‎In this article‎, ‎we apply open-plus-closed-loop control method to drive the synchronization between two fractional-order Lorenz-Stenflo systems‎. ‎Based on the stability theorems‎, ‎sufficient condition for synchronization is proposed‎. Numerical simulations are presented to demonstrate the application of the theoretical results.‎

In this paper, we construct a numerical ‎ method‎ to ‎ the ‎ solution ‎ of‎ the Black-Scholes partial differential equation modeling the European option pricing problem with regard to a single asset‎.‎ We use an explicit spline-difference scheme which is based on using a finite difference approximation for the temporal derivative and a cubic B-spline collocation for spatial derivatives.‎ The derived method leads to a tri-diagonal linear system. The stability of this method has been discussed and shown to be unconditionally stable. The computational performance of the proposed scheme is compared with those obtained by using a scheme based on the radial basis function.

The ₂F₁ hypergeometric function appears in a wide variety of applications in probability and statistics. This article is motivated from an important application that arises in stochastic processes. A birth and death process that starts with a single species and birth rate larger than the death rate may lead to trees of species or genera. Under standard assumptions on the process, the probability of the size of a genera is a discrete distribution that can easily be computed at time t from the origin. However, when the process leads to several sub-line ages of trees that originated at different times from the origin, then the distribution of the size of the population commonly referred to as taxon size distribution of the genus size must be weighted by the different times that the genera originate. It was shown in Moschopoulos and Shpak (2010), that under the exponential distribution for time t the taxon distribution may be expressed as an ₂F₁(𝑥, 𝑏; 𝑥 +𝑏; 𝜃) hypergeometric function. This form has not been exploited in the literature. In this article we consider an asymptotic expansion of this ₂F₁ for large x in terms of generalized Bernoulli polynomials that can be computed upto any degree of accuracy.