Exact solution of the heat equation with boundary condition of the fourth kind by He’s variational iteration method

AbstractIn this paper, solutions of the heat equation with the boundary condition of the fourth kind are presented. The proposed solution is based on He’s variational iteration method, after the application of which the exact solution of the problem is obtained

Families of Increasing Sequences Possessing the Harmonic Series Property

We prove in this paper that any maximal, with respect to inclusion, subset of N – the family of all increasing sequences of positive integers – possessing the harmonic series property has the cardinality of the continuum. Moreover, we prove that for any countable (infinite) set exists an "orthogonal" family such that it hold some facts. All facts are proved constructively, by using the modified version of the classical Sierpiński family of increasing sequences having the cardinality of the continuum

Inhomogenity of the grain size of aircraft engine turbine polycrystalline blades

The determination of the behaviour of inhomogeneous materials with a complex microstructure requires taking into account the inhomogeneity of the grain size, as it is the basis for the process of designing and modelling effective behaviours. Therefore, the functional description of the inhomogeneity is becoming an important issue. The paper presents an analytical approach to the grain size inhomogeneity, based on the derivative of a logarithmic-logistic function. The solution applied enabled an effective evaluation of the inhomogeneity of two macrostructures of aircraft engine turbine blades, characterized by a high degree of diversity in the grain size. For the investigated single-modal and bimodal grain size distributions on a perpendicular projection and for grains with a non-planar surface, we identified the parameters that describe the degree of inhomogeneity of the constituents of weight distributions and we als o derived a formula describing the overall degree of homogeneity of bimodal distributions. The solution presented in the paper is of a general nature and it can be used to describe the degree of inhomogeneity of multi-modal distributions. All the calculations were performed using the Mathematica® package

Distributions of Grain Parameters on the Surface of Aircraft Engine Turbine Blades

In the quality assurance system for components cast using the lost wax method, the object of evaluation is the grain size on the surface of the casting. This paper describes a new method for evaluating the primary grain parameters on the surface of aircraft engine turbine blades. Effectiveness of the method has been tested on two macrostructures distinguished by a high degree of diversity in the grain size. The grounds for evaluating the grain parameters consist of geometric measurement of the turbine blade using a laser profilometer and of approximation of the measurement results using a polynomial of a proper degree. The so obtained analytical non-planar surface serves as a reference point for an assessment of the parameters of grains observed on the real blade surface of a variable curvature. The aspects subjected to evaluation included: the grain areas, shape and elongation coefficients of grains on a non-planar surface of the blade airfoil, using measurements taken on a perpendicular projection by means of a stereoscopic microscope and image analysis methods, and by making calculations using the Mathematica® package

A stronger version of the second mean value theorem for integrals

AbstractWe prove a stronger version of the classic second mean value theorem for integrals

Solution of the Stefan problem by involving material shrinkage

In this paper we describe an algorithm for solving the one-dimensional Stefan problem by involving metal shrinkage. In this algorithm we use the finite element method supplemented by the procedures allowing to define the position of moving interface and the change material size associated with shrinkage. We present also some examples illustrating the precision of the presented method

Comparison of the Adomian decomposition method and the variational iteration method in solving the moving boundary problem

AbstractIn this paper, a comparison between two methods: the Adomian decomposition method and the variational iteration method, used for solving the moving boundary problem, is presented. Both of the methods consist in constructing the appropriate iterative or recurrence formulas, on the basis of the equation considered and additional conditions, enabling one to determine the successive elements of a series or sequence approximating the function sought. The precision and speed of convergence of the procedures compared are verified with an example

ON SIMILARITIES BETWEEN EXPONENTIAL POLYNOMIALS AND HERMITE POLYNOMIALS

Abstract. The aim of this paper is to introduce and compare some fundamental analytical properties of the title polynomials. Many similarities between them are emphasized in the paper. Moreover, the authors present many isolated results, new proofs and identities