Volume 73, Issue 1

By the trial equation method and the complete discrimination system for polynomial method, some exact solutions to Zakharov-Kuznetsov equation with variable coefficients are obtained. These solutions include solitary solutions, rational solutions, periodic solution and double periodic solutions.

The local structures and the spin Hamiltonian parameters (SHPs) for Cu 2+ in (90– x )TeO 2 –10GeO 2 – x WO 3 glasses are theoretically investigated at various WO 3 concentrations ( x =7.5, 15, 22.5 and 30 mol%). Subject to the Jahn-Teller effect, the [CuO 6 ] 10− groups are found to experience the small or moderate tetragonal elongation distortions (characterised by the relative tetragonal elongation ratios ρ ≈0.35–3.09%) in C 4 axis. With only three adjusted coefficients a , b and ω , the relevant model parameters (Dq, k and ρ ) are described by the Fourier type and linear functions, respectively, and the measured concentration dependences of the d–d transition bands and SHPs are reproduced. The maximum of g ∥ and the minimum of | A ∥ | at x =15 mol% are illustrated from the abrupt decrease of the copper–oxygen electron cloud admixtures or covalency and the obvious decline of the copper 3d–3s (4s) orbital admixtures due to the decreasing electron cloud density around oxygen ligands spontaneously bonding with Cu 2+ and Te 4+ (W 6+ ), respectively.

Cobalt-doped ZnO thin films have been deposited using a sol–gel route by changing the number of coats on the substrate from 6 to 18. This project deals with various film thicknesses by increasing the number of deposited coats. The effect of thickness on structural, magnetic, surface morphology and optical properties of Co-doped ZnO thin film was studied. The crystal structure of the Co-doped ZnO films was investigated by X-ray diffraction. The films have polycrystalline wurtzite hexagonal structures. A Co 2+ ion takes the place of a Zn 2+ ion in the lattice without creating any distortion in its hexagonal wurtzite structure. An examination of the optical transmission spectra showed that the energy band gap of the Co-doped ZnO films increased from 3.87 to 3.97 eV with an increase in the number of coatings on the substrate. Ferromagnetic behaviour was confirmed by measurements using a vibrating sample magnetometer. The surface morphology of thin films was assessed by scanning electron microscope. The grain size on the surface of thin films increased with an increase in the number of coats.

In this study, we developed the two-dimensional Legendre wavelet modified Petrov–Galerkin method for solving the two-dimensional moving boundary problem arising during melting of solid whose one surface is kept under most generalised boundary condition, and other two surfaces are insulated. The particular cases when surface subjected to the boundary condition of first, second and third kinds are discussed in detail. For validity of the present method, we have plotted graphs between residual (obtained from the original differential equation and its associated boundary conditions) and x- axis and found the effect of an error on moving layer thickness and y coordinate, respectively. Furthermore, we proved the convergence analysis of present method. The effect of parameters (Predvoditelev number, Kirpichev number, Biot number) on the moving layer thickness is discussed in detail. The whole analysis is presented in a dimensionless form.

The problem of conjugate free convection from a vertical fin embedded in a fluid-saturated porous medium is investigated. The governing nonlinear equations are solved iteratively by a highly implicit finite difference scheme. In this paper, the results based on four models, viz the Darcy model, the Brinkman model, the non-Darcian model with nonlinear inertia and viscous terms, and also the non-Darcian model with viscous, nonlinear inertia and velocity square terms, are compared. It is seen that fin cooling is more effective at higher Grashof or Darcy numbers due to stronger convection effects. The local Nusselt number is observed to increase with the Grashof or Darcy numbers and decrease slightly with the conduction–convection parameter. The limitation of the Darcy’s law is observed at higher values of permeability when the non-Darcian models are more relevant.

A kinky breather-soliton solution and kinky periodic-soliton solution are obtained using Hirota’s bilinear method and homoclinic test approach for the (3+1)-dimensional Jimbo-Miwa equation. Based on these two exact solutions, some lump-type solutions are emerged by limit behaviour. Meanwhile, two kinds of new dynamical phenomena, kinky breather degeneracy and kinky periodic degeneracy, are discussed and presented. Finally, the interaction between a stripe soliton and a lump-type soliton is discussed by the standardisation of the lump-type solution; the fusion and fission phenomena of soliton solutions are investigated and simulated by three-dimensional plots.

For functionalisation of a representative C 30 fullerene nanostructure by pyrrole- n -carboxylic acid (P n CA; n =2, 3) their stabilities and properties were investigated based on density functional theory calculations. Parallel calculations were also done for C 60 fullerene as evidence for comparing the results. Non-covalent interactions are considered to make the functionalised structures. In contrast with the spherical shape of C 60 , the shape of C 30 fullerene is elliptical; therefore, the functionalisation processes were done for both axial and equatorial elliptical positions (AC 30 and EC 30 ). The results indicated that both the positions of C 30 have almost equivalent chances to be functionalised by P n CA; but functionalisation by P2CA is slightly more favourable than P3CA, either for C 60 . The illustrated molecular orbitals’ distributions indicated that the direction of charge transfer could be considered from P n CA counterparts to fullerene counterparts. The molecular properties indicated more reactivity for C 30 than for C 60 fullerene. Finally, the atomic scale quadrupole coupling constants indicated different roles for N and O atoms of P n CA in the functionalised models.

As an increasing variety of composite materials with complex interfaces are emerging, we develop a theory to investigate composite beams and shed some light on new physical insights into composite beams with sinusoidal periodically varying interfaces. For the natural vibration of composite beams with continuous or periodically varying interfaces, the governing equation has been derived according to the generalised Hamiltonian principle. For composite beams having different boundary conditions, we transform the governing equations into integral equations and solve them by using the sinusoidal functions as test functions as well as the basis of the vibration modes. Due to the orthogonality of the sinusoidal functions, expansion coefficients in closed form can be found. Therefore, the proposed iterative schemes, with the help of the Rayleigh quotient and boundary functions, can quickly find the eigenvalues and free vibration modes. The obtained natural frequencies agree well with those obtained using the finite element method. In addition, the proposed method can be extended easily to laminated composite beams in more general cases or complex components and geometries in vibration engineering. The effects of different material properties of the upper and lower components and varying interface geometry function on the frequency of the composite beams are examined. According to our investigation, the natural frequency of a laminated beam with a continuous or periodically varying interface can be changed by altering the density or elastic modulus. We also show the responses of the frequencies of the components to the varying periodic interface.

The concept of forming harmonious coalitions is introduced to both classical and quantum symmetric cooperative games. In both the cases, players are motivated to form coalitions. Also, the main feature of the cooperative game is conserved.

The electron paramagnetic resonance (EPR) spectra of [Cu( l -tyrosine) 2 ] n (CuA) were interpreted based on the fourth-order perturbation treatments where the contributions due to the local distortion, ligand orbit and spin-orbit coupling were included. The calculated band transitions dx2−y2${{\text{d}}_{{{\text{x}}^2} - {{\text{y}}^2}}}$ to d xy (≈16412 cm −1 ) and dz2${{\text{d}}_{{{\text{z}}^2}}}$ (≈14845 cm −1 ) agree well with the band analysis results (dx2−y2→dxy${\text{(}}{{\text{d}}_{{{\text{x}}^2} - {{\text{y}}^2}}} \to {{\text{d}}_{{\text{xy}}}}$≈16410 and dx2−y2→dz2${{\text{d}}_{{{\text{x}}^2} - {{\text{y}}^2}}} \to {{\text{d}}_{{{\text{z}}^2}}}$≈14850 cm −1 ). The unresolved separations dx2−y2→dxz${{\text{d}}_{{{\text{x}}^2} - {{\text{y}}^2}}} \to {{\text{d}}_{{\text{xz}}}}$ and dx2−y2→dyz${{\text{d}}_{{{\text{x}}^2} - {{\text{y}}^2}}} \to {{\text{d}}_{{\text{yz}}}}$ in the absorption spectra were evaluated as 26283 and 26262 cm −1 , respectively. For CuA, copper chromophores in 1,3-diaminorpropane isophtalate copper(II) complex (CuB) and N-methyl-1,2-diaminoetaane-bis copper(II) polymer (CuC), the transition dx2−y2→dxy${{\text{d}}_{{{\text{x}}^2} - {{\text{y}}^2}}} \to {{\text{d}}_{{\text{xy}}}}$ (=E 1 ≈10D q ) suffered an increase with a decrease in R ̅ L which was evaluated as the mean value of the copper-ligand bond lengths. The correlations between the tetragonal elongation ratio ρ (=( R z – R ̅ L )/ R ̅ L ) (or the ratio G =( g z – g e )/(( g x + g y )/2– g e )) and the g isotropy g av (=( g x + g y + g z )/3) (or the covalency factor N ) for CuA, CuB and CuC were acquired and all the results were discussed.

The electrical properties of composite electrolytes (suspensions) composed of α-Al 2 O 3 powder and molten carbonate eutectic (Li 2 CO 3 –Na 2 CO 3 –K 2 CO 3 ) eut or molten carbonate–chloride mixture 0.72(Li 2 CO 3 –Na 2 CO 3 –K 2 CO 3 ) eut –0.28NaCl have been investigated by AC impedance method. This system shows a dependence of the electrical conductivity upon the temperature and the α-Al 2 O 3 content. The specific electrical conductivity of the α-Al 2 O 3 /(Li 2 CO 3 –Na 2 CO 3 –K 2 CO 3 ) eut system can be adequately described by the Maxwell equation for two-phase heterogeneous materials. The regression equation for the dependence of the specific conductivity of the α-Al 2 O 3 /(Li 2 CO 3 –Na 2 CO 3 –K 2 CO 3 ) eut composite on the aluminium oxide concentration and temperature was obtained.

The structural and corresponding magnetic properties of Ni/Cu films sputtered at low and high deposition rates were investigated as there is a limited number of related studies in this field. 5[Ni(10 nm)/Cu(30 nm)] multilayer thin films were deposited using two DC sputtering sources at low (0.02 nm/s) and high (0.10 nm/s) deposition rates of Ni layers. A face centered cubic phase was detected for both films. The surface of the film sputtered at the low deposition rate has a lot of micro-grains distributed uniformly and with sizes from 0.1 to 0.4 μm. Also, it has a vertical acicular morphology. At high deposition rate, the number of micro-grains considerably decreased, and some of their sizes increased up to 1 μm. The surface of the Ni/Cu multilayer deposited at the low rate has a relatively more grainy and rugged structure, whereas the surface of the film deposited at the high rate has a relatively larger lateral size of surface grains with a relatively fine morphology. Saturation magnetisation, M s , values were 90 and 138 emu/cm 3 for deposition rates of 0.02 and 0.10 nm/s, respectively. Remanence, M r , values were also found to be 48 and 71 emu/cm 3 for the low and high deposition rates, respectively. The coercivity, H c , values were 46 and 65 Oe for the low and high Ni deposition rates, respectively. The changes in the film surfaces provoked the changes in the H c values. The M s , M r , and H c values of the 5[Ni(10 nm)/Cu(30 nm)] films can be adjusted considering the surface morphologies and film contents caused by the different Ni deposition rates.