69 research outputs found
A Nonlinear Splitting Algorithm for Systems of Partial Differential Equations with self-Diffusion
Systems of reaction-diffusion equations are commonly used in biological
models of food chains. The populations and their complicated interactions
present numerous challenges in theory and in numerical approximation. In
particular, self-diffusion is a nonlinear term that models overcrowding of a
particular species. The nonlinearity complicates attempts to construct
efficient and accurate numerical approximations of the underlying systems of
equations. In this paper, a new nonlinear splitting algorithm is designed for a
partial differential equation that incorporates self-diffusion. We present a
general model that incorporates self-diffusion and develop a numerical
approximation. The numerical analysis of the approximation provides criteria
for stability and convergence. Numerical examples are used to illustrate the
theoretical results
Biological control via "ecological" damping: An approach that attenuates non-target effects
In this work we develop and analyze a mathematical model of biological
control to prevent or attenuate the explosive increase of an invasive species
population in a three-species food chain. We allow for finite time blow-up in
the model as a mathematical construct to mimic the explosive increase in
population, enabling the species to reach "disastrous" levels, in a finite
time. We next propose various controls to drive down the invasive population
growth and, in certain cases, eliminate blow-up. The controls avoid chemical
treatments and/or natural enemy introduction, thus eliminating various
non-target effects associated with such classical methods. We refer to these
new controls as "ecological damping", as their inclusion dampens the invasive
species population growth. Further, we improve prior results on the regularity
and Turing instability of the three-species model that were derived in earlier
work. Lastly, we confirm the existence of spatio-temporal chaos
Quenching estimates for a non-Newtonian filtration equation with singular boundary conditions
In this paper, the quenching behavior of the non-Newtonian filtration equation (Ο(u))t = (|ux| rβ2 ux)x with singular boundary conditions, ux (0, t) = u βp (0, t), ux (a, t) = (1 β u(a, t))βq is investigated. Various conditions on the initial condition are shown to guarantee quenching at either the left or right boundary. Theoretical quenching rates and lower bounds to the quenching time are determined when Ο(u) = u and r = 2. Numerical experiments are provided to illustrate and provide additional validation of the theoretical estimates to the quenching rates and times
A variable nonlinear splitting algorithm for reaction diffusion systems with self- and cross-diffusion
Self- and cross-diffusion are important nonlinear spatial derivative terms that are included into biological models of predator-prey interactions. Self-diffusion models overcrowding effects, while cross-diffusion incorporates the response of one species in light of the concentration of another. In this paper, a novel nonlinear operator splitting method is presented that directly incorporates both self- and cross-diffusion into a computational efficient design. The numerical analysis guarantees the accuracy and demonstrates appropriate criteria for stability. Numerical experiments display its efficiency and accurac
A variable nonlinear splitting algorithm for reaction diffusion systems with self- and cross-diffusion
Self- and cross-diffusion are important nonlinear spatial derivative terms that are included into biological models of predator-prey interactions. Self-diffusion models overcrowding effects, while cross-diffusion incorporates the response of one species in light of the concentration of another. In this paper, a novel nonlinear operator splitting method is presented that directly incorporates both self- and cross-diffusion into a computational efficient design. The numerical analysis guarantees the accuracy and demonstrates appropriate criteria for stability. Numerical experiments display its efficiency and accurac
CASIMIR ENERGIES IN SPHERICALLY SYMMETRIC BACKGROUND POTENTIALS REVISITED
In this paper we reconsider the formulation for the computation of the Casimir energy in spherically symmetric background potentials. Compared to the previous analysis, the technicalities are much easier to handle and final answers are surprisingly simple
Casimir energies in spherically symmetric background potentials revisited
In this paper we reconsider the formulation for the computation of the
Casimir energy in spherically symmetric background potentials. Compared to the
previous analysis, the technicalities are much easier to handle and final
answers are surprisingly simple.Comment: 14 pages, 6 figure
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