On the optimization of higher eigenvaluesSur l'optimisation de valeurs propres supérieures

A new approach is proposed for the study of the classical Lagrange problem on the optimal form of a column with clamped ends and a fixed volume. It is proved that there exists a column with the maximal possible value of the kth eigenvalue and that such a column is unique. A method for the numerical solution is proposed. The method is based on the analysis of the critical points of a non-linear functional related to the Lagrange problem. To cite this article: Y.V. Egorov, C. R. Mecanique 332 (2004).     

A two-dimensional lagrangian technique for flow field measurement under high dynamic pressure

In this paper, a two-dimensional (2-D) Langrangian technique for flow field measurement under high dynamic pressure is presented, which included a set of experimental device and 2-D Lagrange composite manganin-constantan ring gages. With this kind of gage, the histories of pressure and radial displacement can be measured simultaneously at different Lagrange positions in an axisymmetric shock loading flow field. The technique has some advantages over the 1-D one, such as, simplified loading device, continuously adjustable output pressure, no restriction on sample length and the availability for the study of lateral rarefaction in shock propogation. As a preliminary application, the processes of 2-D shock initiation and attenuation in compacted TNT are measured.     

Gegenbauer spectral method for time‐fractional convection–diffusion equations with variable coefficients

In this paper, we study the numerical solution to time‐fractional partial differential equations with variable coefficients that involve temporal Caputo derivative. A spectral method based on Gegenbauer polynomials is taken for approximating the solution of the given time‐fractional partial differential equation in time and a collocation method in space. The suggested method reduces this type of equation to the solution of a linear algebraic system. Finally, some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.     

An adaptation of homotopy analysis method for reliable treatment of strongly nonlinear problems: construction of homotopy polynomials

In this paper, a new adaption of homotopy analysis method is presented to handle nonlinear problems. The proposed approach is capable of reducing the size of calculations and easily overcome the difficulty arising in calculating complicated integrals. Furthermore, the homotopy polynomials that decompose the nonlinear term of the problem as a series of polynomials are introduced. Then, an algorithm of calculating such polynomials, which makes the solution procedure more straightforward and more effective, is constructed. Numerical examples are examined to highlight the significant features of the developed techniques. The algorithms described in this paper are expected to be further employed to solve nonlinear problems in mathematical physics. Copyright © 2014 John Wiley & Sons, Ltd.     

Positivity of the generalized translation associated with the q-Hankel transform and applications

A new formulation of the Graf-type addition formula related to the third Jackson q-Bessel function gives a solution for the problem of the positivity of the generalized q-translation operator associated with the q-Hankel transform. Next some applications in q-theory are treated, for instance the relationship between the q-Bessel- positive and negative-definite functions. We also show how the positivity of the q-Bessel translation operator plays a central role in q-Fourier analysis, namely in the study of Markov operators in the q-context. The paper concludes with the nonnegative product linearization of the q^?2-Lommel polynomials.     

Complex versus real orthogonal polynomials of two variables

Orthogonal polynomials of two real variables can often be represented in complex variables. We explore the connection between the two types of representations and study the structural relations of complex orthogonal polynomials. The complex Hermite orthogonal polynomials and the disk polynomials are used as illustrating examples.