On Unconditionally Stable New Modified Fractional Group Iterative Scheme for the Solution of 2D Time-Fractional Telegraph Model

In this study, a new modified group iterative scheme for solving the two-dimensional (2D) fractional hyperbolic telegraph differential equation with Dirichlet boundary conditions is obtained from the 2h-spaced standard and rotated Crank–Nicolson FD approximations. The findings of new four-point modified explicit group relaxation method demonstrates the rapid rate of convergence of proposed method as compared to the existing schemes. Numerical tests are performed to test the capability of the group iterative scheme in comparison with the point iterative scheme counterparts. The stability of the derived modified group method is proven by the matrix norm algorithm. The obtained results are tabulated and concluded that exact solutions are exactly symmetric with approximate solutions.

Studying the impact of different additives on the properties of straight-run diesel fuels with various hydrocarbon compositions

One of the most widely used way to improve low-temperature properties of diesel fuels is the use of additives. However, a variety of additives and the effect of susceptibility make it difficult to select additive for a particular composition of diesel fuel and operating conditions. The laws of interaction between functional groups of additives and hydrocarbons of the diesel fraction have not been investigated yet. The article discusses the influence of fractional, group and structural-group composition of straight-run diesel fuels on the effectiveness of cold flow improvers. The effect of additives concentration on the effectiveness of their action is considered. It was shown that when selecting a cold flow improver for diesel fuel and determining its optimal concentration, it is necessary to take into account the optimal content of various groups of hydrocarbons in diesel fuel, at which a cold flow improver is most effective.

The Effect of the Fractional Carbon Dioxide Laser on Improving Minoxidil Delivery for the Treatment of Androgenetic Alopecia

Introduction: Recently, laser treatment for hair loss has become very popular. Laser-assisted drug delivery (LAD) is an evolving technology with potentially broad clinical applications. This work aims at inspecting the effect of the fractional carbon dioxide laser (CO2 ) on improving the delivery of minoxidil in patients with androgenetic alopecia and detecting the role of the fractional CO2 laser in its treatment. Methods: We enrolled 45 Egyptians with male androgenetic alopecia (MAGA); clinical grading was assessed based on Norwood-Hamilton classification. The patients were divided into 3 groups: the first group (combined group) received the fractional CO2 laser session followed by topical application of minoxidil and also in between sessions; the second group received fractional CO2 laser sessions only and 6 sessions with 2-week intervals were performed; the third group applied topical minoxidil only for 3 months. Global photographs and dermoscopic assessments were performed before treatment and 3 months after the treatment. Results: Several dermoscopic findings were detected, including peripilar sign, hair diversity, yellow spot, white dots, and arborizing red lines. The number of double hair units significantly increased after the treatment in the combined group. The mean number of hair after the treatment in the 3 groups significantly increased, mostly in the combined group. The hair thickness (thin & thick) significantly increased after the treatment in the combined group and the fractional group; however, in the minoxidil group, only thin hair thickness increased. In all the 3 groups, there was a significant improvement in hair count and thickness. Conclusion: The ablative fractional CO2 laser alone or combined with minoxidil may serve as an additional treatment for MAGA.

Finite Presentation of a Linear-Fractional Group

In this paper, we show that the coset diagram for the action of a linear-fractional group generated by [Formula: see text] and [Formula: see text] on the rational projective line is transitive and the coset diagram for the action is connected. Using the coset diagram, we show that [Formula: see text] are defining relations for the group.