ObjectiveTo compare foam bubble size and bubble size distribution, stability, and degradation rate of commercially available polidocanol endovenous microfoam (Varithena®) and physician-compounded foams using a number of laboratory tests.MethodsFoam properties of polidocanol endovenous microfoam and physician-compounded foams were measured and compared using a glass-plate method and a Sympatec QICPIC image analysis method to measure bubble size and bubble size distribution, Turbiscan™ LAB for foam half time and drainage and a novel biomimetic vein model to measure foam stability. Physician-compounded foams composed of polidocanol and room air, CO2, or mixtures of oxygen and carbon dioxide (O2:CO2) were generated by different methods.ResultsPolidocanol endovenous microfoam was found to have a narrow bubble size distribution with no large (>500 µm) bubbles. Physician-compounded foams made with the Tessari method had broader bubble size distribution and large bubbles, which have an impact on foam stability. Polidocanol endovenous microfoam had a lower degradation rate than any physician-compounded foams, including foams made using room air (p < 0.035). The same result was obtained at different liquid to gas ratios (1:4 and 1:7) for physician-compounded foams. In all tests performed, CO2 foams were the least stable and different O2:CO2 mixtures had intermediate performance. In the biomimetic vein model, polidocanol endovenous microfoam had the slowest degradation rate and longest calculated dwell time, which represents the length of time the foam is in contact with the vein, almost twice that of physician-compounded foams using room air and eight times better than physician-compounded foams prepared using equivalent gas mixes.ConclusionBubble size, bubble size distribution and stability of various sclerosing foam formulations show that polidocanol endovenous microfoam results in better overall performance compared with physician-compounded foams. Polidocanol endovenous microfoam offers better stability and cohesive properties in a biomimetic vein model compared to physician-compounded foams. Polidocanol endovenous microfoam, which is indicated in the United States for treatment of great saphenous vein system incompetence, provides clinicians with a consistent product with enhanced handling properties.

In the present article, fractional-order heat and wave equations are solved by using the natural transform decomposition method. The series form solutions are obtained for fractional-order heat and wave equations, using the proposed method. Some numerical examples are presented to understand the procedure of natural transform decomposition method. The natural transform decomposition method procedure has shown that less volume of calculations and a high rate of convergence can be easily applied to other nonlinear problems. Therefore, the natural transform decomposition method is considered to be one of the best analytical techniques, in order to solve fractional-order linear and nonlinear Partial deferential equations, particularly fractional-order heat and wave equation.

A divergence measure plays a crucial part in discriminating two probability distributions and drawing inferences constructed on such discrimination. The intention of this study is to propose such a divergence measure based on Jensen inequality and exponential entropy in the settings of probability theory. Further, the idea has been generalized to fuzzy sets to familiarize a novel picture fuzzy divergence measure. Besides proposing the validity, some of its key properties are also deliberated. Finally, two illustrative examples are solved based on the proposed picture fuzzy divergence measure which shows the expediency and effectiveness of the proposed approach.

In the present article, we related the analytical solution of the fractional-order dispersive partial differential equations, using the Laplace–Adomian decomposition method. The Caputo operator is used to define the derivative of fractional-order. Laplace–Adomian decomposition method solutions for both fractional and integer orders are obtained in series form, showing higher convergence of the proposed method. Illustrative examples are considered to confirm the validity of the present method. The fractional order solutions that are convergent to integer order solutions are also investigated.

In the current article, we consider certain subfamilies S e ∗ and C e of univalent functions associated with exponential functions which are symmetric along real axis in the region of open unit disk. For these classes our aim is to find the bounds of Hankel determinant of order three. Further, the estimate of third Hankel determinant for the family S e ∗ in this work improve the bounds which was investigated recently. Moreover, the same bounds have been investigated for 2-fold symmetric and 3-fold symmetric functions.

In this research paper, a hybrid method called Laplace Adomian Decomposition Method (LADM) is used for the analytical solution of the system of time fractional Navier-Stokes equation. The solution of this system can be obtained with the help of Maple software, which provide LADM algorithm for the given problem. Moreover, the results of the proposed method are compared with the exact solution of the problems, which has confirmed, that as the terms of the series increases the approximate solutions are convergent to the exact solution of each problem. The accuracy of the method is examined with help of some examples. The LADM, results have shown that, the proposed method has higher rate of convergence as compare to ADM and HPM. ). Although it is considered an old topic, but for the last few decade, fractional calculus is launched as an important topic by the scientists and researchers [10,11].The Navier-Stokes equation is known as Newton second Law for fluid substance, has been derived in 1822 by Claude Louis Navier and Gabriel Stokes. Navier-Stokes equation is an important model to describe many physical phenomena in applied sciences. This model have the capacity of modelling weather, ocean current, water flow in pipes and air flow around a wing. A very special case was considered, which has established the relationship between pressure and external forces acting on the fluid to the responses of fluid flow [12]. The Navier-Stock equation is also used to derive the connection between viscous fluid with rigid bodies and considered a best tool in the field of thermohydraulics, meteorology, petroleum industry, plasma physics and technology [13].Several mathematicians have applied different techniques for the solution of Navier-Stock equation. Among these methods, Kumar et al. have implemented modified Laplace decomposition method for the analytical solution of fractional Navier-Stokes equation [14] coupled method is the combination of He-Laplace transform (HLT) and Fractional Complex Transform (FCT) is used to solve Navier-Stock equation [15]. Fractional Reduced Differential Transformation Method (FRDM) is also implemented for the numerical solution of time fractional Navier-Stock equation [16], see also [17]. Definitions and Preliminaries ConceptsIn this unit, among few definitions of fractional calculus, presented in the article due to Riemann Liouville, Grunwald Letnikov, Caputo, etc., first folks simple descriptions and introductions are reconsidered, which we want to comprehend our education.

Spherical fuzzy set (SFS) is one of the most important and extensive concept to accommodate more uncertainties than existing fuzzy set structures. In this article, we will describe a novel enhanced TOPSIS-based procedure for tackling multi attribute group decision making (MAGDM) issues under spherical fuzzy setting, in which the weights of both decision-makers (DMs) and criteria are totally unknown. First, we study the notion of SFSs, the score and accuracy functions of SFSs and their basic operating laws. In addition, defined the generalized distance measure for SFSs based on spherical fuzzy entropy measure to compute the unknown weights information. Secondly, the spherical fuzzy information-based decision-making technique for MAGDM is presented. Lastly, an illustrative example is delivered with robot selection to reveal the efficiency of the proposed spherical fuzzy decision support approach, along with the discussion of comparative results, to prove that their results are feasible and credible.

In the present article, fractional-order partial differential equations with proportional delay, including generalized Burger equations with proportional delay are solved by using Natural transform decomposition method. Natural transform decomposition method solutions for both fractional and integer orders are obtained in series form, showing higher convergence of the proposed method. Illustrative examples are considered to confirm the validity of the present method. Therefore, Natural transform decomposition method is considered to be one of the best analytical technique, to solve fractional-order linear and non-linear Partial deferential equations particularly fractional-order partial differential equations with proportional delay.