Narrow Results

In this paper, we present an analysis and modeling of the electrical diffusion equation using the fractional calculus approach. This alternative representation for the current density is expressed in terms of the Caputo derivatives, the order for the space domain is $0<\beta \le 1$ and for the time domain is $0<\gamma \le 2$. We present solutions for the full fractional equation involving space and time fractional derivatives using numerical methods based on Fourier variable separation. The case with spatial fractional derivatives leads to Levy flight type phenomena, while the time fractional equation is related to sub- or super diffusion. We show that the mathematical concept of fractional derivatives can be useful to understand the behavior of semiconductors, the design of solar panels, electrochemical phenomena and the description of anomalous complex processes.

We study memory-based random walk models to understand diffusive motion in crowded heterogeneous environments. The models considered are non-Markovian as the current move of the random walk is determined by randomly selecting a move from history. At each step, particle can take right, left or stay moves which is correlated with the randomly selected past step. There is a perfect stay–stay correlation which ensures that the particle does not move if the randomly selected past step is a stay move. The probability of traversing the same direction as the chosen history or reversing it depends on the current time and the time or position of the history selected. The time- or position-dependent biasing in moves implicitly corresponds to the heterogeneity of the environment and dictates the long-time behavior of the dynamics that can be diffusive, sub or superdiffusive. A combination of analytical solution and Monte Carlo (MC) simulation of different random walk models gives rich insight on the effects of correlations on the dynamics of a system in heterogeneous environment.

We consider a rather general class of non-local in time Fokker–Planck equations and show by means of the entropy method that as $t\to \infty$, the solution converges in $L^1$ to the unique steady state. Important special cases are the time-fractional and ultraslow diffusion case. We also prove estimates for the rate of decay. In contrast to the classical (local) case, where the usual time derivative appears in the Fokker–Planck equation, the obtained decay rate depends on the entropy, which is related to the integrability of the initial datum. It seems that higher integrability of the initial datum leads to better decay rates and that the optimal decay rate is reached, as we show, when the initial datum belongs to a certain weighted $L^2$ space. We also show how our estimates can be adapted to the discrete-time case thereby improving known decay rates from the literature.

From the defects-free self-assembled organic layers (SAMs) of CH₃(${\text{CH}}_2{)}_{n-1}$SH molecules with short chain lengths ($n=4,6,8$) electrodeposited on the (111) surface of monocrystalline gold previously prepared, monitored defects (pinholes) were potential-induced from cyclic partial reduction of SAMs at an appropriate potential. Electrochemical impedance measurements were in-situ conducted and [Fe(CN)₆]${}^{3-∕4-}$ ions were used as probes for mass and charge transfer. Interface evolution was modeled with an equivalent electrical circuit containing two distinct constant-phase elements (CPEs). One is a generalized semi-infinite Warburg element in series with a charge transfer resistance attributed to subdiffusion phenomenon through leaky sublayers at low frequencies; the other CPE is used for characterizing the interface heterogeneity at medium and high frequencies. At low frequencies, electrochemical impedance measurements show subdiffusion phenomenon, which depends on the remaining sublayer and its thickness. When the defect density increases, diffusion tends to be ordinary, obeying the Fick’s law.

Particles driven through a periodic potential by an external constant force are known to exhibit a pronounced peak of the diffusion around the critical deterministic force that defines the transition between locked and running states. It has recently been shown both experimentally and numerically that this peak is greatly enhanced if some amount of spatial disorder is superimposed on the periodic potential. Here, we show that this enhancement is a fingerprint of a broad phenomenology that goes well beyond a simple augmentation. For some values of the model parameters, including the characteristic distances associated with the periodic and random components of the potential, the magnitude of the external force, and the temperature, the system can exhibit a rich variety of regimes from normal diffusion to superdiffusion, subdiffusion and even subtransport.

We consider a basic one-dimensional model which allows to obtain a diversity of diffusive regimes whose speed depends on the moments of a per-site trapping time. This models a discrete subordinated random walk, closely related to the continuous time random walks widely studied in the literature. The model we consider lends itself to a detailed elementary treatment, based on the study of recurrence relation for the time-$t$ dispersion of the process, making it possible to study deviations from normality due to finite time effects.

This study investigates a new formula for option pricing with transaction costs in a discrete time setting. The value of the financial assets is based on time-changed mixed fractional Brownian motion $(MFBM)$ model. The pricing method is obtained for European call option using the time-changed MFBM model in a discrete time setting. Particularly, the minimal value $C_{min}(t,S_t)$ of an option respect to transaction costs is obtained. Furthermore, the new model for pricing currency option is presented by utilizing the time-changed MFBM model. In addition, the impact of time step $\mathrm{\Delta }t$, Hurst parameter H and transaction costs $\alpha$ are also investigated, which substantiate that these parameters play a significant role in our pricing formula. Finally, the empirical studies and the simulation findings corroborate the theoretical bases and indicate the time-changed MFBM is a satisfactory model.

In this paper, we extend the classical constant elasticity of variance (CEV) model to a subdiffusive CEV model, where the underlying CEV process is time changed by an inverse $\alpha$-stable subordinator. The new model can capture the subdiffusive characteristics of financial markets. We find the corresponding fractional Fokker–Planck equation governing the PDF of the new process. We also derive the analytical formula for option prices in terms of eigenfunction expansion. This method avoids the evaluation of PDF of an inverse $\alpha$-stable variable and also eliminates the need for numerical integration to calculate the option prices. We numerically investigate the sensitivities of the option prices to the key parameters of the newly developed model.