yarns connect everywhere, but for loose fabrics or scrims this approach cannot be kept. Specifically when one is interested in tracking an active component released by the fibers, the yarn level plays an important role. This is because the saturation vapor pressure will influence the release rate from the fibers, and its value will vary over the yarn cross-section. Therefore, in this work we present a three step multiscale model: the active component is tracked in the fiber, the yarn, and finally at the fabric level. At the fiber level a onedimensional reduction to a non-linear diffusion equation is performed, and solved on a as needed basis. At the yarn level both a two-dimensional or a one-dimensional model can be applied, and finally the yarn result is upscaled to the fabric level.

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“…From the great mathematicians books [5,6,8,9, 10], we can find many and many concrete problems among partial differential equations, eigenfunctions, integral transforms and reproducing kernels which are admitting the application of these results.…”

Section: Concrete Realization Of the Reproducing Kernel Hilbert Pacesmentioning

confidence: 99%

Initial Value Problems in Linear Integral Operator Equations

Castro

Rodrigues

Saitoh³

2014

Topics in Mathematical Analysis and Applications

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For some general linear integral operator equations, we investigate consequent initial value problems by using the theory of reproducing kernels. A new method is proposed which -in particular -generates a new field among initial value problems, linear integral operators, eigenfunctions and values, integral transforms and reproducing kernels. In particular, examples are worked out for the integral equations of Lalesco-Picard, Dixon and Tricomi types.

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Handbook of Linear Partial Differential Equations for Engineers and Scientists

Cited by 469 publications

References 0 publications

Predicting oscillatory dynamics in the movement of territorial animals

Predicting oscillatory dynamics in the movement of territorial animals

Diffusion of Active Ingredients in Textiles

Initial Value Problems in Linear Integral Operator Equations

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