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.

Photo-cross-linked poly(D,L-lactide)-based networks. Structural characterization by HR-MAS NMR spectroscopy and hydrolytic degradation behavior

To date, biodegradable networks and particularly their kinetic chain lengths have been characterized by analysis of their degradation products in solution. We characterize the network itself by NMR analysis in the solvent-swollen state under magic angle spinning conditions. The networks were prepared by photoinitiated cross-linking of poly(dl-lactide)−dimethacrylate macromers (5 kg/mol) in the presence of an unreactive diluent. Using diffusion filtering and 2D correlation spectroscopy techniques, all network components are identified. By quantification of network-bound photoinitiator fragments, an average kinetic chain length of 9 ± 2 methacrylate units is determined. The PDLLA macromer solution was also used with a dye to prepare computer-designed structures by stereolithography. For these networks structures, the average kinetic chain length is 24 ± 4 methacrylate units. In all cases the calculated molecular weights of the polymethacrylate chains after degradation are maximally 8.8 kg/mol, which is far below the threshold for renal clearance. Upon incubation in phosphate buffered saline at 37 °C, the networks show a similar mass loss profile in time as linear high-molecular-weight PDLLA (HMW PDLLA). The mechanical properties are preserved longer for the PDLLA networks than for HMW PDLLA. The initial tensile strength of 47 ± 2 MPa does not decrease significantly for the first 15 weeks, while HMW PDLLA lost 85 ± 5% of its strength within 5 weeks. The physical properties, kinetic chain length, and degradation profile of these photo-cross-linked PDLLA networks make them most suited materials for orthopedic applications and use in (bone) tissue engineering.

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.