Calculation of Free Energy of the Integrable Landau—Lifshitz model

The generalized Bethe-ansatz method of thermodynamic analysis of integrable systems was employed to compute the free energy of a classical integrable model,i.e.the Landau-Lifshitz model.Using the action-angle variables of the model and by imposing a periodic boundary condition.we derive a phase-shifted density of states for the excitations of the system.The free energy,in the thermodynamic limit,can be expressed analytic in terms of two coupled nonlinear integral equations of the finie temperature excited energy for effective phonons and kinks (antikinks).we solve these equations iteratively for a special case that the model is in the limit of anisotropic strong yz coupling.     

High-pressure phonon dispersion of copper by using the modified analytic embedded atom method

By using the Born-von Krmn theory of lattice dynamics and the modified analytic embedded atom method, we reproduce the experimental results of the phonon dispersion in fcc metal Cu at zero pressure along three high symmetry directions and four off-symmetry directions, and then simulate the phonon dispersion curves of Cu at high pressures of 50, 100, and 150 GPa. The results show that the shapes of dispersion curves at high pressures are very similar to that at zero pressure. All the vibration frequencies of Cu in all vibration branches at high pressures are larger than the results at zero pressure, and increase correspondingly as pressure reaches 50, 100, and 150 GPa sequentially. Moreover, on the basis of phonon dispersion, we calculate the values of specific heat of Cu at different pressures. The prediction of thermodynamic quantities lays a significant foundation for guiding and judging experiments of thermodynamic properties of solids under high pressures.     

A New Pseudospectral Method for Calculations of Hydrogen Atom in Arbitrary External Fields

A new pseudospectral method was introduced to calculate wavefunctions and energy levels of hydrogen atom in arbitrary potential.Some results of hydrogen atom in uniform magnetic fields were presented,high accuracy of results was obtained with simple calculations,and our calculations show very fast convergence.It suggests a new method for calculations of hydrogen atom in external fields.     

A new non-perturbative approach in quantum mechanics for time-independent Schr?dinger equations

A new non-perturbative approach is proposed to solve time-independent Schr?dinger equations in quantum mechanics.It is based on the homotopy analysis method(HAM)that was developed by the author in 1992 for highly nonlinear equations and has been widely applied in many fields.Unlike perturbative methods,this HAM-based approach has nothing to do with small/large physical parameters.Besides,convergent series solution can be obtained even if the disturbance is far from the known status.A nonlinear harmonic oscillator is used as an example to illustrate the validity of this approach for disturbances that might be one thousand times larger than the possible superior limit of the perturbative approach.This HAM-based approach could provide us rigorous theoretical results in quantum mechanics,which can be directly compared with experimental data.Obviously,this is of great benefit not only for improving the accuracy of experimental measurements but also for validating physical theories.     

圆薄板大撓度问题

Equations for the large deflection of thin plate established by Th. von Karman has been well known for many years. But so far there have been only a few iproblems studied with numerical certainty. S. Way was the first to apply these equations to solve the problem of a clamped plate under uniform pressure by the method of power series. After this, S. Levy found the solution of the simply supported rectangular plate under uniform load by the method of double trigonometric series. Both methods are too labourious to be applicable to other more important cases. Lately, Chien Wei-zang treated Way's problem again by means of the perturbation method and obtained excellent results. By the method as given by Chien Wei-zang, Yeh Kai-yuan worked out the problem of circular plate with a central hole under central concentrated load.In this paper, more results are given for various circular plates under various edge conditions. These include uniformly loaded circular plate under various edge conditions (section 2) and central concentrated loaded circular plate under various edge conditions (section 3). Such edge conditions are: (1) simply supported, (2) simply hinged, (3) rigidly clamped, (4) clamped but free to slip, (5) edge clamped but with possible slipping in horizontal direction, (6) edge simply supported but elastically fastened, and (7) edge clamped in elastic wall.All these results are presented in such a form that direct application in design problem is possible. In particular cases, under edge conditions (1) to (4), as σ=0.3, design formulae and curves for central deflection, radial tensile stress and radial bending stress are presented.     

Nonlinear Waves in Two-Dimensional （2D） Square Lattice Frenkel-Kontorova （FK） System

A 2D square lattice is studied. By using the continuum approximation, we set up the differential equations of motion for an arbitrary particle in the square lattice which subjects to an external periodic substrate potential. The exact solitary waves of the system are found for special cases. We conclude that the adhesive force f and the angle between propagation directions of upper and lower layers can affect these waves.     

Elastic analysis of an elliptic notch in quasicrystals of point group 10 subjected to shear loading

Based on the stress potential and complex variable function method, this paper makes an elastic analysis of an elliptic notch subjected to uniform shear stress at infinity in quasicrystals with point group 10. With the aid of conformal transformation, an exact solution for the elliptic notch of the quasicrystals is obtained. The solution of the mode {\rm II} Griffith crack as a special case is constructed. The stress intensity factor and energy release rate have been also obtained as a direct result of the crack solution.     

An analysis of the incompressible viscous flows problem by meshless method

Generally the incompressible viscous flow problem is described by the Navier--Stokes equation. Based on the weighted residual method the discrete formulation of element-free Galerkin is inferred in this paper. By the step-by-step computation in the field of time, and adopting the least-square estimation of the-same-order shift, this paper has calculated both velocity and pressure from the decoupling independent equations. Each time fraction Newton--Raphson iterative method is applied for the velocity and pressure. Finally, this paper puts the method into practice of the shear-drive cavity flow, verifying the validity, high accuracy and stability.     

Simulation of Variable Viscosity and Jeffrey Fluid Model for Blood Flow Through a Tapered Artery with a Stenosis

Non-Newtonian fluid model for blood flow through a tapered artery with a stenosis and variable viscosity by modeling blood as Jeffrey fluid has been studied in this paper.The Jeffrey fluid has two parameters,the relaxation time λ1 and retardation time λ2.The governing equations are simplified using the case of mild stenosis.Perturbation method is used to solve the resulting equations.The effects of non-Newtonian nature of blood on velocity profile,temperature profile,wall shear stress,shearing stress at the stenotsis throat and impedance of the artery are discussed.The results for Newtonian fluid are obtained as special case from this model.     

A quasi-vector finite difference mode solver for optical waveguides with step-index profiles

A finite difference scheme based on the polynomial interpolation is constructed to solve the quasi-vector equations for optical waveguides with step-index profiles. The discontinuities of the normal components of the electric field across abrupt dielectric interfaces are taken into account. The numerical results include the polarization effects, but the memory requirement is the same as in solving the scalar wave equation. Moreover, the proposed finite difference scheme can be applied to both uniform and non-uniform mesh grids. The modal propagation constants and field distributions for a buried rectangular waveguide and a rib waveguide are presented. Solutions are compared favorably with those obtained by the numerical approaches published earlier.     

Atomic and field dynamics in the time-dependent Jaynes-Cu mmings model with arbitrary detuning

We propose a general numerical method for solving the time -dependent Jaynes-Cummings model with arbitrary detuning by integrating relate d groups of coupled equations using Runge-Kutta numerical technique. The atomic and field dynamics such as evolution of atomic population inversion and second -order correlation function of photons are studied in case of different detunin g. The results show that the field tends to exhibit bunching effect due to the i ncreasing of detuning.     

Variational iteration method for solving time-fractional diffusion equations in porous the medium

<正>The variational iteration method is successfully extended to the case of solving fractional differential equations, and the Lagrange multiplier of the method is identified in a more accurate way.Some diffusion models with fractional derivatives are investigated analytically,and the results show the efficiency of the new Lagrange multiplier for fractional differential equations of arbitrary order.     

Approximate solutions of Klein-Gordon equation with improved Manning-Rosen potential in D-dimensions using SUSYQM

In this paper, we present solutions of the Klein–Gordon equation for the improved Manning–Rosen potential for arbitrary l state in d-dimensions using the supersymmetric shape invariance method. We obtained the energy levels and the corresponding wave functions expressed in terms of Jacobi polynomial in a closed form for arbitrary l state. We also calculate the oscillator strength for the potential.     

A set of Boussinesq-type equations for interfacial internal waves in two-layer stratified fluid

Many new forms of Boussinesq-type equations have been developed to extend the range of applicability of the classical Boussinesq equations to deeper water in the study of the surface waves. One approach was used by Nwogu (1993. J. Wtrw. Port Coastal and Oc. Eng. 119, 618--638) to improve the linear dispersion characteristics of the classical Boussinesq equations by using the velocity at an arbitrary level as the velocity variable in derived equations and obtain a new form of Boussinesq-type equations, in which the dispersion property can be optimized by choosing the velocity variable at an adequate level. In this paper, a set of Boussinesq-type equations describing the motions of the interfacial waves propagating alone the interface between two homogeneous incompressible and inviscid fluids of different densities with a free surface and a variable water depth were derived using a method similar to that used by Nwogu (1993. J. Wtrw. Port Coastal and Oc. Eng. 119, 618--638) for surface waves. The equations were expressed in terms of the displacements of free surface and density-interface, and the velocity vectors at arbitrary vertical locations in the upper layer and the lower layer (or depth-averaged velocity vector across each layer) of a two-layer fluid. As expected, the equations derived in the present work include as special cases those obtained by Nwogu (1993, J. Wtrw. Port Coastal and Oc. Eng. 119, 618-638) and Peregrine (1967, J. Fluid Mech. 27, 815-827) for surface waves when the density of the upper fluid is taken as zero.     

A quasi-vector finite difference mode solver for optical waveguides with step-index profiles

A finite difference scheme based on the polynomial interpolation is constructed to solve the quasi-vector equations for optical waveguides with step-index profiles. The discontinuities of the normal components of the electric field across abrupt dielectric interfaces are taken into account. The numerical results include the polarization effects, but the memory requirement is the same as in solving the scalar wave equation. Moreover, the proposed finite difference scheme can be applied to both uniform and non-uniform mesh grids. The modal propagation constants and field distributions for a buried rectangular waveguide and a rib waveguide are presented. Solutions are compared favorably with those obtained by the numerical approaches published earlier.     

Dirac equation for the Hulthén potential within the Yukawa-type tensor interaction

Using the Nikiforov-Uvarov (NU) method, pseudospin and spin symmetric solutions of the Dirac equation for the scalar and vector Hulthén potentials with the Yukawa-type tensor potential are obtained for an arbitrary spin-orbit coupling quantum number κ. We deduce the energy eigenvalue equations and corresponding upper- and lower-spinor wave functions in both the pseudospin and spin symmetry cases. Numerical results of the energy eigenvalue equations and the upper- and lower-spinor wave functions are presented to show the effects of the external potential and particle mass parameters as well as pseudospin and spin symmetric constants on the bound-state energies and wave functions in the absence and presence of the tensor interaction.     

An implicit algorithm based on iterative modified approximate factorization method coupling with characteristic boundary conditions for solving subsonic viscous flows

An algorithm composed of an iterative modified approximate factorization(MAF(k)) method with Navier-Stokes characteristic boundary conditions(NSCBC) is proposed for solving subsonic viscous flows.A transformation on the matrix equation in MAF(k) is made in order to impose the implicit boundary conditions properly.To be in consistent with the implicit solver for the interior domain,an implicit scheme for NSCBC is formulated.The performance of the developed algorithm is investigated using spatially evolving zero pressure gradient boundary layer over a flat plate and a wall jet mixing with a cross flow over a flat plate with a square hole as the test cases.The numerical results are compared to the existing experimental datasets and a number of general correlations,together with other available numerical solutions,which demonstrate that the developed algorithm possesses promising capacity for simulating the subsonic viscous flows with large CFL number.     

A scaled boundary node method applied to two-dimensional crack problems

A boundary-type meshless method called the scaled boundary node method(SBNM) is developed to directly evaluate mixed mode stress intensity factors(SIFs) without extra post-processing.The SBNM combines the scaled boundary equations with the moving Kriging(MK) interpolation to retain the dimensionality advantage of the former and the meshless attribute of the latter.As a result,the SBNM requires only a set of scattered nodes on the boundary,and the displacement field is approximated by using the MK interpolation technique,which possesses the δ function property.This makes the developed method efficient and straightforward in imposing the essential boundary conditions,and no special treatment techniques are required.Besides,the SBNM works by weakening the governing differential equations in the circumferential direction and then solving the weakened equations analytically in the radial direction.Therefore,the SBNM permits an accurate representation of the singularities in the radial direction when the scaling center is located at the crack tip.Numerical examples using the SBNM for computing the SIFs are presented.Good agreements with available results in the literature are obtained.     

Recursion-transform approach to compute the resistance of a resistor network with an arbitrary boundary

We consider a profound problem of two-point resistance in the resistor network with a null resistor edge and an arbitrary boundary,which has not been solved before because the Green’s function technique and the Laplacian matrix approach are invalid in this case.Looking for the exact solutions of resistance is important but difficult in the case of the arbitrary boundary since the boundary is a wall or trap which affects the behavior of a finite network.In this paper,we give a general resistance formula that is composed of a single summation by using the recursion-transform method.Meanwhile,several interesting results are derived by the general formula.Further,the current distribution is given explicitly as a byproduct of the method.     

An improved local radial point interpolation method for transient heat conduction analysis

The smoothing thin plate spline （STPS） interpolation using the penalty function method according to the optimization theory is presented to deal with transient heat conduction problems. The smooth conditions of the shape functions and derivatives can be satisfied so that the distortions hardly occur. Local weak forms are developed using the weighted residual method locally from the partial differential equations of the transient heat conduction. Here the Heaviside step function is used as the test function in each sub-domain to avoid the need for a domain integral. Essential boundary conditions can be implemented like the finite element method （FEM） as the shape functions possess the Kronecker delta property. The traditional two-point difference method is selected for the time discretization scheme. Three selected numerical examples are presented in this paper to demonstrate the availability and accuracy of the present approach comparing with the traditional thin plate spline （TPS） radial basis functions.