Stability and convergence of an effective numerical method for the time-space fractional Fokker-Planck equation with a nonlinear source term

Fractional Fokker-Planck equations (FFPEs) have gained much interest recently for describing transport dynamics in complex systems that are governed by anomalous diffusion and nonexponential relaxation patterns. However, effective numerical methods and analytic techniques for the FFPE are still in their embryonic state. In this paper, we consider a class of time-space fractional Fokker-Planck equations with a nonlinear source term (TSFFPE-NST), which involve the Caputo time fractional derivative (CTFD) of order α ∈ (0, 1) and the symmetric Riesz space fractional derivative (RSFD) of order μ ∈ (1, 2). Approximating the CTFD and RSFD using the L1-algorithm and shifted Grunwald method, respectively, a computationally effective numerical method is presented to solve the TSFFPE-NST. The stability and convergence of the proposed numerical method are investigated. Finally, numerical experiments are carried out to support the theoretical claims.

Encyclopedia contributions: entries A-B

Sections contributed by Jean Sim Agricultural Colleges; p.12 Anzac Park, Townsvile; p.22 Anzac Square, Brisbane; pp.22-23 Benson, Albert Herbert; p.86 Bick, Edward Walter; p.88 Bougainvillea Gardens; p.101 Bowen Park; pp.101-102 Boyd, A.J.; p.103 Brisbane Botanic Gardens; pp.104-105 Bush-house; pp.119-121

Encyclopedia contributions: entries C-G in Oxford Companion to Australian Gardens

Caulfield, Harold William; p.131 Cowan, Alexander; p.164 Cowley, Ebenezer; p.164 East Talgai Station; p.193 Eaves, S.H.; p.193-194 Edgar, J.S.; p.196 Everist, Selwyn; p.206 Experimental Farms and Gardens; pp.207-208 Government Houses - Queensland; pp.267-268

Encyclopedia contributions: Entries H-M in Oxford Companion to Australian Gardens

The Oxford Companion to Australian Gardens is the first comprehensive reference book to cover all aspects of Australian gardens, and the history of gardening and garden design in Australia. The book is comprised of over 400 thematic, bibliographic and place based entries, and is extensively illustrated and cross referenced to ensure ease of use and thorough coverage of the field. The Companion contributes to the understanding of gardens and gardening by including entries on designed landscapes, agriculture, architecture, art, botany, ecology, forestry, horticulture, landscape architecture, town planning and viticulture and will become the standard reference on the subject. Herbert,

Intrinsically photosensitive melanopsin retinal ganglion cell contributions to the pupillary light reflex and circadian rhythm

Recently discovered intrinsically photosensitive melanopsin retinal ganglion cells contribute to the maintenance of pupil diameter, recovery and post-illumination components of the pupillary light reflex and provide the primary environmental light input to the suprachiasmatic nucleus for photoentrainment of the circadian rhythm. This review summarises recent progress in understanding intrinsically photosensitive ganglion cell histology and physiological properties in the context of their contribution to the pupillary and circadian functions and introduces a clinical framework for using the pupillary light reflex to evaluate inner retinal (intrinsically photosensitive melanopsin ganglion cell) and outer retinal (rod and cone photoreceptor) function in the detection of retinal eye disease.

Encyclopedia contributions: entries N-R in Oxford Companion to Australian

New Farm Park; p.436 Oakman, H.O. (with Jan Seto); p.449 Paranella, Jose; p.463 Perrott family; p.469 Pink, James; p.475 Queensland; pp.495-497 The Queenslander; pp.497-498 The Queensland Horticulturist; p.498 Rawson, Mina; p.502 Rockhampton Botanic Gardens; p.572

Oxford Companion to Australian Gardens [contributions: entries S-V]

Shelton, Edward Mason; p.548 Sherwood Arboretum; p.550 Soutter, William; pp.563-564 Styles or gardens and designed landscapes; pp.575-576 Summer-house; pp.579-580 Trapnell, Walter George; p.602 Tropical Gardens; pp.604-605 Wickham Park; p.642 Wijaya, Made; p.642 Williams, George; p.644 Williams, Keith A.W.; p.644 Verandah Garadening; p.614

Bias in the nonlinear filter generator output sequence

Nonlinear filter generators are common components used in the keystream generators for stream ciphers and more recently for authentication mechanisms. They consist of a Linear Feedback Shift Register (LFSR) and a nonlinear Boolean function to mask the linearity of the LFSR output. Properties of the output of a nonlinear filter are not well studied. Anderson noted that the m-tuple output of a nonlinear filter with consecutive taps to the filter function is unevenly distributed. Current designs use taps which are not consecutive. We examine m-tuple outputs from nonlinear filter generators constructed using various LFSRs and Boolean functions for both consecutive and uneven (full positive difference sets where possible) tap positions. The investigation reveals that in both cases, the m-tuple output is not uniform. However, consecutive tap positions result in a more biased distribution than uneven tap positions, with some m-tuples not occurring at all. These biased distributions indicate a potential flaw that could be exploited for cryptanalysis

Nonlinear diffusion and exclusion processes with contact interactions

Exclusion processes on a regular lattice are used to model many biological and physical systems at a discrete level. The average properties of an exclusion process may be described by a continuum model given by a partial differential equation. We combine a general class of contact interactions with an exclusion process. We determine that many different types of contact interactions at the agent-level always give rise to a nonlinear diffusion equation, with a vast variety of diffusion functions D(C). We find that these functions may be dependent on the chosen lattice and the defined neighborhood of the contact interactions. Mild to moderate contact interaction strength generally results in good agreement between discrete and continuum models, while strong interactions often show discrepancies between the two, particularly when D(C) takes on negative values. We present a measure to predict the goodness of fit between the discrete and continuous model, and thus the validity of the continuum description of a motile, contact-interacting population of agents. This work has implications for modeling cell motility and interpreting cell motility assays, giving the ability to incorporate biologically realistic cell-cell interactions and develop global measures of discrete microscopic data.