Introduction Many problems which appear in different sciences such as physics, engineering, biology, applied mathematics and different branches can be modeled by using deterministic integral equations. Weakly singular integral equation is one of the principle type of integral equations which was introduced by Abel for the first time. These problems are often dependent on a noise source which are neglected due to poor computational tools. ...

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.

In the paper, the authors view some ordinary differential equations and their solutions from the angle of (generalized) derivative polynomials and simplify some known identities for the Bernoulli numbers and polynomials, the Frobenius– Euler polynomials, the Euler numbers and polynomials, in terms of the Stirling numbers of the first and second kinds.

Exponential partial Bell polynomials are a main tool for computing explicit formula for a large family of numbers and polynomials. Using these polynomials, we review most of the polynomials already studied in the book of Djordjevic-Milovanovic (2014). This study allows to unify the writing of these polynomials and obtain other extensions. Consequently the obtained results find their application in Number Theory,notably for the arithmetical properties of power products, Fermat numbers, Mersenne numbers and Fermat Last Theorem.

This paper presents the results of a two phase Euler-Euler simulation of seepage flow through granular media for upward flow conditions in soils. The simulation first deals with the fluidization process in a stable granular soil. The first fluidization in the sand column is studied and the stable fluidization condition is reached by the development of particle repulsion and the effect of inertia on the resistance of the granular bed. Considering Richardson's expansion law, porosity changes with the flow rate variation are described based on the rarefaction waves. In this study the variation of water and soil phases before and after the critical hydraulic gradient are evaluated. The results of the simulation reveal that the variation of soil pressure before and after the critical hydraulic gradient are inverse and the Terzaghi's effective stress theory is valid only to the onset of the first fluidization. In the second part of the paper, the fluidization and internal erosion processes in the artesian conditions in internally unstable soils are discussed. In internally unstable soils the finer soil particles are able to move through the constrictions of the coarse particles. This may lead to internal erosion and as a result the critical hydraulic gradient drastically increases. The erosion phenomenon is described using the theory of continuum mechanics for the soil consisting of stationary grains, movable grains and the fluid. The continuity and momentum equations are derived for the soil and based on this theory a numerical two phase model for the erosion and fluidization of particles are presented. The results reveal that by the commencement of erosion, the porosity of the soil decreases and the flux of eroded particles increases with time.

In this paper we demonstrate the existence of a set of polynomials Pi, 1£ i £ n , which are positive semi-definite on an interval [a, b] and satisfy, partially, the conditions of polynomials found in the Lagrange interpolation process. In other words, if a=a1<…<an=b is a given finite sequence of real numbers, then Pi (aj)=dij (dij is the Kronecker delta symbol); moreover, the sum of Pi 's is identically 1.

Zernike circle polynomials use for wavefront analysis because of their orthogonality over a circular. ‏‎ So far, different representations have been presented for these polynomials. In this paper, ‎ ‎‎we introduce Zernike polynomials and their important properties. Also, we present some other orthogonal polynomials such ‎as‎ Bessel ‎functions, ‎ Legendre polynomials and ‎ Jaccobi polynomials ‎and ‎give some important results of them‏. ‎‎ Finally, we express the relationship between Zernike orthogonal polynomials and these well-known orthogonal polynomials. ‏‎ For different cases, the ‎radial‎ polynomials in the Zernike functions are obtained using the well-known orthogonal polynomials‏. These relations frequently used for representing‎ wavefront ‎aberrations.

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.