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The purpose of this paper is to investigate a high order numerical method for solving time-fractional Black-Scholes equation in which the fractional operator is defined by the Caputo fractional derivative. The proposed space-time spectral method employs the Jacobi polynomials for the temporal discretisation and Fourier-like basis functions for the spatial discretisation. The stability and convergence of the numerical scheme are analyzed. Two numerical examples are considered to validate the accuracy and illustrate the practicability of the proposed method. The results agree with the theoretical analysis and this approach can be applied in dealing with option pricing models with smooth payoff functions.

Air quality forecasting for Hong Kong is a challenge. Even taking the advantages of auto-regressive integrated moving average and some state-of-the-art numerical models, a recently-developed hybrid method for one-day (two- and three-day) ahead forecasting performs similarly to (slightly better than) a simple persistence forecasting. Long-term forecasting also remains an important issue, especially for policy decision for better control of air pollution and for evaluation of the long-term impacts on public health. Given the well-recognized negative effects of PM2.5, NO2 and O3 on public health, we study their time series under the multi-scale framework with empirical mode decomposition and nonstationary oscillation resampling to explore the possibility of long-term forecasting and to improve short-term forecasts in Hong Kong. Applied to a dataset from January 2016 to December 2018, the long-term forecasting (with lead time about 100 days) of the multi-scale framework has the root-mean-square error (RMSE) comparable with that of the short-term (with lead time of one or two days) forecasting by the persistence method, while its improvement for short-term forecasting (with lead time of one, two or three days) is quite substantial over the persistence forecasting, with RMSEs reduced by respectively 44%–47%, 30%–45%, and 40%–60% for PM2.5, NO2, and O3. Compared to the hybrid method, it turns out that, for short-term forecasting for the same data, the multi-scale framework can reduce RMSE by about 25% (respectively 30%) for PM2.5 (respectively NO2 and O3). In addition, we find no significant difference in the forecasting performance of the multi-scale framework among different types of stations. The multi-scale framework is feasible for time series forecasting and applicable to other pollutants in other cities.

In magnetic resonance imaging of the human brain, the diffusion process of tissue water is considered in the complex tissue environment of cells, membranes and connective tissue. Models based on fractional order Bloch–Torrey equations are known to provide insights into tissue structures and the microenvironment. In this paper, we consider new two-dimensional multi-term time and space fractional Bloch–Torrey equations with variable coefficients on irregular convex domains, which involve the Caputo time fractional derivative and the Riemann–Liouville space fractional derivative. An unstructured-mesh Galerkin finite element method is used to discretize the spatial fractional derivative, while for each time fractional derivative we use the L1 scheme on a temporal graded mesh. The stability and convergence of the fully discrete scheme are proved. Numerical examples are given to verify the efficiency of our method.

HIV-1 viruses, which are predominant in the family of HIV viruses, have strong pathogenicity and infectivity. They can evolve into many different variants in a very short time. In this study, we propose a new and effective alignment-free method for the phylogenetic analysis of HIV-1 viruses using complete genome sequences. Our method combines the position distribution information and the counts of the k-mers together. We also propose a metric to determine the optimal k value. We name our method the Position-Weighted k-mers (PWkmer) method. Validation and comparison with the Robinson-Foulds distance method and the modified bootstrap method on a benchmark dataset show that our method is reliable for the phylogenetic analysis of HIV-1 viruses. PWkmer can resolve within-group variations for different known subtypes of Group M of HIV-1 viruses. This method is simple and computationally fast for whole genome phylogenetic analysis.

In this paper, we consider a two-dimensional multi-term time-fractional Oldroyd-B equation on a rectangular domain. Its analytical solution is obtained by the method of separation of variables. We employ the finite difference method with a discretization of the Caputo time-fractional derivative to obtain an implicit difference approximation for the equation. Stability and convergence of the approximation scheme are established in the L ∞ -norm. Two examples are given to illustrate the theoretical analysis and analytical solution. The results indicate that the present numerical method is effective for this general two-dimensional multi-term time-fractional Oldroyd-B model.

In this paper, we investigate the characteristics and develop a stochastic model for the horizontal component Bx of the magnetic field at 22 stations of the global near-real-time magnetic observatory network INTERMAGNET. The model is in the form of a fractional stochastic differential equation. A method to estimate the parameters on the basis of observed data and to simulate the data using the model is given. The degree of fractional differentiation and the alpha-stability exponent of the process are employed to cluster the stations. The Bx time series possess pronounced local trends, which must be removed before modeling and simulation can be performed. This trend removal is carried out by an empirical mode decomposition. An outcome is an efficient method to simulate the Bx time series by empirical mode decomposition and fractional stochastic differential equation. The numerical results indicate the existence of two distinct clusters of the INTERMAGNET: one in the mid- and low latitudes consistent with the Dst index, and the other above geomagnetic latitude 60°N consistent with the AE index. This clustering corresponds to the inner magnetosphere and the outer magnetosphere, respectively.