Purpose: The purpose of this paper is to develop a set of identities for EULER type sums of products of harmonic numbers and reciprocal binomial coefficients.Method: We use analytical methods to obtain our results.Results: We obtain identities for variant EULER sums of the type ¥Sn=1H2n/n (n+k)k, and its finite counterpart, which generalize some results obtained by other authors.Conclusions: Identities are successfully achieved for the sums under investigation. Some published results have been successfully generalized.

We study the calculus of variations problem in the presence of a system of differential-integral (D-I) equations. In order to identify the necessary optimality conditions for this problem, we derive the so-called D-I EULER–Lagrange equations. We also generalize this problem to other cases, such as the case of higher orders, the problem of optimal control, and we derive the so-called D-I Pontryagin equations. In special cases, these formulations lead to classical EULER–Lagrange equations. To illustrate our results, we provide simple examples and applications such as obtaining the minimumpower for an RLC circuit.

This paper presents a comparative assessment of low Reynolds number k- models against standard k- model in an EULERian framework. Three different low-Re number k- models: Launder-Sharma (LS), YangShih (YS) and AbeKondoh-Nagano (AKN) have been used for the description of bubble plume behaviour in stratified water. The contribution of the gas phase movement into the liquid phase turbulence has been achieved by using the Dispersed with Bubble Induced Turbulence approach (DIS+BIT). The results reveal that the oscillation frequency of gas-liquid flow are correctly reproduced by standard k- and LS models. In fact, we found for standard K- and LS a clear dominant peak at a frequency equal to 0. 1 Hz. On the other hand, YS and AKN models have predicted chaotic oscillations. The oscillation amplitude of the bubble plume predicted from LS model seems to be in good agreement with the PIV measurements of Besbes et al. (2015). However, for the standard K- model the oscillation amplitude is low. The air-water interface shows that the bubble plume mixing with the stratified water is predicted to be stronger compared to standard k- model.

The accuracy and efficiency of the elements proposed by finite element method (FEM) considerably depend on theinterpolating functions namely shape functions used to formulate the displacement field within the element. In thepresent study, novel functions, namely basic displacements functions (BDFs), are introduced and exploited forstructural analysis of nanobeams using finite element method based on Eringen’ s nonlocal elasticity and EULER-Bernoulli beam theory. BDFs are obtained through solving the governing differential equation of motion of nanobeamsusing the power series method. Unlike the conventional methods which are almost categorized as displacement-basedmethods, the flexibility basis of the method ensures true satisfaction of equilibrium equations at any interior point ofthe element. Accordingly, shape functions and structural matrices are achieved in terms of BDFs by application ofmerely mechanical principles. In order to evaluate the competency and accuracy of the proposed method with differentboundary conditions, several numerical examples with various boundary conditions are scrutinized. Carrying outseveral numerical examples, the results in stability analysis, free longitudinal vibration and free transverse vibrationshow a complete accordance with those in literature.