Transport theory for light propagation in biological tissue

We study light propagation in biological tissue using the radiative transport equation. The Green's function is the fundamental solution to the radiative transport equation from which all other solutions can be computed. We compute the Green's function as an expansion in plane-wave modes. We calculate these plane-wave modes numerically using the discrete-ordinate method. When scattering is sharply peaked, calculating the plane-wave modes for the transport equation is difficult. For that case we replace it with the Fokker-Planck equation since the latter gives a good approximation to the transport equation and requires less work to solve. We calculate the plane-wave modes for the Fokker-Planck equation numerically using a finite-difference approximation. The method of computing the Green's function for it is the same as for the transport equation. We demonstrate the use of the Green's function for the transport and Fokker-Planck equations by computing the point-spread function in a half-space composed of a uniform scattering and absorbing medium.     

Modeling imperfectly repaired system data via grey differential equations with unequal-gapped times

In this paper, we argue that grey differential equation models are useful in repairable system modeling. The arguments starts with the review on GM(1,1) model with equal- and unequal-spaced stopping time sequence. In terms of two-stage GM(1,1) filtering, system stopping time can be partitioned into system intrinsic function and repair effect. Furthermore, we propose an approach to use grey differential equation to specify a semi-statistical membership function for system intrinsic function times. Also, we engage an effort to use GM(1,N) model to model system stopping times and the associated operating covariates and propose an unequal-gapped GM(1,N) model for such analysis. Finally, we investigate the GM(1,1)-embed systematic grey equation system modeling of imperfectly repaired system operating data. Practical examples are given in step-by-step manner to illustrate the grey differential equation modeling of repairable system data.     

On the solutions of Navier-Stokes equations and the theory of homogeneous isotropic turbulence

This paper is an extension of our earlier work of solving the whole problem of homogeneous isotropic turbulence from the initial period to the final period of decay. An expansion method has been developed to obtain an axially symmetrical solution of the Navier-Stokes equations of motion in the form of an infinite set of nonlinear partial differential equations of the second order. For the present we solve the zeroth order approximation. By using the method of the Fourier transform, we get a nonlinear integro-differential equation for the amplitude function in the wave number space. It is also the dynamical equation for the energy spectrum. By choosing a suitable initial condition, we solve this equation numerically. The energy spectrum function and the energy transfer spectrum function thus calculated satisfy the spectrum form of the Kármán-Howarth equation exactly. We have computed the energy spectrum function, the energy transfer function, the decay of turbulent energy, the integral scale, the Taylor microscale, the double and triple velocity correlations on the whole range from the initial period to the final period of decay. On the whole all these calculated statistical physical quantities agree with experiments very well except for a few cases of small discrepancies at large separations.     

刚／粘塑性梁的强迫振动

本文依据粘塑性梁强迫振动的非齐次方程与非线性本构方程,提出采用分离变量的位移方法求解,获得该问题的应力和位移解.     

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.     

Davey-Stewartson Ⅰ方程新的精确周期解

Davey-Stewartson方程描述了有限深度的水中水波的运动,它的第一种类型称为(Davey-Stewartson I)是椭圆一双曲型方程。在物理学中,微分方程的精确解对考察非线性现象起着非常重要的作用,为了揭示Davey-Stewartson I方程的运动性质,本文研究它的精确周期解。应用F-代数方法并通过一个高阶辅助微分方程,获得了Davey-Stewartson I方程的一系列新的精确周期解,包括三角函数周期解,Jacobi椭圆函数周期解。     

Linearized thermoviscoelasticity with high temperature variations and related periodic problems

In the first part of this paper we develop a linearization of the equations of the thermoviscoelastic field in the case of great temperature variations. The possibility of uncoupling the heat equation from the motion equation is discussed in Section 3. After recalling some results on duality and virtual work principle we then study the motion equation with temperature as data, i.e. a given function of time and space variables. More precisely we study existence, uniqueness, regularity and asymptotic stability of a T-periodic (stress) solution of the motion equation in the dynamical case (Section 7) and in the quasi-static case (Section 8), when the temperature field is T-periodic in time and with a constitutive equation of Maxwell type where the stiffness and viscosity matrix are temperature dependent and thus are T-periodic functions of time. In the proof of the theorems we use frequently an inequality of monotony which means that the material is dissipative on a period. This inequality hold if the stiffness is a slowly varying function of time (the temperature has a little effect on the stiffness), on the other hand, fortunately, there is no condition on the viscosity.     

Alternative DRM formulations

In this paper, we present two DRM formulations. For a pseudo-Poisson equation, if the right-hand side is a linear operation on the dependent variable, we can derive a new DRM formulation. In comparison with the traditional DRM formulation for the same equation, the proposed approach is much easier and more efficient. For the axisymmetric Poisson equation, we construct a DRM formulation by using the linear axisymmetric radial basis function. The particular solution involved is obtained in a closed form, and thus speeds up the evaluation of the particular solution. Three numerical examples demonstrate the accuracy and efficiency of these formulations.     

Quasiparticle Kinetic Equation in a Trapped Bose Gas at Low Temperatures

Recently the authors used the Kadanoff–Baym non-equilibrium Green's function formalism to derive kinetic equation for the non-condensate atoms, in conjunction with a consistent generalization of the Gross–Pitaevskii equation for the Bose condensate wavefunction. This work was limited to high temperatures, where the excited atoms could be described by a Hartree–Fock particle-like spectrum. Following the approach of Kane and Kadanoff in 1965, we present the generalization of our recent work which is valid at low temperatures, where the input single-particle spectrum is now described by the Bogoliubov–Popov approximation. We derive a kinetic equation for the quasiparticle distribution function with collision integrals describing scattering between quasiparticles and the condensate atoms. From the general expression for the collision integral for the scattering between quasiparticle excitations, we find the quasiparticle distribution function corresponding to local equilibrium. This expression includes a quasiparticle chemical potential that controls the non-diffusive equilibrium between condensate atoms and the quasiparticle excitations. We derive a generalized Gross–Pitaevskii equation for the condensate wavefunction that also includes the damping effects due to collisions between atoms in the condensate and the thermally excited quasiparticles. For a uniform Bose gas, our kinetic equation for the thermally excited quasiparticles reduces to that found by Eckern, as well as by Kirkpatrick and Dorfman.     

基于广义等值面提取的多视场深度像融合

建立等值面方程是隐式曲面造型问题的一个重要方面。一旦等值面方程被建立起来，可以设计相应的算法从深度数据中提取等值面，从而实现多视场深度像的融合。笔者提出一种广义的等值面方程，其理论依据是“空间点对应的向量图解”和“最小二乘逼近的法则”。在广义的等值面方程中注入了采样点的方向信息，提供了一个清晰的物理和几何含义的描述，由此可以解释设置“权函数”的标准以及隐式曲面造型中其他的一些几何关系。它为设计等值面提取算法提供了一个新的方法。     

A model of eddy-current probes with ferrite cores

A model of a three-dimensional axisymmetric probe coil with a ferrite core in the presence of a conducting half-space (the workpiece) is developed. The half-space is accounted for by computing the appropriate Green's function by using Bessel transforms. Upon introducing equivalent Amperian currents within the core, we derive a volume integral equation, whose unknown is either the magnetic induction field, or induced magnetization, and whose kernel is the Green's function that was previously derived. The integral equation is transformed via the method of moments into a vector-matrix equation, which is then solved using a linear equation solver. This allows the computation of the magnetic induction field within the core, the driving-point impedance of the coil-core combination, and the induced eddy currents within the workpiece.     

Radiative transfer in inhomogeneous stratified scattering media with use of the auxiliary function method

The auxiliary function method consists of taking full advantage of the expansion of the phase function on spherical harmonics in order to deduce an integral equation from the radiative transfer equation. In contrast to the discrete-ordinate method, it is free of the channel concept, the unknowns being a function only of the optical depth. After presenting the method, we show that it is very accurate and particularly well fitted when the scattering medium is continuously inhomogeneous in albedo and phase function and also for sublayers with different refractive index.     

An equation to calculate oxygen content relating to copper oxidation level of doped cuprates

A new equation relating oxygen contenty to copper valence 2+p has been derived and proposed. Compared to the previous equation, there is a corrected function in the new equation. The corrected function depends on dopant contentx and copper oxidation levelp. As a metalM is substituted for the metal of the sample, the corrected function has a different form, and the function is discussed in detail. For each case, taking high-T _c cuprate for example, itsp is determined andy calculated by the new and previous equations. Changes in the corrected function are shown in the figures. All the discussions show that it is necessary to use the new equation to calculatey if we are to get accurate experiment results. The new equation can be widely applied to calculate the oxygen content of multicomponent doped or undoped cuprates, and the exact stoichiometries and superconducting or impurity phases of samples need not be known.     

Solution to the topology optimization problem using a time-evolution equation

This article describes a numerical solution to the topology optimization problem using a time-evolution equation. The design variables of the topology optimization problem are defined as a mathematical scalar function in a given design domain. The scalar function is projected to the normalized density function. The adjoint variable method is used to determine the gradient defined as the ratio of the variation of the objective function or constraint function to the variation of the design variable. The variation of design variables is obtained using the solution of the time-evolution equation in which the source term and Neumann boundary condition are given as a negative gradient. The distribution of design variables yielding an optimal solution is obtained by time integration of the solution of the time-evolution equation. By solving the topology optimization problem using the proposed method, it is shown that the objective function decreases when the constraints are satisfied. Furthermore, we apply the proposed method to the thermal resistance minimization problem under the total volume constraint and the mean compliance minimization problem under the total volume constraint.     

Solitary Magnon Localization in Antiferromagnet RbFeBr_3

Introducing the Dyson-Maleev transformation. the coherent state ansatz and the time-dependent variation principle, we obtain two partial different equations of motion from Hamiltonian. Employing the method of multiple scales. we reduce these equations into the envelope function equations and force the amplitude function to satisfy a nonlinear Schrodinger equation. Using the inverse- scat-tering transformation, we obtain the single soliton solution and discuss the solitary magnon localization in antiferromagnet RbFeBr3     

A boundary-only meshless method for numerical solution of the Eikonal equation

The radial basis function (RBF) collocation methods for the numerical solution of partial differential equation have been popular in recent years because of their advantage. For instance, they are inherently meshless, integration free and highly accurate. In this article we study the RBF solution of Eikonal equation using boundary knot method and analog equation method. The boundary knot method (BKM) is a meshless boundary-type radial basis function collocation technique. In contrast with the method of fundamental solution (MFS), the BKM uses the non-singular general solution instead of the singular fundamental solution to obtain the homogeneous solution. Similar to MFS, the RBF is employed to approximate the particular solution via the dual reciprocity principle. In the current paper, we applied the idea of analog equation method (AEM). According to AEM, the nonlinear governing operator is replaced by an equivalent nonhomogeneous linear one with known fundamental solution and under the same boundary conditions. Finally numerical results and discussions are presented to show the validity and efficiency of the proposed method.     

Beam propagation in sharply peaked forward scattering media

We calculate the radiance of a light beam propagating in a uniformly scattering and absorbing slab and determine the point-spread function. We do this by solving numerically the governing radiative transport equation by use of plane-wave mode expansions. When scattering is sharply peaked in the forward direction and it becomes difficult to solve the radiative transport equation, we replace it with either the Fokker-Planck or the Leakeas-Larsen equation. We also solve these equations by using plane-wave mode expansions. Numerical results show that these two equations agree with the radiative transport equation for large anisotropy factors. The agreement improves as the optical thickness increases.     

Linearized kinetic equations and relaxation processes of a superconductor near T c

Starting from the equation of Gor'kov and Eliashberg in a form introduced by Eilenberger, we derive a set of linearized equations for the deviation from the equilibrium value of the quasiparticle distribution function as well as of the order parameter. These equations resemble the Boltzmann equation and the Ginzburg-Landau equation, respectively, and they form a set of coupled equations. Two different modes can be distinguished, depending on whether the order parameter changes in magnitude or in phase. The equations are solved for the case of a stationary quasiparticle injection into a superconductor and the change in the electrochemical potential of the quasiparticles is calculated. Furthermore, we treat the problem of a current flowing perpendicular to a superconducting-normal interface in which a normal current is converted into a supercurrent, and we calculate the extra resistance of the interface.     

求解广义Lyapunov方程的非单调谱投影梯度法

本文研究双线性控制系统中的一类广义Lyapunov方程的半正定解．基于凸函数的局部极小解就是全局极小解这一良好性质,首先将广义Lyapunov方程的半正定解问题等价转化为凸优化问题．利用非单调线搜索技术确定步长,构造了非单调谱投影梯度方法求解这一等价问题．最后用数值例子验证了新方法的可行性和有效性．     

Reconstruction of a time-dependent source term in a time-fractional diffusion equation

This paper is devoted to determine a time-dependent source term in a time-fractional diffusion equation by using the usual initial and boundary data and an additional measurement data at an inner point. Based on the separation of variables and Duhamel's principle, we transform the inverse source problem into a first kind Volterra integral equation with the source term as the unknown function and then show the ill-posedness of the problem. Further, we use a boundary element method combined with a generalized Tikhonov regularization to solve the Volterra integral equation of the fist kind. The generalized cross-validation choice rule is applied to find a suitable regularization parameter. Four numerical examples are provided to show the effectiveness and robustness of the proposed method.