New exact traveling wave solutions of the Tzitzéica-type evolution equations arising in non-linear optics

The properties of Tzitzéica equations in non-linear optics have been the subject of many recent studies. In this article, a new and effective modification of Kudryashov method is adopted to study this class of non-linear evolution equations. As an achievement, new exact traveling wave solutions of Tzitzéica, Dodd–Bullough–Mikhailov (DBM) and Tzitzéica–Dodd–Bullough (TDB) equations are formally extracted. It is believed that the modified Kudryashov method along with the symbolic computation package suggests a promising technique to handle non-linear evolution equations in non-linear optics.     

Nonlinear Green's function method for the Tzitzéica and related equations

In this short note we apply the nonlinear Green's function method for the solution of the Tzitzéica type equation hierarchies arising in nonlinear science. Using the travelling wave ansatz, we first transform the nonlinear partial differential equations to nonlinear ordinary differential equations. Then, we establish a general representation formula for nonlinear Green's function of these equations. Eventually, using Frasca's short time expansion, we obtain the exact solution to these equations. Numerical analysis shows that the obtained Green's function solution is sufficiently close to the numerical solution obtained by the well-known method of lines. Finally, we involve the inverse transform and study the full nature of the Tzitzéica equation.     

On exact solutions of the unsteady navierstokes equations—the vortex with instantaneous curvilinear axis

The unsteady pseudo plane motions have been investigated in which each point of the parallel planes is subjected to non-torsional oscillations in their own plane and at any given instant the streamlines are concentric circles. Exact solutions are obtained and the form of the curve Γ, the locus of the centers of these concentric circles, is discussed. The existence of three infinite sets of exact solutions, for the flow in the geometry of an orthogonal rheometer in which the above non-torsional oscillations are superposed on the disks, is established. Three cases arise according to whether ω is greater than, equal to or less than σ, where ω is angular velocity of the basic rotation and σ is the frequency of the superposed oscillations. For a symmetric solution of the flow these solutions reduce to a single unique solution. The nature of the curve Γ is illustrated graphically by considering an example of the flow between coaxial rotating disks.     

Analysis of the compaction behavior of Al–SiC nanocomposites using linear and non-linear compaction equations

The compressibility behavior of Al–SiC nanocomposite powders was examined and the density-pressure data were analyzed by linear and non-linear compaction equations. SiC particles with an average size of 50 nm were mixed with gas-atomized aluminum powder (40 μm average size) at different volume fractions (up to 20 vol%) and compacted in a rigid die at various pressures. In order to highlight the effect of reinforcement particle size, the compressibility of micrometric SiC particles of two sizes (1 and 40 μm) was also examined. Analysis of the compressibility data indicated hindering effect of the hard ceramic particles on the plastic deformability of soft aluminum matrix, particularly at high volume fractions. More pronounced effect on the yield pressure was obtained for the nanometric particles compared with the micrometric ones. Nevertheless, better particles rearrangement was taken place when the ultrafine SiC particles were utilized. In light of the experimental and theoretical analysis, the densification mechanism of aluminum matrix composites and the effect of reinforcement particle size and volume fraction are discussed.     

Fringe-scanning method using a general function for shadow moiré

We describe a new high-resolution three-dimensional measurement method for shadow moiré. The method is based on the principle of using shadow moiré to produce moiré fringes and a fringe-scanning technique. In this method, a general function, instead of an arctangent function, is used for detecting the shape of an object. One can subsequently analyze the general function using numerical analysis with a digital computer. Two systems for static and dynamic measurements are proposed.

Experimental results show that measurement accuracies in static and dynamic measurement systems are obtainable to greater than 1/50 and 1/40 fringes, respectively.

     

Identification of diffusion parameters in a non-linear convection–diffusion equation using adaptive homotopy perturbation method

This paper is concerned with the numerical identification of diffusion parameters in a non-linear convection–diffusion equation, which arises as the saturation equation in the fractional flow formulation of the two-phase porous media flow equations. In order to overcome the defect of the local convergence of traditional methods, an adaptive homotopy perturbation method is applied to solve this parameter identification inverse problem. The adaptive homotopy perturbation method provides a simple way to adapt computational refinement to the choice of the homotopy parameter. Numerical simulations illustrate that the proposed algorithm is globally convergent and computationallyefficient.     

Analytical solutions of delay‐differential equations via a time‐partition method

Abstract

A time‐partition method is used to obtain the analytical solutions of delay‐differential equations by the Laplace transformation technique with a special matrix inversion algorithm.     

Group method solutions of the generalized forms of Burgers, Burgers�CKdV and KdV equations with time-dependent variable coefficients

Solutions for the generalized forms of Burgers, Burgers?CKdV, and KdV equations with time-dependent variable coefficients and with initial and boundary conditions are constructed. The analysis rests mainly on the standard group method. Similarity solutions are found which reduce the nonlinear system of partial differential equations to systems of ordinary differential equations to obtain some exact solutions and others as numerical solutions.     

Tidal current energy potential assessment in the Avilés Port using a three-dimensional CFD method

Clean Technologies and Environmental Policy - Tidal energy is considered as an energy resource of maximum interest in both technical and research fields due to its largely unexploited energy...     

The localized differential quadrature method for two-dimensional stream function formulation of Navier–Stokes equations

Localized differential quadrature (LDQ) method is employed to solve two-dimensional stream function formulation of incompressible Navier–Stokes equations. Being developed by introducing the localization concept to the general differential quadrature (GDQ) method, the employment of LDQ method becomes efficient and flexible, especially for the simulations of large scale computations. By introducing the Lagrange stream function to vorticity transport equation, the governing equation—the fourth-order partial differential equation (PDE)—is derived. To stably obtain the solutions of the fourth-order PDE, a fictitious point method is included to treat the boundary conditions. To examine the present scheme, two different types of classic benchmark fluid flow problems are proposed, including driven cavity flow problems and backward-facing step flow problems. The good agreement of solutions demonstrate the robustness and feasibility of the proposed scheme. Conclusively, the LDQ method is sufficient and appropriate enough to simulate the solutions of stream function formulation of Navier–Stokes equations with various Reynolds numbers.     

Direct measurement of oxygen in lead-based ceramics using the ζ-factor method in an analytical electron microscope

X-ray absorption of oxygen is significant in thin specimens of Pb(Mg_1/3Nb_2/3)O₃-35 mol% PbTiO₃ [PMN-35PT] due to the presence of heavy elements such as Pb and Nb. Therefore, direct measurement of the oxygen concentration in these types of systems can be difficult. Furthermore, assumption of the composition from stoichiometric considerations may not be feasible, particularly if the valence state of one or more of the cation species is variable. Using only XEDS data, the -factor method provides absorption corrected compositional information. In the present study, it was shown that such data were in very good agreement with the nominal values for PMN-35 PT, whereas the uncorrected data underestimated the oxygen content by 300%. In previous work, it was theorized that the swelling of samples containing excess PbO was linked to changes in the composition of the intergranular liquid phase. The -factor technique was used to show that the oxygen to lead ratio of this second phase changes upon annealing.     

Application of the fundamental solutions by Ganowicz in a static analysis of Reissner’s plates by the boundary element method

In this paper the direct non-singular formulation of the boundary element method using the fundamental solutions given by Ganowicz (1966) [6] and its application to a static analysis of plates with intermediate thickness is presented. A more exact calculation of almost singular integrals is possible, thanks to the applied modification of the Gauss integration procedure with the inversed distribution of integration points. The non-singular method is based on an offset of collocation points from a plate boundary. The accuracy of results depends on this offset, so the analysis of a relation between the distance of the collocation point from the plate and the conditioning of the matrix of integral equations is carried out. The optimal distance equal to 0.01 of the boundary element length was determined. The presented approach allows to carry out the static analysis of plates with arbitrary shapes including plates with holes. Solution of thin plates is also possible.     

A dual reciprocity boundary element formulation using the fractional step method for the incompressible Navier–Stokes equations

This paper presents a dual reciprocity boundary element solution method for the unsteady Navier–Stokes equations in two-dimensional incompressible flow, where a fractional step algorithm is utilized for the time advancement. A fully explicit, second-order, Adams–Bashforth scheme is used for the nonlinear convective terms. We performed numerical tests for two examples: the Taylor–Green vortex and the lid-driven square cavity flow for Reynolds numbers up to 400. The results in the former case are compared to the analytical solution, and in the latter to numerical results available in the literature. Overall the agreement is excellent demonstrating the applicability and accuracy of the fractional step, dual reciprocity boundary element solution formulations to the Navier–Stokes equations for incompressible flows.     

A systematic approach to obtaining numerical solutions of Jeffery’s type equations using Spherical Harmonics

This paper extends the work of Bird, Warner, Stewart, Sørensen, Larson, Ottinger, Vukadinovic, and Forest et al., who have applied Spherical Harmonics to numerically solve certain types of partial differential equations on the two-dimensional sphere. We present a systematic approach and implementation for solving such equations with efficient numerical solutions. In particular we are able to solve a wide variety of fiber orientation equations considered before by Jeffery, Folgar and Tucker, and Koch, and include several recently introduced fiber orientation collision models. The main tools used to compute the coefficients for the Spherical Harmonic-based expansion are Rodrigues’ formula and the ladder operators. We show that solutions of the Folgar–Tucker model using our new algorithm retains the accuracy of full simulations of the fiber orientation distribution function with computational efforts that are only slightly more than the Advani–Tucker orientation/moment tensor solutions commonly used in industrial applications. The Spherical Harmonic approach requires a computational effort of just three times that of the orientation tensor approach employing the orthotropic closure of VerWeyst, but with less than 1/1000th the computational effort of numerical solutions of the full orientation distribution function obtained using control volume methods.     

Interdiffusion inα solid solutions of the Cu-Zn-Sn system

The interdiffusion coefficients in the f c c phase of Cu-Zn-Sn alloys, , have been determined at 1073 K. The concentration profiles indicate that the diffusion rate of tin is greater than that of zinc in the Cu-Zn-Sn alloy. The diffusion paths show the typical S-shaped curves. All of the four interdiffusion coefficients are positive and they are very sensitive to the solute concentration. The atomic mobilities of the three diffusing elements in Kirkendall planes increase in the order of Cu, Zn, Sn. The interaction energy of the Cu-Sn bond is much larger than that of the Zn-Sn bond. From the results of the present work it seems that the Onsager reciprocal relation holds in the a phase of the Cu-Zn-Sn system.     

Study of the surface structure of butterfly wings using the scanning electron microscopic moiré method

Scanning electron microscopic (SEM) moiré method was used to study the surface structure of three kinds of butterfly wings: Papilio maackii Menetries, Euploea midamus (Linnaeus), and Stichophthalma howqua (Westwood). Gratings composed of curves with different orientations were found on scales. The planar characteristics of gratings and some other planar features of the surface structure of these wings were revealed, respectively, in terms of virtual strain. Experimental results demonstrate that SEM moiré method is a simple, nonlocal, economical, effective technique for determining which grating exists on one whole scale, measuring the dimension and the whole planar structural character of the grating on each scale, as well as characterizing the relationship between gratings on different scales of each butterfly wing. Thus, the SEM moiré method is a useful tool to assist with characterizing the structure of butterfly wings and explaining their excellent properties.     

The method of fundamental solutions with dual reciprocity for diffusion and diffusion–convection using subdomains

The method of fundamental solutions (MFS), first proposed in the 1960s, has recently reappeared in the literature and solutions of an extraordinary accuracy have been reported using relatively few data points. The method requires no mesh and therefore no integration, and has been recently combined with dual reciprocity method (DRM) for treating inhomogeneous terms. The objective of this paper is the combination of the two methods for treating convective terms which are derivatives of the problem variable. First the formulation of the methods for mixed Neumann–Dirichlet boundary conditions is considered, as both these types of boundary condition are necessary for this type of problem. Next a formulation for the usual Crank–Nickleson and Galerkin time-stepping procedures is obtained for both diffusion and diffusion–convection and the use of the subdomain technique with MFS is considered. Finally results obtained for some test problems are presented including a diffusion convection problem with variable velocity using both a single domain and a division into subregions, the convective terms being modeled using DRM. Results are compared with exact solutions and in some cases with DRBEM examples from the literature.     

Optical quantities of multi-layer systems with randomly rough boundaries calculated using the exact approach of the Rayleigh–Rice theory

In this paper, the exact approach of the Rayleigh–Rice theory enabling us to calculate optical quantities of multi-layer systems with boundaries exhibiting slight random roughness is presented. This approach is exact in the sense that it takes into account the propagation of perturbed electromagnetic fields (waves) among randomly rough boundaries including all cross-correlation and auto-correlation effects. The restriction to the second order of perturbation, which is the lowest order that gives nonzero corrections to coherent waves (obeying the Snell’s law), represents the only approximation used in our calculations. It is assumed that the layers and the substrates are formed by optically homogeneous and isotropic materials. The formulae obtained in the theoretical part are used to investigate the influence of layer thicknesses and roughness parameters on reflectances and associated ellipsometric parameters of the selected numerical examples of a three-layer system. The presented approach represents the generalization of the exact approach for single-layer systems and the improvement of the approximate approach for multi-layer systems published earlier. The exact approach of the RRT has a substantial importance for the optical characterization of multi-layer systems occurring in applied research and optics industry applications.