Volume 65, Issue 1-2

In this article, two powerful analytical methods called the variational iteration method (VIM) and the variational homotopy perturbation method (VHPM) are introduced to obtain the exact and the numerical solutions of the (2+1)-dimensional Korteweg-de Vries-Burgers (KdVB) equation and the (1+1)-dimensional Sharma-Tasso-Olver equation. The main objective of the present article is to propose alternative methods of solutions, which avoid linearization and physical unrealistic assumptions. The results show that these methods are very efficient, convenient and can be applied to a large class of nonlinear problems.

In this paper, two equations of state (EOSs) (Sun Jiu-Xun-Morse with parameters n = 3 and 4, designated by SMS3 and SMS4) with two parameters are proposed to satisfy four merits proposed previously and give improved results for the cohesive energy. By applying ten typical EOSs to fit experimental compression data of 50 materials, it is shown that the SMS4 EOS gives the best results; the Baonza and Morse EOSs give the second best results; the SMS3 and modified generalized Lennard-Jones (mGLJ) EOSs give the third best results. However, the Baonza and mGLJ EOSs cannot give physically reasonable values of cohesive energy and P-V curves in the expansion region; the SMS3 and SMS4 EOS give fairly good results, and have some advantages over the Baonza and mGLJ EOSs in practical applications.

We derive series solution of a nonlinear problem which models the magnetohydrodynamic (MHD) shrinking flow due to a porous plate in a rotating frame of reference. The governing partial differential equations are first converted into ordinary differential equations and then solved by homotopy analysis method. The convergence of the derived series solution is carefully analyzed. Graphical results are presented to examine the role of various interesting parameters.

In this paper, the approximate analytical solutions of a general diffusion equation with fractional time derivative in the presence of a linear external force are obtained with the help of the homotopy perturbation method (HPM). The explicit solutions of the problem for the initial condition as a function of x have been obtained. It reveals that a few iterations are needed to obtain accurate approximate analytical solutions. The numerical calculations are carried out when the initial conditions are like exponential and periodic functions and the results are depicted through graphs. The examples prove that the method is extremely effective due to its simplistic approach and performance.

In this paper, we suggest a new reliable application of He’s homotopy perturbation method to study some evolution equations. The new application accelerates the rapid convergence of the series solutions and is used for analytic treatment of these equations. Some illustrative examples are given to further highlight the reliability and flexibility of the homotopy perturbation method.

In this paper, we present an efficient modification of the homotopy perturbation method by using Chebyshev’s polynomials and He’s polynomials to solve some nonlinear differential equations. Some illustrative examples are given to demonstrate the efficiency and reliability of the modified homotopy perturbation method.

In this paper, with the aid of symbolic computation the bright soliton solutions of two variablecoefficient coupled nonlinear Schr¨odinger equations are obtained by Hirota’s method. Some figures are plotted to illustrate the properties of the obtained solutions. The properties are meaningful for the investigation on the stability of soliton propagation in the optical soliton communications.

In this paper, we apply a relatively new technique which is called the exp-function method to construct generalized solitary and periodic solutions of Calogero-Degasperis-Fokas (CDF) equation which plays a very important role in mathematical physics, applied and engineering sciences. The suggested algorithm is quite efficient and is practically well suited for use in these problems. Numerical results clearly indicate the reliability and efficiency of the proposed method.

A four-component dusty plasma consisting of electrons, ions, and negatively as well as positively charged dust grains has been considered. Shock waves may exist in such a four-component dusty plasma. The basic characteristics of shock waves have been theoretically investigated by employing reductive perturbation technique (RPT). It is found that negative as well as positive shock potentials are present in such dusty plasma. The present results may be useful for understanding the existence of nonlinear potential structures that are observed in different regions of space (viz. cometary tails, lower and upper mesosphere, Jupiter’s magnetosphere, interstellar media, etc).

The propagation of dust ion acoustic waves (DIAWs) in a weakly inhomogeneous, weakly coupled, collisionless, and unmagnetized four components dusty plasma are examined. The fluid system considered in this work consists of cold positive ions, cold negatively and positively charged dust particles associated with isothermal electrons. For nonlinear (DIAW) waves, a reductive perturbation method was employed to obtain the variable coefficients Kortewege-de Vries (KdV) equation for the first-order potential. For local inhomogenity, the present system admits the coexistence of rarefactive and compressive solitons. As a matter of fact, when the wave amplitude enlarged, the width and velocity of the wave deviate from the prediction of the KdV equation. It means that we have to extend our analysis to obtain the variable coefficients Kortewege-de Vries (KdV) equation with fifth-order dispersion term. For locally constant parameters, the higher-order solution for the resulting equation has been achieved via what is called perturbation technique. The effects of positive and negative dust charge fluctuations on the higher-order soliton amplitude and width of electrostatic solitary structures are outlined.

In this paper, we deal with the inverse problem of reconstructing the diffusion equation on a finite interval. We prove that a dense subset of nodal points uniquely determine the boundary conditions and the coefficients of the diffusion equation. We also provide constructive procedure for them.

The molecular structures, vibrational frequencies, and corresponding vibrational assignments of 2-amino-3-, 4-, and 5-nitropyridine have been calculated by using ab initio Hartree-Fock (HF) and density functional theory (B3LYP) methods with 6-311++G(d,p) basis set level. The calculated vibrational frequencies and optimized geometric parameters (bond lengths and bond angles) were found to be in well agreement with the experimental data. The comparison of the observed and the calculated results showed that the scaled B3LYP method is superior to the scaled HF method for both the vibrational frequencies and the geometric parameters. For well fitting the calculated and the experimental frequencies we used scale factors obtained from the ratio of the frequency values of the strongest peaks in the calculated and the experimental spectra. These obtained scales seem to cause the better agreement of the gained vibrations to the experimental data.

Quantum-chemical ab initio calculations up to the ZPE+CCSD(T)/aug-cc-pVTZ//MP2/6- 311++G** level were performed on three possible structural isomers of diborabenzene (C4H4B2). All three molecules were found to be local minima on the C4H4B2 energy surface and to have closed shell singlet ground states. While the ground states of the 1,3- and 1,4-isomer are planar and of C2v and D2h symmetry, respectively, 1,2-diborabenzene is non-planar with a C2 axis passing through the center of the BB bond and the middle of the opposite carbon-carbon bond as the only symmetry element. The energetically most favourable 1,3-diborabenzene was found to be about 19 and 36 kcal/mol lower in energy than the 1,2- and the 1,4-isomer. Planar 1,3- and 1,4-diborabenzene have three doubly occupied π orbitals while non-planar 1,2-diborabenzene has also three doubly occupied orbitals which can be derived from the π orbitals of its 3.7 kcal/mol energetically less favourable planar form (“π-like” orbitals). The lowest unoccupied orbitals of all three isomers have σ symmetry with large coefficients at the two boron atoms. These orbitals are lower in energy than the lowest unoccupied molecular orbitals (LUMOs) of e. g. benzene and pyridine and might cause pronounced acceptor properties which could be one of the reasons for the elusiveness of the title compounds. The results of bond separation reactions show that cyclic conjugation stabilizes all three diborabenzenes relative to their isolated fragments. The most effective stabilization energy of about 24 kcal/mol was found for the energetically lowest 1,3-isomer. This value amounts to approximately one third of the experimental value for the bond separation energy of pyridine. In all cases the energetically lowest triplet states are significantly (16 - 24 kcal/mol) higher in energy than the singlet ground states. Also among the triplets the 1,3-isomer is the energetically most fabourable species.

A local short-to-intermediate range order of liquid Al 80 Co 10 Ni 10 , Al 72.5 Co 14.5 Ni 13 , and Al 65 Co 17.5 Ni 17.5 alloys was examined by X-ray diffraction and the reverse Monte Carlo modelling. The comprehensive analysis of three-dimensional models of the liquid ternary alloys was performed by means of the Voronoi-Delaunay method. The existence of a prepeak on the S(Q) function of the liquid alloys is caused by medium range ordering of 3d-transition metal atoms in dense-packed polytetrahedral clusters at temperatures close to the liquidus. The non-crystalline clusters, represented by aggregates of pentagons that consist of good tetrahedra, and chemical short-range order lead to the formation of the medium range order in the liquid binary Al-Ni, Al-Co and ternary Al-Ni-Co alloys.

We have studied the stirring effect on the time-delayed bifurcations of transient oscillations in the Belousov-Zhabotinsky (BZ) oscillating chemical reaction in a closed system. Experiments show that oscillations disappear through the time-delayed Hopf bifurcations, whose parameters depend on the stirring rate. The explanation of the stirring effect is based on the theories of diffusion-controlled reactions and hydrodynamic turbulence. We show that an increase of the stirring rate leads to an increase of the rate constant for the diffusion-controlled reaction. We propose a kinetic scheme that describes the effect observed in the experiments. A good agreement between the experimental data and the simulations is obtained.

Using standard standing wave microwave X-band technique and following Gopala Krishna’s single frequency (9.90 GHz) concentration variational method, the dielectric relaxation times τ and the dipole moments μ of dilute solutions of Tetrahydrofuran (THF), N-methylformamide (NMF), and THF+NMF binary mixtures in benzene solutions have been calculated at different temperatures (25°C, 30°C, 35°C, and 40°C). The energy parameters (ΔH ε , ΔF ε , ΔS ε ) for the dielectric relaxation process for the THF+NMF binary mixture containing 30 mol% THF have been calculated at 25°C, 30°C, 35°C, and 40°C and compared with the corresponding viscosity parameters. A good agreement between the free energy of activation from these two sets of values shows that the dielectric relaxation process like the viscous flow process can be treated as a rate process. From relaxation time behaviour of THF and NMF binary mixture in benzene solution, solute-solute types of the molecular association has been proposed.