Analytical Solution of Second-Order Hyperbolic Telegraph Equation by Variational Iteration and Homotopy Perturbation Methods

In this research, two analytical methods, namely homotopy perturbation method and variational iteration method are introduced to obtain solutions of the initial value problem of hyperbolic type which is called telegraph equation. Some illustrative examples are presented to show the efficiency of the methods.     

Construction of amperometric biosensor modified with conducting polymer/carbon dots for the analysis of catechol

Phenolic compounds used in food industries and pesticide industry, are environmentally toxic and pollute the rivers and ground water. For that reason, detection of phenolic compounds such as catechol by using simple, efficient and cost-effective devices have been becoming increasingly popular. In this study, a suitable and a novel matrix was composed using a novel conjugated polymer, namely poly[1-(5-(4,8-bis(5-(2-ethylhexyl)thiophen-2-yl)benzo[1,2-b:4,5-b']dithiophen-2-yl)furan-2-yl)-5-(2-ethylhexyl)-3-(furan-2-yl)-4H thieno[3,4-c]pyrrole-4,6(5H)-dione] (PFTBDT) and carbon dots (CDs) to detect catechol. PFTBDT and CDs were synthesized and the optoelectronic properties of PFTBDT were investigated via electrochemical and spectroelectrochemical studies. Laccase enzyme was immobilized onto the constructed film matrix on the graphite electrode. The proposed biosensor was found to have a low detection limit (1.23 μM) and a high sensitivity (737.44 μA/mM.cm⁻²) with a linear range of 1.25–175 μM. Finally, the applicability of the proposed enzymatic biosensor was evaluated in a tap water sample and a satisfactory recovery (96–104%) was obtained for catechol determination.     

Multiplication operators on Cesàro second order function spaces

Positivity - In this article, we investigate bounded, invertible and compact multiplication operators on the second order Cesàro function spaces.     

An approximate analytical solution of time-fractional telegraph equation

In this article, the powerful, easy-to-use and effective approximate analytical mathematical tool like homotopy analysis method (HAM) is used to solve the telegraph equation with fractional time derivative α (1 < α ? 2). By using initial values, the explicit solutions of telegraph equation for different particular cases have been derived. The numerical solutions show that only a few iterations are needed to obtain accurate approximate solutions. The method performs extremely well in terms of efficiency and simplicity to solve this historical model.     

HOLDITCH THEOREM FOR THE CLOSED SPACE CURVES IN LORENTZIAN 3-SPACE

In this article, we give the area formula of the closed projection curve of a closed space curve in Lorentzian 3-space L3. For the 1-parameter closed Lorentzian space motion in L3, we obtain a Holditch Theorem taking into account the Lorentzian matrix multiplication for the closed space curves by using their othogonal projections onto the Euclidean plane in the fixed Lorentzian space. Moreover, we generalize this Holditch Theorem for noncollinear three fixed points of the moving Lorentzian space and any other fixed point on the plane which is determined by these three fixed points.     

Diffusion-controlled evaporation of sodium dodecyl sulfate solution drops placed on a hydrophobic substrate

In this work, the effect of SDS anionic surfactant on the diffusion-controlled evaporation rate of aqueous solution drops placed on TEFLON-FEP substrate was investigated with 11 different SDS concentrations. Drop evaporation was monitored in a closed chamber having a constant RH of 54-57% by a video camera. The initial contact angle, θ(i) decreased from 104±2° down to 68±1° due to the adsorption of SDS both at the water-air and the solid-water interfaces. The adsorption of SDS on the solid surface was found to be 76% of that of its adsorption at the water-air interface by applying Lucassen-Reynders approach. An equation was developed for the comparison of the evaporation rates of drops having different θ(i) on the same substrate. It was found that the addition of SDS did not alter the drop evaporation rate considerably for the first 1200 s for all the SDS concentrations. The main difference was found to be the change of the mode of drop evaporation by varying the SDS concentration. The constant θ mode was operative up to 80 mM SDS concentration, whereas constant contact area mode was operative after 200 mM SDS concentrations due to rapid drop pining on the substrate.     

Synthesis of bishomoinositols and an entry for construction of a substituted 3-oxabicyclo[3.3.1]nonane skeleton

1,3,3a,7a-Tetrahydro-2-benzofuran was used as key compound for the synthesis of various bishomoinositol derivatives. The diene was subjected to an epoxidation reaction for further functionalization of the diene unit. The bisepoxide obtained was submitted to a ring-opening reaction with acid in the presence of water. Various bishomoinositols were synthesized. However, when the reaction was carried out in the presence of acetic anhydride, a substituted 3-oxabicyclo[3.3.1]nonane skeleton was formed. The mechanism of the formation of the products is discussed.     

Numerical solution of fractional differential equations with a Tau method based on Legendre and Bernstein polynomials

In this paper, we state and prove a new formula expressing explicitly the integratives of Bernstein polynomials (or B‐polynomials) of any degree and for any fractional‐order in terms of B‐polynomials themselves. We derive the transformation matrices that map the Bernstein and Legendre forms of a degree‐n polynomial on [0,1] into each other. By using their transformation matrices, we derive the operational matrices of integration and product of the Bernstein polynomials. These matrices together with the Tau method are then utilized to reduce the solution of this problem to the solution of a system of algebraic equations. The method is applied to solve linear and nonlinear fractional differential equations. Copyright © 2013 John Wiley & Sons, Ltd.     

On Gauss-Weierstrass semigroups and inversion of potentials associated with the Bessel differential operator

In this paper, we define the generalized Gauss Weierstrass semigroups with Weierstrass kernel, and give some of their properties. Using them, we study the inversion formulas for the generalized Riesz and Bessel potentials, generated by the generalized shift operators and associated with the Laplace Bessel differential operator.