345 research outputs found
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
Qualitative modeling of chaotic logical circuits and walking droplets: a dynamical systems approach
Logical circuits and wave-particle duality have been studied for most of the 20th century. During the current century scientists have been thinking differently about these well-studied systems. Specifically, there has been great interest in chaotic logical circuits and hydrodynamic quantum analogs.
Traditional logical circuits are designed with minimal uncertainty. While this is straightforward to achieve with electronic logic, other logic families such as fluidic, chemical, and biological, naturally exhibit uncertainties due to their inherent nonlinearity. In recent years, engineers have been designing electronic logical systems via chaotic circuits. While traditional boolean circuits have easily determined outputs, which renders dynamical models unnecessary, chaotic logical circuits employ components that behave erratically for certain inputs.
There has been an equally dramatic paradigm shift for studying wave-particle systems. In recent years, experiments with bouncing droplets (called walkers) on a vibrating fluid bath have shown that quantum analogs can be studied at the macro scale. These analogs help us ask questions about quantum mechanics that otherwise would have been inaccessible. They may eventually reveal some unforeseen properties of quantum mechanics that would close the gap between philosophical interpretations and scientific results.
Both chaotic logical circuits and walking droplets have been modeled as differential equations. While many of these models are very good in reproducing the behavior observed in experiments, the equations are often too complex to analyze in detail and sometimes even too complex for tractable numerical solution. These problems can be simplified if the models are reduced to discrete dynamical systems. Fortunately, both systems are very naturally time-discrete. For the circuits, the states change very rapidly and therefore the information during the process of change is not of importance. And for the walkers, the position when a wave is produced is important, but the dynamics of the droplets in the air are not.
This dissertation is an amalgam of results on chaotic logical circuits and walking droplets in the form of experimental investigations, mathematical modeling, and dynamical systems analysis. Furthermore, this thesis makes connections between the two topics and the various scientific disciplines involved in their studies
Detailed balance in Horava-Lifshitz gravity
We study Horava-Lifshitz gravity in the presence of a scalar field. When the
detailed balance condition is implemented, a new term in the gravitational
sector is added in order to maintain ultraviolet stability. The
four-dimensional theory is of a scalar-tensor type with a positive cosmological
constant and gravity is nonminimally coupled with the scalar and its gradient
terms. The scalar field has a double-well potential and, if required to play
the role of the inflation, can produce a scale-invariant spectrum. The total
action is rather complicated and there is no analog of the Einstein frame where
Lorentz invariance is recovered in the infrared. For these reasons it may be
necessary to abandon detailed balance. We comment on open problems and future
directions in anisotropic critical models of gravity.Comment: 10 pages. v2: discussion expanded and improved, section on
generalizations added, typos corrected, references added, conclusions
unchange
New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus
This reprint focuses on exploring new developments in both pure and applied mathematics as a result of fractional behaviour. It covers the range of ongoing activities in the context of fractional calculus by offering alternate viewpoints, workable solutions, new derivatives, and methods to solve real-world problems. It is impossible to deny that fractional behaviour exists in nature. Any phenomenon that has a pulse, rhythm, or pattern appears to be a fractal. The 17 papers that were published and are part of this volume provide credence to that claim. A variety of topics illustrate the use of fractional calculus in a range of disciplines and offer sufficient coverage to pique every reader's attention
Computational and numerical analysis of differential equations using spectral based collocation method.
Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.In this thesis, we develop accurate and computationally eļ¬cient spectral collocation-based methods,
both modiļ¬ed and new, and apply them to solve diļ¬erential equations. Spectral collocation-based
methods are the most commonly used methods for approximating smooth solutions of diļ¬erential
equations deļ¬ned over simple geometries. Procedurally, these methods entail transforming the gov
erning diļ¬erential equation(s) into a system of linear algebraic equations that can be solved directly.
Owing to the complexity of expanding the numerical algorithms to higher dimensions, as reported
in the literature, researchers often transform their models to reduce the number of variables or
narrow them down to problems with fewer dimensions. Such a process is accomplished by making
a series of assumptions that limit the scope of the study. To address this deļ¬ciency, the present
study explores the development of numerical algorithms for solving ordinary and partial diļ¬erential
equations deļ¬ned over simple geometries. The solutions of the diļ¬erential equations considered are
approximated using interpolating polynomials that satisfy the given diļ¬erential equation at se
lected distinct collocation points preferably the Chebyshev-Gauss-Lobatto points. The size of the
computational domain is particularly emphasized as it plays a key role in determining the number
of grid points that are used; a feature that dictates the accuracy and the computational expense of
the spectral method. To solve diļ¬erential equations deļ¬ned on large computational domains much
eļ¬ort is devoted to the development and application of new multidomain approaches, based on
decomposing large spatial domain(s) into a sequence of overlapping subintervals and a large time
interval into equal non-overlapping subintervals. The rigorous analysis of the numerical results con
ļ¬rms the superiority of these multiple domain techniques in terms of accuracy and computational
eļ¬ciency over the single domain approach when applied to problems deļ¬ned over large domains.
The structure of the thesis indicates a smooth sequence of constructing spectral collocation method
algorithms for problems across diļ¬erent dimensions. The process of switching between dimensions
is explained by presenting the work in chronological order from a simple one-dimensional problem
to more complex higher-dimensional problems. The preliminary chapter explores solutions of or
dinary diļ¬erential equations. Subsequent chapters then build on solutions to partial diļ¬erential
i
equations in order of increasing computational complexity. The transition between intermediate
dimensions is demonstrated and reinforced while highlighting the computational complexities in
volved. Discussions of the numerical methods terminate with development and application of a
new method namely; the trivariate spectral collocation method for solving two-dimensional initial
boundary value problems. Finally, the new error bound theorems on polynomial interpolation are
presented with rigorous proofs in each chapter to benchmark the adoption of the diļ¬erent numerical
algorithms. The numerical results of the study conļ¬rm that incorporating domain decomposition
techniques in spectral collocation methods work eļ¬ectively for all dimensions, as we report highly
accurate results obtained in a computationally eļ¬cient manner for problems deļ¬ned on large do
mains. The ļ¬ndings of this study thus lay a solid foundation to overcome major challenges that
numerical analysts might encounter
Adomian decomposition method, nonlinear equations and spectral solutions of burgers equation
Tese de doutoramento. CiĆŖncias da Engenharia. 2006. Faculdade de Engenharia. Universidade do Porto, Instituto Superior TĆ©cnico. Universidade TĆ©cnica de Lisbo
Differential Equation Models in Applied Mathematics
The present book contains the articles published in the Special Issue āDifferential Equation Models in Applied Mathematics: Theoretical and Numerical Challengesā of the MDPI journal Mathematics. The Special Issue aimed to highlight old and new challenges in the formulation, solution, understanding, and interpretation of models of differential equations (DEs) in different real world applications. The technical topics covered in the seven articles published in this book include: asymptotic properties of high order nonlinear DEs, analysis of backward bifurcation, and stability analysis of fractional-order differential systems. Models oriented to real applications consider the chemotactic between cell species, the mechanism of on-off intermittency in food chain models, and the occurrence of hysteresis in marketing. Numerical aspects deal with the preservation of mass and positivity and the efficient solution of Boundary Value Problems (BVPs) for optimal control problems. I hope that this collection will be useful for those working in the area of modelling real-word applications through differential equations and those who care about an accurate numerical approximation of their solutions. The reading is also addressed to those willing to become familiar with differential equations which, due to their predictive abilities, represent the main mathematical tool for applying scenario analysis to our changing world
Bivariate pseudospectral collocation algorithms for nonlinear partial differential equations.
Doctor of Philosophy in Applied Matheatics. University of KwaZulu-Natal, Pietermaritzburg 2016.Abstract available in PDF file
Poster abstract research showcase College of Science and Technology
Welcome to the College of Science and Technology Research and Innovation Showcase 2014, an
event which celebrates the research achievements of our science disciplines.
Our research brings together scientists from architecture and the built environment through to
computing, engineering, mathematics and physics and biology, geography and environmental
science. We are committed to build on our strengths, and our key vision is to drive research growth
and impact through exploitation of the synergy between research, innovation and enterprise.
This yearās showcase event includes over 70 posters illustrating the excellent research being
pursued, a Dean's prize recognising the achievements of an early career researcher, prizes for the
best student and best studentsā posters and journal papers, and this proceedings of abstracts
showing the high quality and range of research in the College of Science and Technology
- ā¦