A Quasi-Boundary Semi-Analytical Approach for Two-Dimensional Backward Heat Conduction Problems

In this article, we propose a semi-analytical method to tackle the two-dimensional backward heat conduction problem (BHCP) by using a quasi-boundary idea. First, the Fourier series expansion technique is employed to calculate the temperature field u(x, y, t) at any time t < T. Second, we consider a direct regularization by adding an extra termau(x, y, 0) to reach a second-kind Fredholm integral equation for u(x, y, 0). The termwise separable property of the kernel function permits us to obtain a closed-form regularized solution. Besides, a strategy to choose the regularization parameter is suggested. When several numerical examples were tested, we find that the proposed scheme is robust and applicable to the two-dimensional BHCP.     

A Fictitious Time Integration Method for Multi-Dimensional Backward Heat Conduction Problems

In this article, we propose a new numerical approach for solving these multi-dimensional nonlinear and nonhomogeneous backward heat conduction problems (BHCPs). A fictitious time t is employed to transform the dependent variable u(x, y, z, t) into a new one by (1+t)u(x, y, z, t)=: v(x, y, z, t, t), such that the original nonlinear and nonhomogeneous heat conduction equation is written as a new parabolic type partial differential equation in the space of (x, y, z, t, t). In addition, a fictitious viscous damping coefficient can be used to strengthen the stability of numerical integration of the discretized equations by utilizing a group preserving scheme. Six numerical experiments illustrate that the present algorism can be employed to recover the initial data very well. Even under the very large noisy final data, the fictitious time integration method is also robust against noise.     

A Fictitious Time Integration Method for Multi-Dimensional Backward Wave Problems

We address a new numerical approach to deal with these multi-dimensional backward wave problems (BWPs) in this study. A fictitious time τ is utilized to transform the dependent variable u(x, y, z, t) into a new one by (1+τ)u(x, y, z, t)=: v(x, y, z, t, τ), such that the original wave equation is written as a new hyperbolic type partial differential equation in the space of (x, y, z, t, τ). Besides, a fictitious viscous damping coefficient can be employed to strengthen the stability of numerical integration of the discretized equations by using a group preserving scheme. Several numerical instances demonstrate that the present scheme can be utilized to retrieve the initial wave very well. Even though the noisy final data are very large, the fictitious time integration method is also robust against disturbance.     

Mathematical modeling of a transient heat-conduction process

A numerical algorithm is proposed for solution of the transient heat-conduction equation by the Monte Carlo method. The calculated values of the temperature are compared with experimental data.Notation U temperature - t time - a thermal diffusivity - x, y, z coordinates - (x, y, z, t) temperature on the surface of the body - g(x, y, z) initial temperature - T finite time of investigation of the process - (x, y, z, t) = F (x, y, z, t)/c where F(x, y, z, t) is an internal heat source - c specific heat - density Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 4, pp. 745–747, October, 1980.     

A Quasi-Boundary Semi-Analytical Approach for Two-Dimensional Backward Advection-Dispersion Equation

In this study, we employ a semi-analytical approach to solve a two-dimensional advection-dispersion equation (ADE) for identifying the contamination problems. First, the Fourier series expansion technique is used to calculate the concentration field C(x, y, t) at any time t < T. Then, we ponder a direct regularization by adding an extra termaC(x, y, 0) on the final time data C(x, y, T), to reach a second-kind Fredholm integral equation. The termwise separable property of kernel function allows us obtaining a closed-form solution of the Fourier coefficients. A strategy to choose the regularization parameter is offered. The solver utilized in this work can retrieve the spatial distribution of the groundwater contaminant concentration. Several numerical examples are scrutinized to display that the new method can recover all the past data very well, and is good enough to deal with heterogeneous parameters, even though the final time data are noised seriously.     

一个非标准热方程反向问题的最优滤波正则化方法

本文讨论一类非标准反向热传导问题。它是严重不适定的,即如果问题的解存在,其解将不连续依赖于数据。为了获得稳定的数值解,我们给出了一种最优滤波正则化方法,并对空间无界和有界两种情形进行了研究。我们分别对空间无界和有界情形采用了Fourier变换技术和分离变量方法,并均获得了最优的稳定性误差估计。此外,我们还给出了两个有趣的数值例子验证了所提出的正则化方法的有效性。     

Three-dimensional finite element computation of laser cavity eigenmodes [corrected

A new method for computing eigenmodes of a laser resonator by the use of finite element analysis is presented. For this purpose, the scalar wave equation (delta + k2)E(x, y, z) = 0 is transformed into a solvable three-dimensional eigenvalue problem by the separation of the propagation factor exp(-ikz) from the phasor amplitude E(x, y, z) of the time-harmonic electrical field. For standing wave resonators, the beam inside the cavity is represented by a two-wave ansatz. For cavities with parabolic optical elements, the new approach has successfully been verified by the use of the Gaussian mode algorithm. For a diode-pumped solid-state laser with a thermally lensing crystal inside the cavity, the expected deviation between Gaussian approximation and numerical solution could be demonstrated clearly.     

A Quasi-Boundary Semi-Analytical Method for Backward in Time Advection-Dispersion Equation

In this paper, we take the advantage of an analytical method to solve the advection-dispersion equation (ADE) for identifying the contamination problems. First, the Fourier series expansion technique is employed to calculate the concentration field C(x, t) at any time t< T. Then, we consider a direct regularization by adding an extra term αC(x,0) on the final condition to carry off a second kind Fredholm integral equation. The termwise separable property of the kernel function permits us to transform itinto a two-point boundary value problem. The uniform convergence and error estimate of the regularized solution C^α(x,t) are provided and a strategy to select the regularized parameter is suggested. The solver used in this work can recover the spatial distribution of the groundwater contaminant concentration. Several numerical examples are examined to show that the new approach can retrieve all past data very well and is good enough to cope with heterogeneous parameters’ problems, even though the final data are noised seriously.     

一类四阶方程边值问题正解的存在性与多重性

本文得到了四阶方程边值问题u^（4）=f（u（x）），u′（0）=u″（0）=u′″（0）=0，ku（1）=u′″（1）相应的Green函数G（x，t）。从而将该问题转化为Hammerstein型积分方程，并利用锥拉伸与锥压缩不动点定理，讨论了其正解的存在性与多重性，得到了相应的结论。     

Imaging and transforming transmission through a medium with nonrotation-symmetric gradient index

The objective of this paper is to show that it is possible to transmit a paraxial optical image and transform through a dielectric inhomogeneous medium whose refractive index is given by n2 = n2(1)(z) + n2(0)[hz(z)x + h2(z)y - g2(z)(x2 + y2)], where n0 = n1(0), and n1, g, h1 and h2 are arbitrary functions of z. The optical image transmission, with a scaling factor F = H2(zm), m being an integer, is obtained at planes z = zm such that H1(zm) = 0 (the image condition), and the optical transform transmission is obtained at planes z = zm such that H2(zm) = 0 (the transform condition), where Hz(z) and H2(z) are two independent solutions of the paraxial ray equation H(z) + g2(z)H(z) = 0 with the initial conditions H1(0) = 0,H1(0) = 1,H2(0) = 1, and H2(0) = 0, where the point denotes the derivative with respect to z. Finally, we show that this medium can be represented by a transmittance function similar to the spherical-lens transmittance function and thus can be an element of image-forming systems.     

A Lie-Group Adaptive Method to Identify the Radiative Coefficients in Parabolic Partial Differential Equations

We consider two inverse problems for estimating radiative coefficients α(x) and α(x, y), respectively, in T_t(x, t) = T_xx(x, t)-α(x)T(x, t), and T_t(x, y, t) = T_xx(x, y, t) + T_yy(x, y, t)-α(x, y)T(x, y, t), where a are assumed to be continuous functions of space variables. A Lie-group adaptive method is developed, which can be used to find a at the spatially discretized points, where we only utilize the initial condition and boundary conditions, such as those for a typical direct problem. This point is quite different from other methods, which need the overspecified final time data. Three-fold advantages can be gained by the present Lie-group adaptive method (LGAM): (i) no a priori information of radiative coefficients is required, (ii) no extra data are measured, and (iii) no complicated procedure is involved. The accuracy and efficiency of present method are confirmed by comparing the estimated results with some exact solutions for 1-D and 2-D cases.     

Three-dimensional polarization states of monochromatic light fields

The 3×1 generalized Jones vectors (GJVs) [E(x) E(y) E(z)](t) (t indicates the transpose) that describe the linear, circular, and elliptical polarization states of an arbitrary three-dimensional (3-D) monochromatic light field are determined in terms of the geometrical parameters of the 3-D vibration of the time-harmonic electric field. In three dimensions, there are as many distinct linear polarization states as there are points on the surface of a hemisphere, and the number of distinct 3-D circular polarization states equals that of all two-dimensional (2-D) polarization states on the Poincaré sphere, of which only two are circular states. The subset of 3-D polarization states that results from the superposition of three mutually orthogonal x, y, and z field components of equal amplitude is considered as a function of their relative phases. Interesting contours of equal ellipticity and equal inclination of the normal to the polarization ellipse with respect to the x axis are obtained in 2-D phase space. Finally, the 3×3 generalized Jones calculus, in which elastic scattering (e.g., by a nano-object in the near field) is characterized by the 3-D linear transformation E(s)=T E(i), is briefly introduced. In such a matrix transformation, E(i) and E(s) are the 3×1 GJVs of the incident and scattered waves and T is the 3×3 generalized Jones matrix of the scatterer at a given frequency and for given directions of incidence and scattering.     

Analysing one isolated single walled carbon nanotube in the near-field domain with selective nanovolume Raman spectroscopy

In this paper, we describe a new method to the selective nanovolume analysing of one isolated single walled carbon nanotube (SWNT). This concept is based on actually available imaging micro-spectrometry systems for working in near-field domain combined with a stigmatic solid immersion lens. This combination of different analytical methods, and modified and configured equipment entitles us to expand the functionality toward a three-dimensional (3D) nanovolume Raman mapping and photoluminescence intensity with a possible discrimination in polarization, as well as photoluminescence decaytime constant mapping with their unique combination. Subsequently, selective spectra can be acquired from the same location on the samples. By spectrally selecting a SWNT, we registered the spatial distribution of the emitted photons in x, y, z vectors to determine the position of a SWNT in the near-field domain. For the SWNTs that are localized with an accuracy better than 18 nm in the x, y and <1 nm in the z directions, we demonstrate an analytical sensitivity close to a single nanotube with unity throughput. This near-field capability is applied to resolve local variations unambiguously in the Raman spectrum along one single SWNT. Finally, in this paper, we report what we believe to be the first evidence of Raman mapping and 3D real optical imaging of carbon nanotubes with near-field resolution.     

Large-scale three-dimensional object measurement: a practical coordinate mapping and image data-patching method

In a practical three-dimensional (3-D) sensing system, the measurement of a large-scale object cannot be completed in only one operation. A relieflike object is generally divided into several subregions, an optical sensor positioned at each of these locations, and the shape of the whole object obtained by patching together all the 3-D data of the subregions. It is important to have accurate 3-D coordinates (x, y, z) for each subregion. We propose a new phase-to-height mapping algorithm and an accurate lateral coordinate calibration method with which to obtain the 3-D coordinates. After all the subregions are measured, it is necessary to transform the local coordinates into global world coordinates; here we present a new image data-patching method based on a flood algorithm. This method provides the optimal path along which to patch all the subregions into the shape of the entire object. We have measured and successfully patched a large sandy pool (9 m x 5 m), and the reliability and feasibility of our method have been demonstrated by experiment.     

调和映射约束下的超分辨率图像重建

针对超分辨率图像重建过程中的正则化约束问题,本文提出采用p(x)调和映射进行正则化重建,根据超分辨率图像观察模型及正则约束,给出相应的能量泛函,并采用动态偏微分方程演化来求解能量泛函.该算法在重建的过程中能够根据图像空间特性自适应地采用不同的p(x)范数进行正则化,在图像的平滑区域采用近似2次范数进行正则化,而在图像的边缘区域采用近似1次范数进行正则化.实验结果均表明该算法不仅能有效地重建图像边缘,而且能有效地改善一次范教约束重建的分片常数效应.     

Spatial nonstationary heat-conduction problem for a prism with a coordinate-dependent heat-transfer coefficient

We present an efficient method for the determination of three-dimensional nonsteady-state fields of bodies of simple shapes, when the heat-transfer coefficient from their surface changes locally.Notation t temperature field - dimensionless temperature field - x, y, and z dimensional coordinates - X, Y, and Z dimensionless cordinates - time - thermal conductivity - a thermal diffusivity Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 42, No. 1, pp. 133–137, January, 1982.     

非化学计量比对(K0.465Na0.465Li0.07)(Nb0.95Sb0.05)O3无铅压电陶瓷结构及性能的影响

通过传统固相合成工艺制备了(K0.465+xNa0.465+yLi0.07)(Nb0.95-zSb0.05)O3(x,y,z=-0.01～0.02)无铅压电陶瓷,研究了非化学计量比对陶瓷结构及压电性能的影响。结果表明符合化学计量比的陶瓷具有四方钙钛矿结构;在实验范围内,K和Nb的过量或少量均不会改变体系的相结构,而Na过量将会导致体系正交-四方相变温度升高到室温以上,并且正交-四方相变温度随y的增加而升高;过量添加约0.5%(摩尔分数)的K或Na便能补偿在高温烧结时的碱金属元素的挥发损失,进而提高陶瓷的压电性能;该体系陶瓷的组分在较大范围内变化时(如当y=z=0时,x=0～0.02;当x=z=0时,y=0～0.01;以及当x=y=0时,z=-0.01～0.005),仍然能保持d33>200pC/N和kP>40%这样较好的性能。上述结果不仅有利于在研究中材料制备工艺的重复,而且有利于当材料在器件应用时所面临的规模化生产。     

Measured and modeled radiometric quantities in coastal waters: toward a closure

Accurate radiative transfer modeling in the coupled atmosphere-sea system is increasing in importance for the development of advanced remote-sensing applications. Aiming to quantify the uncertainties in the modeling of coastal water radiometric quantities, we performed a closure experiment to intercompare theoretical and experimental data as a function of wavelength lambda and water depth z. Specifically, the study focused on above-water downward irradiance E(d)(lambda, 0+) and in-water spectral profiles of upward nadir radiance L(u)(lambda, z), upward irradiance E(u)(lambda, z), downward irradiance E(d)(lambda, z), the E(u)(lambda, z)/L(u)(lambda, z) ratio (the nadir Q factor), and the E(u)(lambda, z)/E(d)(lambda, z) ratio (the irradiance reflectance). The theoretical data were produced with the finite-element method radiative transfer code ingesting in situ atmospheric and marine inherent optical properties. The experimental data were taken from a comprehensive coastal shallow-water data set collected in the northern Adriatic Sea. Under various measurement conditions, differences between theoretical and experimental data for the above-water E(d)(lambda, 0+) and subsurface E(d)(lambda, 0-) as well as for the in-water profiles of the nadir Q factor were generally less than 15%. In contrast, the in-water profiles of L(u)(lambda, z), E(d)(lambda, z), E(u)(lambda, z) and of the irradiance reflectance exhibited larger differences [to approximately 60% for L(u)(lambda, z) and E(u)(lambda, z), 30% for E(d)(lambda, z), and 50% for the irradiance reflectance]. These differences showed a high sensitivity to experimental uncertainties in a few input quantities used for the simulations: the seawater absorption coefficient; the hydrosol phase function backscattering probability; and, mainly for clear water, the bottom reflectance.     

Three-dimensional particle tracking via bifocal imaging

We introduce a bifocal imaging method that enables three-dimensional (3D) tracking of both fluorescent and nonfluorescent particles. We accomplish this by simultaneously imaging a focused plane, for in-plane position (x,y), and a defocused plane, for out-of-plane position (z), of a molecule using a single CCD camera. We applied our method to several systems including in vivo melanosome tracking and phagocytosed fluorescent bead tracking. We have achieved 2-5 nm accuracy with a 2-50 ms time resolution.     

Optical and electrical H/sub 2/- and NO/sub 2/-sensing properties of Au/In/sub x/O/sub y/N/sub z/ films

In this paper, we describe the optical and electrical gas-sensing properties of In/sub x/O/sub y/N/sub z/ films with an ultrathin gold promoter overlayer. We have fabricated In/sub x/O/sub y/N/sub z/ films with a nanocrystalline porous structure by RF-sputtering in Ar/N/sub 2/ followed by an annealing process. Gold particles with 20-30-nm diameter have been formed on top of the In/sub x/O/sub y/N/sub z/ films by dc sputtering and an annealing process. We have investigated the optical H/sub 2/and NO/sub 2/-sensing properties (change of absorbance) and also the electrical sensing effect (change of electrical resistance) for these two gases. A combined optical/electrical sensor for H/sub 2//NO/sub 2/ is proposed.