On symmetries, conservation laws and invariant solutions of the foam-drainage equation

This study deals with symmetry group properties and conservation laws of the foam-drainage equation. Firstly, we study the classical Lie symmetries, optimal systems, similarity reductions and similarity solutions of the foam-drainage equation which are obtained through the Lie group method of infinitesimal transformations. Secondly, using the new general theorem on non-local conservation laws and partial Lagrangian approach, local and non-local conservation laws are also studied and, finally, non-classical symmetries are derived.     

Compact submanifolds supporting singular interactions

A quantum particle moving under the influence of singular interactions on embedded surfaces furnish an interesting example from the spectral point of view. In these problems, the possible occurrence of a bound-state is perhaps the most important aspect. Such systems can be introduced as quadratic forms and generically they do not require renormalization. Yet an alternative path through the resolvent is also beneficial to study various properties. In the present work, we address these issues for compact surfaces embedded in a class of ambient manifolds. We discover that there is an exact bound state solution written in terms of the heat kernel of the ambient manifold for a range of coupling strengths. Moreover, we develop techniques to estimate bounds on the ground state energy when several surfaces, each of which admits a bound state solution, coexist.     

Sensitive and Rapid Flow Injection Amperometric Hydrazine Sensor using an Electrodeposited Gold Nanoparticle Graphite Pencil Electrode

A gold (Au) nanoparticle-modified graphite pencil electrode was prepared by an electrodeposition procedure for the sensitive and rapid flow injection amperometric determination of hydrazine (N₂H₄). The electrodeposited Au nanoparticles on the pretreated graphite pencil electrode surface were characterized by scanning electron microscopy, energy-dispersive X-ray spectroscopy, X-ray diffraction spectroscopy, and electrochemical impedance spectroscopy. Cyclic voltammograms showed that the Au nanoparticle-modified pretreated graphite pencil electrode exhibits excellent electrocatalytic activity toward oxidation of hydrazine because the highly irreversibly and broadly observed oxidation peak at +600?mV at the pretreated graphite pencil electrode shifted to ?167?mV at the Au nanoparticle pretreated graphite pencil electrode; in addition, a significant enhancement in the oxidation peak current was obtained. Thus, the flow-injection (FI) amperometric hydrazine sensor was constructed based on its electrocatalytic oxidation at the Au nanoparticle-modified pretreated graphite pencil electrode. The Au nanoparticle-modified pretreated graphite pencil electrode exhibits a linear calibration curve between the flow injection amperometric current and hydrazine concentration within the concentration range from 0.01 to 100?µM with a detection limit of 0.002?µM. The flow injection amperometric sensor has been successfully used for the determination of N₂H₄ in water samples with good accuracy and precision.     

Conservation laws for one-layer shallow water wave systems

The problem of correspondence between symmetries and conservation laws for one-layer shallow water wave systems in the plane flow, axisymmetric flow and dispersive waves is investigated from the composite variational principle of view in the development of the study [N.H. Ibragimov, A new conservation theorem, Journal of Mathematical Analysis and Applications, 333(1) (2007) 311–328]. This method is devoted to construction of conservation laws of non-Lagrangian systems. Composite principle means that in addition to original variables of a given system, one should introduce a set of adjoint variables in order to obtain a system of Euler–Lagrange equations for some variational functional. After studying Lie point and Lie–Bäcklund symmetries, we obtain new local and nonlocal conservation laws. Nonlocal conservation laws comprise nonlocal variables defined by the adjoint equations to shallow water wave systems. In particular, we obtain infinite local conservation laws and potential symmetries for the plane flow case.     

Stress concentration effects in micropolar elasticity

Zusammenfassung Die vorliegende Notiz behandelt das Problem der Spannungskonzentration an einem kreisförmigen Loch in der Theorie der mikropolaren Elastizität. Es wird gezeigt, dass der Konzentrationsfaktor im Gegensatz zum klassischen Fall nicht konstant ist, sondern von drei dimensionslosen Parametern abhängt, welche mit den Materialkonstanten verknüpft sind. Bei passender Wahl dieses Parameters stellt sich heraus, dass der Konzentrationsfaktor um wenig vom klassischen Wert abweicht.     

Some basic viscous flows in micropolar fluids

Summary This paper analyzes some basic viscous flows of micropolar fluids. The problems ofCouette andPoiseuille flows between two parallel plates and a rotating fluid with a free surface, are solved using the theory of micropolar fluids. The results are presented graphically and compared with the classical ones, and the differences are discussed.     

New exact solutions to the CDF equations

New exact soliton solutions to the Cologero–Degasperies–Fokas (CDF) equations in (1+1)-dimension and (2+1)-dimension by using the improved tanh method are investigated. First, the (1+1)-dimensional CDF equation is analyzed. By the improved tanh method, the corresponding nonlinear partial differential equation is reduced to the nonlinear ordinary differential equations and then the different types of exact solutions to the original equation are obtained based on the solutions of the Riccati equation. For the case of (2+1)-dimensional CDF equation the same computation procedure is carried out. It is presented that one could obtain new exact explicit solutions, which are traveling wave solutions, to (2+1)-dimensional CDF equation. Additionally, some graphical representations of the solitary and periodic solutions are presented.