#### Sample records for boundary analytical solution

1. Analytical Solutions for Beams Passing Apertures with Sharp Boundaries

Luz, Eitam; Malomed, Boris A

2016-01-01

An approximation is elaborated for the paraxial propagation of diffracted beams, with both one- and two-dimensional cross sections, which are released from apertures with sharp boundaries. The approximation applies to any beam under the condition that the thickness of its edges is much smaller than any other length scale in the beam's initial profile. The approximation can be easily generalized for any beam whose initial profile has several sharp features. Therefore, this method can be used as a tool to investigate the diffraction of beams on complex obstacles. The analytical results are compared to numerical solutions and experimental findings, which demonstrates high accuracy of the approximation. For an initially uniform field confined by sharp boundaries, this solution becomes exact for any propagation distance and any sharpness of the edges. Thus, it can be used as an efficient tool to represent the beams, produced by series of slits with a complex structure, by a simple but exact analytical solution.

2. THE ANALYTICAL SOLUTION FOR SEDIMENT REACTION AND DIFFUSION EQUATION WITH GENERALIZED INITIAL-BOUNDARY CONDITIONS

熊岳山; 韦永康

2001-01-01

The sediment reaction and diffusion equation with generalized initial and boundary condition is studied. By using Laplace transform and Jordan lemma , an analytical solution is got, which is an extension of analytical solution provided by Cheng Kwokming James ( only diffusion was considered in analytical solution of Cheng ). Some problems arisen in the computation of analytical solution formula are also analysed.

3. Analytical solution for 1D consolidation of unsaturated soil with mixed boundary condition

Zhen-dong SHAN; Dao-sheng LING; Hao-jiang DING

2013-01-01

Based on consolidation equations proposed for unsaturated soil,an analytical solution for 1D consolidation of an unsaturated single-layer soil with nonhomogeneous mixed boundary condition is developed.The mixed boundary condition can be used for special applications,such as tests occur in laboratory.The analytical solution is obtained by assuming all material parameters remain constant during consolidation.In the derivation of the analytical solution,the nonhomogeneous boundary condition is first transformed into a homogeneous boundary condition.Then,the eigenfunction and eigenvalue are derived according to the consolidation equations and the new boundary condition.Finally,using the method of undetermined coefficients and the orthogonal relation of the eigenfunction,the analytical solution for the new boundary condition is obtained.The present method is applicable to various types of boundary conditions.Several numerical examples are provided to investigate the consolidation behavior of an unsaturated single-layer soil with mixed boundary condition.

4. Amendment to "Analytical Solution for the Convectively-Mixed Atmospheric Boundary Layer": Inclusion of Subsidence

Ouwersloot, H.G.; Arellano, de J.V.G.

2013-01-01

In Ouwersloot and Vila-Guerau de Arellano (Boundary-Layer Meteorol. doi: 10. 1007/s10546-013-9816-z, 2013, this issue), the analytical solutions for the boundary-layer height and scalar evolutions are derived for the convective boundary layer, based on the prognostic equations of mixed-layer slab mo

5. An analytical solution for the Marangoni mixed convection boundary layer flow

Moghimi, M. A.; Kimiaeifar, Amin; Rahimpour, M.; Bagheri, G. H.

2010-01-01

In this article, an analytical solution for a Marangoni mixed convection boundary layer flow is presented. A similarity transform reduces the Navier-Stokes equations to a set of nonlinear ordinary differential equations, which are solved analytically by means of the homotopy analysis method (HAM...... control the convergence of the solution. The numerical solution of the similarity equations is developed and the results are in good agreement with the analytical results based on the HAM....

6. Analytical Solution of Boundary Integral Equations for 2-D Steady Linear Wave Problems

J.M. Chuang

2005-01-01

Based on the Fourier transform, the analytical solution of boundary integral equations formulated for the complex velocity of a 2-D steady linear surface flow is derived. It has been found that before the radiation condition is imposed,free waves appear both far upstream and downstream. In order to cancel the free waves in far upstream regions, the eigensolution of a specific eigenvalue, which satisfies the homogeneous boundary integral equation, is found and superposed to the analytical solution. An example, a submerged vortex, is used to demonstrate the derived analytical solution. Furthermore,an analytical approach to imposing the radiation condition in the numerical solution of boundary integral equations for 2-D steady linear wave problems is proposed.

7. Analytic solution of an initial-value problem from Stokes flow with free boundary

Xuming Xie

2008-01-01

We study an initial-value problem arising from Stokes flow with free boundary. If the initial data is analytic in disk $mathcal{R}_r$ containing the unit disk, it is proved that unique solution, which is analytic in $mathcal{R}_s$ for $sin (1,r)$, exists locally in time.

8. Analytical solution of second Stokes problem of behaviour of rarefied gas with Cercignani boundary accomodation conditions

Latyshev, A. V.; Yushkanov, A. A.

2012-01-01

Analytical solution of second Stokes problem of behaviour of rarefied gas with Cercignani boundary accomodation conditions The second Stokes problem about behaviour of rarefied gas filling half-space is analytically solved. A plane, limiting half-space, makes harmonious fluctuations in the plane. The kinetic BGK-equation (Bhatnagar, Gross, Krook) is used. The boundary accomodation conditions of Cercignani of reflexion gaseous molecules from a wall are considered. Distribution function of the ...

9. Approximate Analytical Solutions for a Class of Laminar Boundary-Layer Equations

Seripah Awang Kechil; Ishak Hashim; Sim Siaw Jiet

2007-01-01

A simple and efficient approximate analytical technique is presented to obtain solutions to a class of two-point boundary value similarity problems in fluid mechanics. This technique is based on the decomposition method which yields a general analytic solution in the form of a convergent infinite series with easily computable terms. Comparative study is carried out to show the accuracy and effectiveness of the technique.

10. A semi-analytical solution for viscothermal wave propagation in narrow gaps with arbitrary boundary conditions.

Wijnant, Ysbrand; Spiering, Ruud; Blijderveen, van Maarten; Boer, de André

2006-01-01

Previous research has shown that viscothermal wave propagation in narrow gaps can efficiently be described by means of the low reduced frequency model. For simple geometries and boundary conditions, analytical solutions are available. For example, Beltman [4] gives the acoustic pressure in the gap b

11. Radionuclides migration in fractured porous rock: Analytical solutions for various boundary conditions

Analytical solutions based on Laplace transform are developed for the problem of radionuclides transport along a discrete planar fracture in porous rock. The solutions take into account advective transport in the fracture, longitudinal hydrodynamic dispersion along the fracture axis, molecular diffusion from the fracture into the rock matrix, sorption within the rock matrix, sorption onto the surface of the fracture and radioactive decay. The initial concentration in both the fracture and the rock matrix is assumed to be zero. Four boundary conditions, constant concentration, exponentially decaying concentration, exponentially decaying flux and kinetic solubility-limited dissolution are assumed. All these analytical solutions are in a form of a single integral that is evaluated by a Gauss-Legendre quadrature for each point in space and time. A comparison between the concentration profiles with a constant concentration inlet boundary condition and those with a decaying concentration inlet boundary condition shows that the concentration profile is strongly influenced by the inlet boundary condition when the retardation factor of the matrix is high. As the dissolution rate constant approaches infinity, the inlet boundary condition of the kinetic solubility-limited dissolution model can be replaced by the boundary condition of constant concentration

12. Analytical solution of second Stokes problem of behaviour of rarefied gas with Cercignani boundary accomodation conditions

Latyshev, A V

2012-01-01

Analytical solution of second Stokes problem of behaviour of rarefied gas with Cercignani boundary accomodation conditions The second Stokes problem about behaviour of rarefied gas filling half-space is analytically solved. A plane, limiting half-space, makes harmonious fluctuations in the plane. The kinetic BGK-equation (Bhatnagar, Gross, Krook) is used. The boundary accomodation conditions of Cercignani of reflexion gaseous molecules from a wall are considered. Distribution function of the gaseous molecules is constructed. The velocity of gas in half-space is found, also its value direct at a wall is found. The force resistance operating from gas on border is found. Besides, the capacity of dissipation of the energy falling to unit of area of the fluctuating plate limiting gas is obtained.

13. Analytical solutions for two-dimensional soil heat flow with radiation surface boundary conditions

Heat flow add temperature variations in soil are important in agriculture, forestry, and ecology. Nonuniform surface cover and variability in soil properties result in two-dimensional soil heat flow. This study derives analytical solutions for unsteady two-dimensional soil heat transfer problems with standard (constant temperature coefficient) and modified (temperature coefficient varies with position) radiation surface boundary conditions. Solutions are periodic in time and horizontal direction. The structure of the solutions guarantees that soil temperatures are smooth functions of position and time, even if the temperature coefficient or forcing function in the radiation boundary condition are discontinuous. Calculated soil temperature heat flux densities, and surface energy balance components for bare wet strips alternating with strips covered with either chalk, black plastic, or clear plastic were found to vary strongly with time and position. For diurnal variations, lateral heat flow only significantly affected temperatures in the middle of strips narrower than approximately 0.2 m. Sensitivity of soil temperature to changes in soil thermal properties increased as the temperature coefficient in the surface boundary condition decreased. Both cases showed that spatial differences in albedo, surface resistance, and serodynamic resistance spatially alter the surface energy balance and soil thermal regimes, including surface temperature and heat flux density

14. UNSTEADY BOUNDARY LAYER FLOW ALONG A STRETCHING CYLINDER AN ANALYTICAL SOLUTION

M. Y. Akl

2014-01-01

Full Text Available The axisymetric laminar boundary layer unsteady flow along a continuously stretching cylinder immersed in a viscous and incompressible fluid is studied. The governing partial boundary layer equations in cylindrical form are first transformed into ordinary differential equations these equations are solved analytically using the optimal modified Homotopy Asymptotic method in order to get a closed form solution for the dimensionless functions f and è. The main object of this study is to investigate the effect of an unsteady motion of a stretching cylinder on the flow and heat transfer characteristics such as surface skin friction and surface heat flux. These characteristics have a direct effect on the quality of the final product of the fiber manufacturing and extrusion processes. Considerable effects were found for the dynamic parameter (γ, the curvature parameter (ρ and the prandtl number (pr on the velocity and the heat transfer.

15. Analytical solutions for thermal forcing vortices in boundary layer and its applications

LIU Xiao-ran; LI Guo-ping

2007-01-01

Using the Boussinesq approximation, the vortex in the boundary layer is assumed to be axisymmetrical and thermal-wind balanced system forced by diabatic heating and friction, and is solved as an initial-value problem of linearized vortex equation set in cylindrical coordinates. The impacts of thermal forcing on the flow field structure of vortex are analyzed. It is found that thermal forcing has significant impacts on the flow field structure, and the material representative forms of these impacts are closely related to the radial distribution of heating. The discussion for the analytical solutions for the vortex in the boundary layer can explain some main structures of the vortex over the Tibetan Plateau.

16. Approximate analytical solution to the Boussinesq equation with a sloping water-land boundary

Tang, Yuehao; Jiang, Qinghui; Zhou, Chuangbing

2016-04-01

An approximate solution is presented to the 1-D Boussinesq equation (BEQ) characterizing transient groundwater flow in an unconfined aquifer subject to a constant water variation at the sloping water-land boundary. The flow equation is decomposed to a linearized BEQ and a head correction equation. The linearized BEQ is solved using a Laplace transform. By means of the frozen-coefficient technique and Gauss function method, the approximate solution for the head correction equation can be obtained, which is further simplified to a closed-form expression under the condition of local energy equilibrium. The solutions of the linearized and head correction equations are discussed from physical concepts. Especially for the head correction equation, the well posedness of the approximate solution obtained by the frozen-coefficient method is verified to demonstrate its boundedness, which can be further embodied as the upper and lower error bounds to the exact solution of the head correction by statistical analysis. The advantage of this approximate solution is in its simplicity while preserving the inherent nonlinearity of the physical phenomenon. Comparisons between the analytical and numerical solutions of the BEQ validate that the approximation method can achieve desirable precisions, even in the cases with strong nonlinearity. The proposed approximate solution is applied to various hydrological problems, in which the algebraic expressions that quantify the water flow processes are derived from its basic solutions. The results are useful for the quantification of stream-aquifer exchange flow rates, aquifer response due to the sudden reservoir release, bank storage and depletion, and front position and propagation speed.

17. Generalized analytical solution for advection-dispersion equation in finite spatial domain with arbitrary time-dependent inlet boundary condition

J.-S. Chen

2011-04-01

Full Text Available This study presents a generalized analytical solution for one-dimensional solute transport in finite spatial domain subject to arbitrary time-dependent inlet boundary condition. The governing equation includes terms accounting for advection, hydrodynamic dispersion, linear equilibrium sorption and first order decay processes. The generalized analytical solution is derived by using the Laplace transform with respect to time and the generalized integral transform technique with respect to the spatial coordinate. Several special cases are presented and compared to illustrate the robustness of the derived generalized analytical solution. Result shows an excellent agreement. The analytical solutions of the special cases derived in this study have practical applications. Moreover, the derived generalized solution which consists an integral representation is evaluated by the numerical integration to extend its usage. The developed generalized solution offers a convenient tool for further development of analytical solution of specified time-dependent inlet boundary conditions or numerical evaluation of the concentration field for arbitrary time-dependent inlet boundary problem.

18. Analytical solution of non-Fourier heat conduction problem on a fin under periodic boundary conditions

Ahmadikia, H. [University of Isfahan, Isfahan (Iran, Islamic Republic of); Rismanian, M. [Bu-Ali Sina University, Hamadan (Iran, Islamic Republic of)

2011-11-15

Fourier and hyperbolic models of heat transfer on a fin that is subjected to a periodic boundary condition are solved analytically. The differential equation in Fourier and non-Fourier models is solved by the Laplace transform method. The temperature distribution on the fin is obtained using the residual theorem in a complex plan for the inverse Laplace transform method. The thermal shock is generated at the base of the fin, which moves toward the tip of the fin and is reflected from the tip. The current study of various parameters on the thermal shock location shows that relaxation time has a great influence on the temperature distribution on the fin. An unsteady boundary condition in the base fin caused the shock, which is generated continuously from the base and has interacted with the other reflected thermal shocks. Results of the current study show that the hyperbolic heat conduction equation can violate the second thermodynamic law under some unsteady boundary conditions.

19. Analytical solution of non-Fourier heat conduction problem on a fin under periodic boundary conditions

Fourier and hyperbolic models of heat transfer on a fin that is subjected to a periodic boundary condition are solved analytically. The differential equation in Fourier and non-Fourier models is solved by the Laplace transform method. The temperature distribution on the fin is obtained using the residual theorem in a complex plan for the inverse Laplace transform method. The thermal shock is generated at the base of the fin, which moves toward the tip of the fin and is reflected from the tip. The current study of various parameters on the thermal shock location shows that relaxation time has a great influence on the temperature distribution on the fin. An unsteady boundary condition in the base fin caused the shock, which is generated continuously from the base and has interacted with the other reflected thermal shocks. Results of the current study show that the hyperbolic heat conduction equation can violate the second thermodynamic law under some unsteady boundary conditions

20. On the analytical solution of the SN equation in a rectangle assuming an exponential exiting angular flux boundary

In this work, we report an analytical solution for the set of SN equations for the angular flux, in a rectangle, using the double Laplace transform technique. Its main idea comprehends the steps: application of the Laplace transform in one space variable, solution of the resulting equation by the LTSN method and reconstruction of the double Laplace transformed angular flux using the inversion theorem of the Laplace transform. We must emphasize that we perform the Laplace inversion by the LTSN method in the x direction, meanwhile we evaluate the inversion in the y direction performing the calculation of the corresponding line integral solution by the Stefest method. We have also to figure out that the application of Laplace transform to this type of boundary value problem introduces additional unknown functions associated to the partial derivatives of the angular flux at boundary. Based on the good results attained by the nodal LTSN method, we assume that the angular flux at boundary is also approximated by an exponential function. By analytical we mean that no approximation is done along the solution derivation except for the exponential hypothesis for the exiting angular flux at boundary. For sake of completeness, we report numerical comparisons of the obtained results against the ones of the literature. (author)

1. An analytical solution for simulation of the fission gas behaviors with time-dependent piece-wise boundary resolution

Cui Yi; Huo Yongzhong [Department of Mechanics and Engineering Science, Fudan University, Shanghai 200433 (China); Ding Shurong, E-mail: dsr1971@163.com [Department of Mechanics and Engineering Science, Fudan University, Shanghai 200433 (China) and Science and Technology on Reactor System Design Technology Laboratory, Nuclear Power Institution of China, Chengdu 610041, Sichuan (China); Zhang Lin; Li Yuanming [Science and Technology on Reactor System Design Technology Laboratory, Nuclear Power Institution of China, Chengdu 610041, Sichuan (China)

2012-05-15

An analytical solution of gas concentration for the equivalent spherical grain is obtained first in Laplace space, then the inverse-Laplace transformed solution is further developed. The corresponding analytical expressions for the grain boundary gaseous swelling and the fission gas release in UO{sub 2} nuclear fuels are developed in the absence of grain growth. The following phenomena and assumptions are taken into account in our model, including the gas atom diffusion, saturation and the time-varying piece-wise inter-granular resolution. The explicit expression for saturation time of the grain boundary gas atoms is also obtained. Our approximated analytical solutions for the fission gas behaviors are validated through comparison with those solved by finite difference method. Good agreement has been achieved for the cases with different input parameters. Based on the developed analytical solutions, the effects of the grain sizes and the external pressure on the fission gas behaviors are investigated. This study lays a foundation for the multi-scale simulation of the thermo-mechanical behaviors in nuclear fuel elements.

2. An analytical solution for simulation of the fission gas behaviors with time-dependent piece-wise boundary resolution

An analytical solution of gas concentration for the equivalent spherical grain is obtained first in Laplace space, then the inverse-Laplace transformed solution is further developed. The corresponding analytical expressions for the grain boundary gaseous swelling and the fission gas release in UO2 nuclear fuels are developed in the absence of grain growth. The following phenomena and assumptions are taken into account in our model, including the gas atom diffusion, saturation and the time-varying piece-wise inter-granular resolution. The explicit expression for saturation time of the grain boundary gas atoms is also obtained. Our approximated analytical solutions for the fission gas behaviors are validated through comparison with those solved by finite difference method. Good agreement has been achieved for the cases with different input parameters. Based on the developed analytical solutions, the effects of the grain sizes and the external pressure on the fission gas behaviors are investigated. This study lays a foundation for the multi-scale simulation of the thermo-mechanical behaviors in nuclear fuel elements.

3. Analytical solutions of the one-dimensional advection-dispersion solute transport equation subject to time-dependent boundary conditions

Analytical solutions of the advection-dispersion solute transport equation remain useful for a large number of applications in science and engineering. In this paper we extend the Duhamel theorem, originally established for diffusion type problems, to the case of advective-dispersive transport subj...

4. Analytical solution for beam with time-dependent boundary conditions versus response spectrum

Gou, P.F.; Panahi, K.K. [GE Nuclear Energy, San Jose, CA (United States)

2001-07-01

This paper studies the responses of a uniform simple beam for which the supports are subjected to time-dependent conditions. Analytical solution in terms of series was presented for two cases: (1) Two supports of a simple beam are subjected to a harmonic motion, and (2) One of the two supports is stationary while the other is subjected to a harmonic motion. The results of the analytical solution were investigated and compared with the results of conventional response spectrum method using the beam finite element model. One of the applications of the results presented in this paper can be used to assess the adequacy and accuracy of the engineering approaches such as response spectra methods. It has been found that, when the excitation frequency equals the fundamental frequency of the beam, the results from response spectrum method are in good agreement with the exact calculation. The effects of initial conditions on the responses are also examined. It seems that the non-zero initial velocity has pronounced effects on the displacement time histories but it has no effect on the maximum accelerations. (author)

5. An analytical solution for the monoenergetic diffusion equation in a multilayered slab: determination of albedo boundary condition

In this work, we solve analytically, without losing generality, the neutron diffusion equation for monoenergetic neutrons in a multilayered slab. To this end, we initially determine the solution of the neutron diffusion equation for a generic slab using standard results of second order linear ordinary differential equation with constant coefficients. The global solution for the multilayered slab is then determined applying the boundary condition and continuity of the flux and current at interface. Once the neutron flux is known, the albedo boundary condition is straightly obtained for an arbitrary number of layers in the baffle-reflecting regions, by just using the definition of albedo. We also present numerical simulation for the results neutron flux and comparison with the in literature. (author)

6. Analytical Solution of Thermal Wave Models on Skin Tissue Under Arbitrary Periodic Boundary Conditions

2013-01-01

Modeling and understanding the heat transfer in biological tissues is important in medical thermal therapeutic applications. The biothermomechanics of skin involves interdisciplinary features, such as bioheat transfer, biomechanics, and burn damage. The hyperbolic thermal wave model of bioheat transfer and the parabolic Pennes bioheat transfer equations with blood perfusion and metabolic heat generation are applied for the skin tissue as a finite and semi-infinite domain when the skin surface temperature is suddenly exposed to a source of an arbitrary periodic temperature. These equations are solved analytically by Laplace transform methods. The thermal wave model results indicate that a non-Fourier model has predicted the thermal behavior correctly, compared to that of previous experiments. The results of the thermal wave model show that when the first thermal wave moves from the first boundary, the temperature profiles for finite and semi-infinite domains of skin become separated for these phenomena; the discrepancy between these profiles is negligible. The accuracy of the obtained results is validated through comparisons with existing numerical results. The results demonstrate that the non-Fourier model is significant in describing the thermal behavior of skin tissue.

7. Analytic solution for magnetohydrodynamic boundary layer flow of Casson fluid over a stretching/shrinking sheet with wall mass transfer

Krishnendu Bhattacharyya; Tasawar Hayat; Ahmed Alsaedi

2013-01-01

In this analysis,the magnetohydrodynamic boundary layer flow of Casson fluid over a permeable stretching/shrinking sheet in the presence of wall mass transfer is studied.Using similarity transformations,the governing equations are converted to an ordinary differential equation and then solved analytically.The introduction of a magnetic field changes the behavior of the entire flow dynamics in the shrinking sheet case and also has a major impact in the stretching sheet case.The similarity solution is always unique in the stretching case,and in the shrinking case the solution shows dual nature for certain values of the parameters.For stronger magnetic field,the similarity solution for the shrinking sheet case becomes unique.

8. Analytical solutions to the fractional advection-diffusion equation with time-dependent pulses on the boundary

Rubbab, Qammar; Mirza, Itrat Abbas; Qureshi, M. Zubair Akbar

2016-07-01

The time-fractional advection-diffusion equation with Caputo-Fabrizio fractional derivatives (fractional derivatives without singular kernel) is considered under the time-dependent emissions on the boundary and the first order chemical reaction. The non-dimensional problem is formulated by using suitable dimensionless variables and the fundamental solutions to the Dirichlet problem for the fractional advection-diffusion equation are determined using the integral transforms technique. The fundamental solutions for the ordinary advection-diffusion equation, fractional and ordinary diffusion equation are obtained as limiting cases of the previous model. Using Duhamel's principle, the analytical solutions to the Dirichlet problem with time-dependent boundary pulses have been obtained. The influence of the fractional parameter and of the drift parameter on the solute concentration in various spatial positions was analyzed by numerical calculations. It is found that the variation of the fractional parameter has a significant effect on the solute concentration, namely, the memory effects lead to the retardation of the mass transport.

9. An Analytical Solution of the Radiative Transfer Equation for Inhomogeneous Finite Medium with Fresnel Boundary Conditions

The problem of radiation transfer through an inhomogeneous finite medium with internal energy sources and diffusely and specular reflecting boundary conditions is considered (problem 1). The partial heat fluxes for this problem are given in terms of the albedo of the source-free problem with specular-reflecting boundaries (problem 2). We assume that the reflectivity of the surfaces dependent upon the refractive indices and is considered as Fresnel’s reflection probability function. The Pomading-Eddington method is used to calculate first the albedo s for problem 2 and then calculate the partial heat fluxes for problem 1. Results are obtained for isotropic scattering in a homogeneous and an inhomogeneous media for uniform and non-uniform internal sources and are compared with the published calculations.

10. Analytical solutions for the surface response to small amplitude perturbations in boundary data in the shallow-ice-stream approximation

G. H. Gudmundsson

2008-07-01

Full Text Available New analytical solutions describing the effects of small-amplitude perturbations in boundary data on flow in the shallow-ice-stream approximation are presented. These solutions are valid for a non-linear Weertman-type sliding law and for Newtonian ice rheology. Comparison is made with corresponding solutions of the shallow-ice-sheet approximation, and with solutions of the full Stokes equations. The shallow-ice-stream approximation is commonly used to describe large-scale ice stream flow over a weak bed, while the shallow-ice-sheet approximation forms the basis of most current large-scale ice sheet models. It is found that the shallow-ice-stream approximation overestimates the effects of bed topography perturbations on surface profile for wavelengths less than about 5 to 10 ice thicknesses, the exact number depending on values of surface slope and slip ratio. For high slip ratios, the shallow-ice-stream approximation gives a very simple description of the relationship between bed and surface topography, with the corresponding transfer amplitudes being close to unity for any given wavelength. The shallow-ice-stream estimates for the timescales that govern the transient response of ice streams to external perturbations are considerably more accurate than those based on the shallow-ice-sheet approximation. In particular, in contrast to the shallow-ice-sheet approximation, the shallow-ice-stream approximation correctly reproduces the short-wavelength limit of the kinematic phase speed given by solving a linearised version of the full Stokes system. In accordance with the full Stokes solutions, the shallow-ice-sheet approximation predicts surface fields to react weakly to spatial variations in basal slipperiness with wavelengths less than about 10 to 20 ice thicknesses.

11. Indirect boundary element method for three dimensional problems. Analytical solution for contribution to wave field by triangular element; Sanjigen kansetsu kyokai yosoho. Sankakukei yoso no kiyo no kaisekikai

Yokoi, T. [Building Research Institute, Tokyo (Japan); Sanchez-Sesma, F. [Universidad National Autonoma de Mexico, (Mexico). Institute de Ingenieria

1997-05-27

Formulation is introduced for discretizing a boundary integral equation into an indirect boundary element method for the solution of 3-dimensional topographic problems. Yokoi and Takenaka propose an analytical solution-capable reference solution (solution for the half space elastic body with flat free surface) to problems of topographic response to seismic motion in a 2-dimensional in-plane field. That is to say, they propose a boundary integral equation capable of effectively suppressing the non-physical waves that emerge in the result of computation in the wake of the truncation of the discretized ground surface making use of the wave field in a semi-infinite elastic body with flat free surface. They apply the proposed boundary integral equation discretized into the indirect boundary element method to solve some examples, and succeed in proving its validity. In this report, the equation is expanded to deal with 3-dimensional topographic problems. A problem of a P-wave vertically landing on a flat and free surface is solved by the conventional boundary integral equation and the proposed boundary integral equation, and the solutions are compared with each other. It is found that the new method, different from the conventional one, can delete non-physical waves from the analytical result. 4 figs.

12. Existence and Analytic Approximation of Solutions of Duffing Type Nonlinear Integro-Differential Equation with Integral Boundary Conditions

Alsaedi Ahmed

2009-01-01

Full Text Available A generalized quasilinearization technique is developed to obtain a sequence of approximate solutions converging monotonically and quadratically to a unique solution of a boundary value problem involving Duffing type nonlinear integro-differential equation with integral boundary conditions. The convergence of order for the sequence of iterates is also established. It is found that the work presented in this paper not only produces new results but also yields several old results in certain limits.

13. Analytical solution of buoyancy-driven flow and heat transfer in a vertical channel with spatially periodic boundary conditions

Gunes, Hasan [Department of Mechanical Engineering, Istanbul Technical University, Gumussuyu (Turkey)

2003-12-01

In this study, we derive analytical expressions describing the variation of field variables in steady, 2-D and 3-D natural convection in a vertical channel with discrete in-space, flush-mounted heat sources. The expressions are valid for sufficiently small Grasof numbers. The solution are governed by the following dimensionless parameters: aspect ratios defining the geometry of the problem, Prandtl number, Grashof number and dimensionless channel reference temperature. Test case solutions are obtained numerically to assess the accuracy of the derived expressions. For small values Gr, the derived expressions are in excellent agreement with the numerical solutions in the entire computational domain. Analytical expressions for the net volume flow rate through the channel and Nusselt number variation are also given. (orig.)

14. On the analytical solution of the S{sub N} equation in a rectangle assuming an exponential exiting angular flux boundary

Goncalez, Tifani T. [Universidade Federal do Rio Grande do Sul (PROMEC/UFRGS), Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica; Segatto, Cynthia F.; Vilhena, Marco Tullio, E-mail: csegatto@pq.cnpq.b, E-mail: vilhena@pq.cnpq.b [Universidade Federal do Rio Grande do Sul (DMPA/UFRGS), Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Matematica Aplicada

2011-07-01

In this work, we report an analytical solution for the set of S{sub N} equations for the angular flux, in a rectangle, using the double Laplace transform technique. Its main idea comprehends the steps: application of the Laplace transform in one space variable, solution of the resulting equation by the LTS{sub N} method and reconstruction of the double Laplace transformed angular flux using the inversion theorem of the Laplace transform. We must emphasize that we perform the Laplace inversion by the LTS{sub N} method in the x direction, meanwhile we evaluate the inversion in the y direction performing the calculation of the corresponding line integral solution by the Stefest method. We have also to figure out that the application of Laplace transform to this type of boundary value problem introduces additional unknown functions associated to the partial derivatives of the angular flux at boundary. Based on the good results attained by the nodal LTS{sub N} method, we assume that the angular flux at boundary is also approximated by an exponential function. By analytical we mean that no approximation is done along the solution derivation except for the exponential hypothesis for the exiting angular flux at boundary. For sake of completeness, we report numerical comparisons of the obtained results against the ones of the literature. (author)

15. Heat Generation/Absorption Effects in a Boundary Layer Stretched Flow of Maxwell Nanofluid: Analytic and Numeric Solutions.

Full Text Available Analysis has been done to investigate the heat generation/absorption effects in a steady flow of non-Newtonian nanofluid over a surface which is stretching linearly in its own plane. An upper convected Maxwell model (UCM has been utilized as the non-Newtonian fluid model in view of the fact that it can predict relaxation time phenomenon which the Newtonian model cannot. Behavior of the relaxations phenomenon has been presented in terms of Deborah number. Transport phenomenon with convective cooling process has been analyzed. Brownian motion "Db" and thermophoresis effects "Dt" occur in the transport equations. The momentum, energy and nanoparticle concentration profiles are examined with respect to the involved rheological parameters namely the Deborah number, source/sink parameter, the Brownian motion parameters, thermophoresis parameter and Biot number. Both numerical and analytic solutions are presented and found in nice agreement. Comparison with the published data is also made to ensure the validity. Stream lines for Maxwell and Newtonian fluid models are presented in the analysis.

16. Unified analytical solution for various boundary conditions for the coupled flexural-torsional vibration of beams subjected to axial loads and end moments

Joshi, A.; Suryanarayan, S.

1989-03-01

The problem of free vibration of beams having different end conditions and subjected to static initial loads has been studied with the aim of arriving at good closed-form analytical solutions. Elementary beam theory is used as a starting point to obtain the transverse vibration frequencies for various cases of classical homogeneous end conditions and for various values of the static axial load and end moment. These results indicate that it is possible to identify simple algebraic expressions which accurately represent the solution for various boundary conditions. It is also found that reasonably accurate estimates of the predominantly flexural frequency of coupled flexural-torsional vibration can be obtained from the uncoupled flexural vibration frequency of beam-columns. This is achieved by defining an effective axial load parameter, which is a combination of the axial load, the end moment and the slenderness parameter. Finally, the study also brings out that the various expressions, corresponding to different end conditions, can be combined together into a single expression for the predominantly flexural frequency. This expression is common for the boundary conditions considered here and use is made of various normalizing factors which depend on the boundary conditions, and are obtainable from the corresponding free vibration and stability analyses of beam-columns.

17. Analytical Solution of Melting in a Subcooled Semi-infinite Solid with Boundary Conditions of the Second Kind

1993-01-01

The melting problem in a semi-infinite region with constant heat flux boundary condition is solved by a semi-exact method and an integral approximate method .Effect of subcooling on the transient solid-liquid interface location and surface temperature are also discussed in this paper.

18. Comparison of Different Analytic Solutions to Axisymmetric Squeezing Fluid Flow between Two Infinite Parallel Plates with Slip Boundary Conditions

Hamid Khan

2012-01-01

Full Text Available We investigate squeezing flow between two large parallel plates by transforming the basic governing equations of the first grade fluid to an ordinary nonlinear differential equation using the stream functions ur(r,z,t=(1/r(∂ψ/∂z and uz(r,z,t=−(1/r(∂ψ/∂r and a transformation ψ(r,z=r2F(z. The velocity profiles are investigated through various analytical techniques like Adomian decomposition method, new iterative method, homotopy perturbation, optimal homotopy asymptotic method, and differential transform method.

19. Boundary-value problems for x-analytical functions with weighted boundary conditions

Kapshivyi, A.A. [Kiev Univ. (Ukraine)

1994-11-10

We consider boundary-value problems for x-analytical functions of a complex variable z = x + iy in a number of domains. Limit values with the weight (ln x){sup {minus}1} are given for the real part of the x-analytical function on the sections of the boundary that follow the imaginary axis, and simple limits are given for the real part of the x-analytical functions on the part of the boundary outside the imaginary axis. The apparatus of integral representations of x-analytical functions is applied to obtain a solution of the problem in quadratures.

20. Strongly nonlinear oscillators analytical solutions

Cveticanin, Livija

2014-01-01

This book provides the presentation of the motion of pure nonlinear oscillatory systems and various solution procedures which give the approximate solutions of the strong nonlinear oscillator equations. The book presents the original author’s method for the analytical solution procedure of the pure nonlinear oscillator system. After an introduction, the physical explanation of the pure nonlinearity and of the pure nonlinear oscillator is given. The analytical solution for free and forced vibrations of the one-degree-of-freedom strong nonlinear system with constant and time variable parameter is considered. Special attention is given to the one and two mass oscillatory systems with two-degrees-of-freedom. The criteria for the deterministic chaos in ideal and non-ideal pure nonlinear oscillators are derived analytically. The method for suppressing chaos is developed. Important problems are discussed in didactic exercises. The book is self-consistent and suitable as a textbook for students and also for profess...

1. Analytic anisotropic solution for holography

Ren, Jie

2016-01-01

An exact solution to Einstein's equations for holographic models is presented and studied. The IR geometry has a timelike cousin of the Kasner singularity, which is the less generic case of the BKL (Belinski-Khalatnikov-Lifshitz) singularity, and the UV is asymptotically AdS. This solution describes a holographic RG flow between them. The solution's appearance is an interpolation between the planar AdS black hole and the AdS soliton. The causality constraint is always satisfied. The boundary condition for the current-current correlation function and the Laplacian in the IR is examined in detail. There is no infalling wave in the IR, but instead, there is a normalizable solution in the IR. In a special case, a hyperscaling-violating geometry is obtained after a dimension reduction.

2. An analytical model for stress-induced grain growth in the presence of both second-phase particles and solute segregation at grain boundaries

A theoretical framework that incorporates the influence of second-phase particles and solute segregation at grain boundaries (GBs) on stress-induced GB migration and grain rotation is formulated in the present paper. In our work, we modified the well-established Cahn–Taylor model to account for the drag stresses generated by second-phase particles and by solute atoms segregated at GBs. The theoretical framework is then implemented to rationalize GB migration and grain rotation using experimental data from a previously published study on stress-induced grain growth in the presence of both second-phase particles and solute segregation at GBs. The calculated grain growth results are generally consistent with the experimental data, providing support to the proposed theoretical model, despite the various assumptions involved. Moreover, the influence of second-phase particles and solute segregation at GBs on GB migration and grain rotation was also investigated using the model, and our results suggest that both second-phase particles and solute atoms segregated at GBs reduce the velocities of GB migration and grain rotation as compared to those in the case of high-purity Al

3. Analytic prediction for planar turbulent boundary layers

Chen, Xi

2016-01-01

Analytic predictions of mean velocity profile (MVP) and streamwise ($x$) development of related integral quantities are presented for flows in channel and turbulent boundary layer (TBL), based on a symmetry analysis of eddy length and total stress. Specific predictions are the friction velocity $u_\\tau$: ${ U_e/u_\\tau }\\approx 2.22\\ln Re_x+2.86-3.83\\ln(\\ln Re_x)$; the boundary layer thickness $\\delta_e$: $x/\\delta_e \\approx 7.27\\ln Re_x-5.18-12.52\\ln(\\ln Re_x)$; the momentum thickness Reynolds number: $Re_x/Re_\\theta=4.94[{(\\ln {{\\mathop{\\rm Re}\ 4. Analyticity of thermoelastic plates with dynamical boundary conditions ZHANG; Qiong(张琼); HUANG; Falun(黄发伦) 2003-01-01 We consider a thermoelastic plate with dynamical boundary conditions. Using the contradictionargument of Pazy's well-known analyticity criterion and P.D.E. estimates, we prove that the corresponding C0semigroup is analytic, hence exponentially stable. 5. Analytical solutions for problems of bubble dynamics Recently, an asymptotic solution of the Rayleigh equation for an empty bubble in N dimensions has been obtained. Here we give the closed-form general analytical solution of this equation. We also find the general solution of the Rayleigh equation in N dimensions for the case of a gas-filled hyperspherical bubble. In addition, we include a surface tension into consideration. - Highlights: • The Rayleigh equation for bubble's dynamics is considered. • General analytical solutions of the Rayleigh equation are obtained. • Various types of analytical solutions of the Rayleigh equation are studied 6. Analytical solutions to flexural vibration of slender piezoelectric multilayer cantilevers The modeling of vibration of piezoelectric cantilevers has often been based on passive cantilevers of a homogeneous material. Although piezoelectric cantilevers and passive cantilevers share certain characteristics, this method has caused confusion in incorporating the piezoelectric moment into the differential equation of motion. The extended Hamilton’s principle is a fundamental approach to modeling flexural vibration of multilayer piezoelectric cantilevers. Previous works demonstrated derivation of the differential equation of motion using this approach; however, proper analytical solutions were not reported. This was partly due to the fact that the differential equation derived by the extended Hamilton’s principle is a boundary-value problem with nonhomogeneous boundary conditions which cannot be solved by modal analysis. In the present study, an analytical solution to the boundary-value problem was obtained by transforming it into a new problem with homogeneous boundary conditions. After the transformation, modal analysis was used to solve the new boundary-value problem. The analytical solutions for unimorphs and bimorphs were verified with three-dimensional finite element analysis (FEA). Deflection profiles and frequency response functions under voltage, uniform pressure and tip force were compared. Discrepancies between the analytical results and FEA results were within 3.5%. Following model validation, parametric studies were conducted to investigate the effects of thickness of electrodes and piezoelectric layers, and the piezoelectric coupling coefficient d 31 on the performance of piezoelectric cantilever actuators. (paper) 7. Analytical solution to one-dimensional consolidation in unsaturated soils QIN Ai-fang; CHEN Guang-jing; TAN Yong-wei; SUN Dean 2008-01-01 This paper presents an analytical solution of the one-dimensional consolidation in unsaturated soil with a finite thickness under vertical loading and confinements in the lateral directions. The boundary contains the top surface permeable to water and air and the bottom impermeable to water and air. The analytical solution is for Fredlund's one-dimensionai consolidation equation in unsaturated soils. The transfer relationship between the state vectors at top surface and any depth is obtained by using the Laplace transform and Cayley-Hamilton mathematical methods to the governing equations of water and air, Darcy's law and Fick's law. Excess pore-air pressure, excess pore-water pressure and settlement in the Laplace-transformed domain are obtained by using the Laplace transform with the initial conditions and boundary conditions. By performing inverse Laplace transforms, the analytical solutions are obtained in the time domain. A typical example illustrates the consolidation characteristics of unsaturated soft from analytical results. Finally, comparisons between the analytical solutions and results of the finite difference method indicate that the analytical solution is correct. 8. Transient solute drag in migrating grain boundaries Understanding the solute drag in migrating grain boundaries or interfaces has been a topic in materials research since Cahn's seminal paper in 1962. However, mostly steady-state solutions for solute segregation and drag in a migrating interface have been investigated. Here a new concept, based on the thermodynamic extremal principle, is introduced, which allows a detailed study of the transient processes in the migrating interface starting from a given initial configuration. The system is then described by two parameters, the first representing the amount of segregated solute in the grain boundary and the second the grain boundary position. Stability studies are performed using the perturbation concept. The model is demonstrated by simulations for a Fe-0.1 at.% Ni alloy taking different values for the grain boundary mobility and the driving force. 9. Analytical solutions for problems of bubble dynamics Kudryashov, Nikolai A 2016-01-01 Recently, an asymptotic solution of the Rayleigh equation for an empty bubble in$N$dimensions has been obtained. Here we give the closed--from general analytical solution of this equation. We also find the general solution of the Rayleigh equation in$N$dimensions for the case of a gas--filled hyperspherical bubble. In addition, we include a surface tension into consideration. 10. Unique solution to periodic boundary value problems Yong Sun 1991-01-01 Full Text Available Existence of unique solution to periodic boundary value problems of differential equations with continuous or discontinuous right-hand side is considered by utilizing the method of lower and upper solutions and the monotone properties of the operator. This is subject to discussion in the present paper. 11. The boundary value problem for discrete analytic functions Skopenkov, Mikhail 2013-06-01 This paper is on further development of discrete complex analysis introduced by R.Isaacs, J.Ferrand, R.Duffin, and C.Mercat. We consider a graph lying in the complex plane and having quadrilateral faces. A function on the vertices is called discrete analytic, if for each face the difference quotients along the two diagonals are equal.We prove that the Dirichlet boundary value problem for the real part of a discrete analytic function has a unique solution. In the case when each face has orthogonal diagonals we prove that this solution uniformly converges to a harmonic function in the scaling limit. This solves a problem of S.Smirnov from 2010. This was proved earlier by R.Courant-K.Friedrichs-H.Lewy and L.Lusternik for square lattices, by D.Chelkak-S.Smirnov and implicitly by P.G.Ciarlet-P.-A.Raviart for rhombic lattices.In particular, our result implies uniform convergence of the finite element method on Delaunay triangulations. This solves a problem of A.Bobenko from 2011. The methodology is based on energy estimates inspired by alternating-current network theory. © 2013 Elsevier Ltd. 12. Iterative solution of Hermite boundary integral equations Gray, Leonard J [ORNL; Nintcheu Fata, Sylvain [ORNL; Ma, Ding [ORNL 2008-01-01 An efficient iterative method for the solution of the linear equations arising from a Hermite boundary integral approximation has been developed. Along with equations for the boundary unknowns, the Hermite system incorporates equations for the first-order surface derivatives (gradient) of the potential, and is therefore substantially larger than the matrix for a corresponding linear approximation. However, by exploiting the structure of the Hermite matrix, a two-level iterative algorithm has been shown to provide a very efficient solution algorithm. In this approach, the boundary function unknowns are treated separately from the gradient, taking advantage of the sparsity and near-positive definiteness of the gradient equations. In test problems, the new algorithm significantly reduced computation time compared to iterative solution applied to the full matrix. This approach should prove to be even more effective for the larger systems encountered in three-dimensional analysis, and increased efficiency should come from pre-conditioning of the non-sparse matrix component. 13. Analytical solution methods for geodesic motion Hackmann, Eva 2015-01-01 The observation of the motion of particles and light near a gravitating object is until now the only way to explore and to measure the gravitational field. In the case of exact black hole solutions of the Einstein equations the gravitational field is characterized by a small number of parameters which can be read off from the observables related to the orbits of test particles and light rays. Here we review the state of the art of analytical solutions of geodesic equations in various space--times. In particular we consider the four dimensional black hole space--times of Pleba\\'nski--Demia\\'nski type as far as the geodesic equation separates, as well as solutions in higher dimensions, and also solutions with cosmic strings. The mathematical tools used are elliptic and hyperelliptic functions. We present a list of analytic solutions which can be found in the literature. 14. Analytic vortex solutions on compact hyperbolic surfaces Maldonado, Rafael; Manton, Nicholas S. 2015-06-01 We construct, for the first time, abelian Higgs vortices on certain compact surfaces of constant negative curvature. Such surfaces are represented by a tessellation of the hyperbolic plane by regular polygons. The Higgs field is given implicitly in terms of Schwarz triangle functions and analytic solutions are available for certain highly symmetric configurations. 15. Analytic vortex solutions on compact hyperbolic surfaces We construct, for the first time, abelian Higgs vortices on certain compact surfaces of constant negative curvature. Such surfaces are represented by a tessellation of the hyperbolic plane by regular polygons. The Higgs field is given implicitly in terms of Schwarz triangle functions and analytic solutions are available for certain highly symmetric configurations. (paper) 16. Analytic vortex solutions on compact hyperbolic surfaces Maldonado, R 2015-01-01 We construct, for the first time, Abelian-Higgs vortices on certain compact surfaces of constant negative curvature. Such surfaces are represented by a tessellation of the hyperbolic plane by regular polygons. The Higgs field is given implicitly in terms of Schwarz triangle functions and analytic solutions are available for certain highly symmetric configurations. 17. Analytic Solutions of Elastic Tunneling Problems Strack, O.E. 2002-01-01 The complex variable method for solving two dimensional linearly elastic problems is used to obtain several fundamental analytical solutions of tunneling problems. The method is used to derive the general mathematical representation of problems involving resultant forces on holes in a half-plane 18. Analytic solutions of an unclassified artifact / Trent, Bruce C. 2012-03-01 This report provides the technical detail for analytic solutions for the inner and outer profiles of the unclassified CMM Test Artifact (LANL Part Number 157Y-700373, 5/03/2001) in terms of radius and polar angle. Furthermore, analytic solutions are derived for the legacy Sheffield measurement hardware, also in terms of radius and polar angle, using part coordinates, i.e., relative to the analytic profile solutions obtained. The purpose of this work is to determine the exact solution for the “cosine correction” term inherent to measurement with the Sheffield hardware. The cosine correction is required in order to interpret the actual measurements taken by the hardware in terms of an actual part definition, or “knot-point spline definition,” that typically accompanies a component drawing. Specifically, there are two portions of the problem: first an analytic solution must be obtained for any point on the part, e.g., given the radii and the straight lines that define the part, it is required to find an exact solution for the inner and outer profile for any arbitrary polar angle. Next, the problem of the inspection of this part must be solved, i.e., given an arbitrary sphere (representing the inspection hardware) that comes in contact with the part (inner and outer profiles) at any arbitrary polar angle, it is required to determine the exact location of that intersection. This is trivial for the case of concentric circles. In the present case, however, the spherical portion of the profiles is offset from the defined center of the part, making the analysis nontrivial. Here, a simultaneous solution of the part profiles and the sphere was obtained. 19. Can We Remove Secular Terms for Analytical Solution of Groundwater Response under Tidal Influence? Munusamy, Selva Balaji 2016-01-01 This paper presents a secular term removal methodology based on the homotopy perturbation method for analytical solutions of nonlinear problems with periodic boundary condition. The analytical solution for groundwater response to tidal fluctuation in a coastal unconfined aquifer system with the vertical beach is provided as an example. The non-linear one-dimensional Boussinesq's equation is considered as the governing equation for the groundwater flow. An analytical solution is provided for non-dimensional Boussinesq's equation with cosine harmonic boundary condition representing tidal boundary condition. The analytical solution is obtained by using homotopy perturbation method with a virtual embedding parameter. The present approach does not require pre-specified perturbation parameter and also facilitates secular terms elimination in the perturbation solution. The solutions starting from zeroth-order up to third-order are obtained. The non-dimensional expression,$A/D_{\\infty}$emerges as an implicit parame... 20. Non-perturbative solution of free-convective boundary-layer equation by Adomian decomposition method Kechil, Seripah Awang [Department of Mathematics, Universiti Teknologi MARA, 40450 Shah Alam Selangor (Malaysia); Hashim, Ishak [School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi Selangor (Malaysia)]. E-mail: ishak_h@ukm.my 2007-03-19 A free-convective boundary layer flow modeled by a system of nonlinear ordinary differential equations is considered. The system is solved using the Adomian decomposition method (ADM) which yields an analytic solution in the form of a rapidly convergent infinite series with easily computable terms. The analytical solutions and the pertinent features of the illustrations show the efficiency of the method. 1. Analytic Solutions of Elastic Tunneling Problems Strack, O.E. 2002-01-01 The complex variable method for solving two dimensional linearly elastic problems is used to obtain several fundamental analytical solutions of tunneling problems. The method is used to derive the general mathematical representation of problems involving resultant forces on holes in a half-plane. Such problems are encountered in geomechanics during the excavation of tunnels. When tunnels are excavated the removal of the weighted material inside the tunnel causes the ground under the tunnel to... 2. Exact Solutions of Chemically Reactive Solute Distribution in MHD Boundary Layer Flow over a Shrinking Surface An analytical study of the distribution of a reactant solute undergoing a first-order chemical reaction in the boundary layer flow of an electrically conducting incompressible fluid over a linearly shrinking surface is presented. The flow is permeated by an externally applied magnetic field normal to the plane of the flow. The equations governing the flow and concentration field are reduced into a set of nonlinear ordinary differential equations using similarity variables. Closed form exact solutions of the reduced concentration equation are obtained for both prescribed power-law surface concentration (PSC) and power-law wall mass flux (PMF) as boundary conditions. The study reveals that the concentration over a shrinking sheet is significantly different from that of a stretching surface. It is found that the solute boundary layer thickness is enhanced with the increasing values of the Schmidt number and the power-law index parameter, but decreases with enhanced values of magnetic and reaction rate parameters for the PSC case. For the PMF case, the solute boundary layer thickness decreases with the increase of the Schmidt number, magnetic and reaction rate parameter for power-law index parameter n = 0. Negative solute boundary layer thickness is observed for the PMF case when n = 1 and 2, and these facts may not be realized in real-world applications. (fundamental areas of phenomenology(including applications)) 3. Large deflection of clamped circular plate and accuracy of its approximate analytical solutions Zhang, Yin 2016-02-01 A different set of governing equations on the large deflection of plates are derived by the principle of virtual work (PVW), which also leads to a different set of boundary conditions. Boundary conditions play an important role in determining the computation accuracy of the large deflection of plates. Our boundary conditions are shown to be more appropriate by analyzing their difference with the previous ones. The accuracy of approximate analytical solutions is important to the bulge/blister tests and the application of various sensors with the plate structure. Different approximate analytical solutions are presented and their accuracies are evaluated by comparing them with the numerical results. The error sources are also analyzed. A new approximate analytical solution is proposed and shown to have a better approximation. The approximate analytical solution offers a much simpler and more direct framework to study the plate-membrane transition behavior of deflection as compared with the previous approaches of complex numerical integration. 4. Approximate Analytic Solutions for the Two-Phase Stefan Problem Using the Adomian Decomposition Method Xiao-Ying Qin 2014-01-01 Full Text Available An Adomian decomposition method (ADM is applied to solve a two-phase Stefan problem that describes the pure metal solidification process. In contrast to traditional analytical methods, ADM avoids complex mathematical derivations and does not require coordinate transformation for elimination of the unknown moving boundary. Based on polynomial approximations for some known and unknown boundary functions, approximate analytic solutions for the model with undetermined coefficients are obtained using ADM. Substitution of these expressions into other equations and boundary conditions of the model generates some function identities with the undetermined coefficients. By determining these coefficients, approximate analytic solutions for the model are obtained. A concrete example of the solution shows that this method can easily be implemented in MATLAB and has a fast convergence rate. This is an efficient method for finding approximate analytic solutions for the Stefan and the inverse Stefan problems. 5. Comparison between analytical and numerical solution of mathematical drying model Shahari, N.; Rasmani, K.; Jamil, N. 2016-02-01 Drying is often related to the food industry as a process of shifting heat and mass inside food, which helps in preserving food. Previous research using a mass transfer equation showed that the results were mostly concerned with the comparison between the simulation model and the experimental data. In this paper, the finite difference method was used to solve a mass equation during drying using different kinds of boundary condition, which are equilibrium and convective boundary conditions. The results of these two models provide a comparison between the analytical and the numerical solution. The result shows a close match between the two solution curves. It is concluded that the two proposed models produce an accurate solution to describe the moisture distribution content during the drying process. This analysis indicates that we have confidence in the behaviour of moisture in the numerical simulation. This result demonstrated that a combined analytical and numerical approach prove that the system is behaving physically. Based on this assumption, the model of mass transfer was extended to include the temperature transfer, and the result shows a similar trend to those presented in the simpler case. 6. Analytical solutions for anomalous dispersion transport O'Malley, D.; Vesselinov, V. V. 2014-06-01 Groundwater flow and transport often occur in a highly heterogeneous environment (potentially heterogeneous at multiple spatial scales) and is impacted by geochemical reactions, advection, diffusion, and other pore scale processes. All these factors can give rise to large-scale anomalous dispersive behavior that can make complex model representation and prediction of plume concentrations challenging due to difficulties unraveling all the complexities associated with the governing processes, flow medium, and their parameters. An alternative is to use upscaled stochastic models of anomalous dispersion, and this is the approach used here. Within a probabilistic framework, we derive a number of analytical solutions for several anomalous dispersion models. The anomalous dispersion models are allowed to be either non-Gaussian (α-stable Lévy), correlated, or nonstationary from the Lagrangian perspective. A global sensitivity analysis is performed to gain a greater understanding of the extent to which uncertainty in the parameters associated with the anomalous behavior can be narrowed by examining concentration measurements from a network of monitoring wells and to demonstrate the computational speed of the solutions. The developed analytical solutions are encoded and available for use in the open source computational framework MADS (http://mads.lanl.gov). 7. Application of a two energy group analytical solution to the Yalina experiment SC3A The SC3A experiment in the YALINA-Booster facility is described and investigated. For this investigation the very special configuration of YALINA-Booster is analyzed based on HELIOS calculations. To improve the representation to this special configuration a new analytical solution for two energy groups with two sources (central external and boundary source) has been developed starting form the Green's function solution. Very good agreement has been found for these improved analytical solutions. (author) 8. Polynomial and Analytic Boundary Feedback Stabilization of Square Plate Salsabil Nouira 2009-11-01 Full Text Available We consider a boundary feedback stabilization problem of the plateequation in a square, in the case where the geometric condition of Ammari-Tucsnak [6] is not satisﬁed. We prove a polynomial decay for regular initial data. Moreover, we prove an exponential stability result for some subspace of the energy space. Finally, we give a precise estimate on the analyticity of reachable functions wherewe have an exponential stability. 9. Analytical Solutions for Sequentially Reactive Transport with Different Retardation Factors Sun, Y; Buscheck, T A; Mansoor, K; Lu, X 2001-08-01 Integral transforms have been widely used for deriving analytical solutions for solute transport systems. Often, analytical solutions can only be written in closed form in frequency domains and numerical inverse-transforms have to be involved to obtain semi-analytical solutions in the time domain. For this reason, previously published closed form solutions are restricted either to a small number of species or to the same retardation assumption. In this paper, we applied the solution scheme proposed by Bauer et al. in the time domain. Using available analytical solutions of a single species transport with first-order decay without coupling with its parent species concentration as fundamental solutions, a daughter species concentration can be expressed as a linear function of those fundamental solutions. The implementation of the solution scheme is straight forward and exact analytical solutions are derived for one- and three-dimensional transport systems. 10. Analytical Solution to the MHD Flow of Micropolar Fluid Over a Linear Stretching Sheet Siddheshwar P.G. 2015-05-01 Full Text Available The flow due to a linear stretching sheet in a fluid with suspended particles, modeled as a micropolar fluid, is considered. All reported works on the problem use numerical methods of solution or a regular perturbation technique. An analytical solution is presented in the paper for the coupled non-linear differential equations with inhomogeneous boundary conditions. 11. Analytical Solution to the MHD Flow of Micropolar Fluid Over a Linear Stretching Sheet Siddheshwar P.G.; Mahabaleshwar U.S. 2015-01-01 The flow due to a linear stretching sheet in a fluid with suspended particles, modeled as a micropolar fluid, is considered. All reported works on the problem use numerical methods of solution or a regular perturbation technique. An analytical solution is presented in the paper for the coupled non-linear differential equations with inhomogeneous boundary conditions. 12. Analytical Solution to the MHD Flow of Micropolar Fluid Over a Linear Stretching Sheet Siddheshwar, P. G.; Mahabaleshwar, U. S. 2015-05-01 The flow due to a linear stretching sheet in a fluid with suspended particles, modeled as a micropolar fluid, is considered. All reported works on the problem use numerical methods of solution or a regular perturbation technique. An analytical solution is presented in the paper for the coupled non-linear differential equations with inhomogeneous boundary conditions. 13. Analytical Solution of Multicompartment Solute Kinetics for Hemodialysis Przemysław Korohoda 2013-01-01 Full Text Available Objective. To provide an exact solution for variable-volume multicompartment kinetic models with linear volume change, and to apply this solution to a 4-compartment diffusion-adjusted regional blood flow model for both urea and creatinine kinetics in hemodialysis. Methods. A matrix-based approach applicable to linear models encompassing any number of compartments is presented. The procedure requires the inversion of a square matrix and the computation of its eigenvalues λ, assuming they are all distinct. This novel approach bypasses the evaluation of the definite integral to solve the inhomogeneous ordinary differential equation. Results. For urea two out of four eigenvalues describing the changes of concentrations in time are about 105 times larger than the other eigenvalues indicating that the 4-compartment model essentially reduces to the 2-compartment regional blood flow model. In case of creatinine, however, the distribution of eigenvalues is more balanced (a factor of 102 between the largest and the smallest eigenvalue indicating that all four compartments contribute to creatinine kinetics in hemodialysis. Interpretation. Apart from providing an exact analytic solution for practical applications such as the identification of relevant model and treatment parameters, the matrix-based approach reveals characteristic details on model symmetry and complexity for different solutes. 14. ANALYTIC SOLUTIONS OF SYSTEMS OF FUNCTIONAL EQUATIONS LiuXinhe 2003-01-01 Let r be a given positive number.Denote by D=D the closed disc in the complex plane C whose center is the origin and radius is r.For any subset K of C and any integer m ≥1,write A(Dm,K)={f|f:Dm→Kis a continuous map,and f|(Dm)*is analytic).For H∈A(Dm,C)(m≥2),f∈A(D,D)and z∈D,write ψH(f)(z)=H(z,f(z)……fm=1(x)).Suppose F,G∈A(D2n+1,C),and Hk,Kk∈A(Dk,C),k=2,……,n.In this paper,the system of functional equations {F(z,f(z),f2(ψHz(f)(z))…,fn(ψk2(g)(x))… gn(ψKn(g)(z)))=0 G(z,f(z),f2(ψH2(f)(z))…fn(ψHn(f)(z)),g(z),g2(ψk2(g)(x))…,gn(ψkn(g)(z)))=0(z∈D)is studied and some conditions for the system of equations to have a solution or a unique solution in A(D,D)×A（D，D）are given. 15. Exact Solutions of Chemically Reactive Solute Distribution in MHD Boundary Layer Flow over a Shrinking Surface Chandaneswar Midya 2012-01-01 An analytical study of the distribution of a reactant solute undergoing a first-order chemical reaction in the boundary layer flow of an electrically conducting incompressible Buid over a linearly shrinking surface is presented. The Row is permeated by an externally applied magnetic Geld normal to the plane of the flow. The equations governing the Row and concentration Reid are reduced into a set of nonlinear ordinary differential equations using similarity variables. Closed form exact solutions of the reduced concentration equation are obtained for both prescribed power-law surface concentration (PSC) and power-law wall mass flux (PMF) as boundary conditions. The study reveals that the concentration over a shrinking sheet is signiRcantly different from that of a stretching surface. It s found that te solute boundary layer thickness is enhanced with the increasing values of the Schmidt number and the power-law index parameter, but decreases with enhanced vaJues of magnetic and reaction rate parameters for the PSC case. For the PMF case, the solute boundary layer thickness decreases with the increase of the Schmidt number, magnetic and reaction rate parameter for power-law index parameter n = 0. Negative solute boundary layer thickness is observed for the PMF case when n = 1 and 2, and these facts may not be realized in real-world applications.%An analytical study of the distribution of a reactant solute undergoing a first-order chemical reaction in the boundary layer flow of an electrically conducting incompressible fluid over a linearly shrinking surface is presented.The flow is permeated by an externally applied magnetic field normal to the plane of the flow.The equations governing the flow and concentration field are reduced into a set of nonlinear ordinary differential equations using similarity variables.Closed form exact solutions of the reduced concentration equation are obtained for both prescribed power-law surface concentration (PSC) and power-law wall 16. ANALYTICAL SOLUTION OF GROUNDWATER FLUCTUATIONS IN ESTUARINE AQUIFER CHEN Jing; ZHOU Zhi-fang; JIA Suo-bao 2005-01-01 As a basic factor in the environment of estuary, tidal effects in the coastal aquifer have recently attracted much attention because tidal dynamic also greatly influences the solute transport in the coastal aquifer. Previous studies on tidal dynamic of coastal aquifers have focused on the inland propagation of oceanic tides in the cross-shore direction, a configuration that is essentially one-dimensional. Two-dimensional analytical solutions for groundwater level fluctuation in recent papers are localized in presenting the effect of both oceanic tides and estuarine tides in quadrantal aquifer. A two-dimensional model of groundwater fluctuations in estuarine zone in proposed in this paper. Using complex transform, the two-dimensional flow equation subject to periodic boundary condition is changed into time-independent elliptic problem. Based on Green function method, an analytical solution for groundwater fluctuations in fan-shaped aquifer is derived. The response to of groundwater tidal loading in an estuary and ocean is discussed. The result show that its more extensive application than recent studies. 17. Transient Solute Drag in Migrating Grain Boundaries Svoboda, Jiří; Fischer, F. D.; Liendl, M. 2011-01-01 Roč. 59, č. 17 (2011), s. 6556-6562. ISSN 1359-6454 R&D Projects: GA MŠk(CZ) OC10029 Institutional research plan: CEZ:AV0Z20410507 Keywords : Grain boundary diffusion * Grain boundary migration * Grain boundary segregation Subject RIV: BJ - Thermodynamics Impact factor: 3.755, year: 2011 18. Solution of moving boundary problems with implicit boundary condition An algorithm that solves numerically a model for studying one dimensional moving boundary problems, with implicit boundary condition, is described. Landau's transformation is used, in order to work with a fixed number of nodes at each instant. Then, it is necessary to deal with a parabolic partial differential equation, whose diffusive and convective terms have variable coefficients. The partial differential equation is implicitly discretized, using Laasonen's scheme, always stable, instead of employing Crank-Nicholson sheme, as it has been done by Ferris and Hill. Fixed time and space steps (Δt, Δξ) are used, and the iteration is made with variable positions of the interface, i.e. varying δs until a boundary condition is satisfied. The model has the same features of the oxygen diffusion in absorbing tissue. It would be capable of estimating time variant radiation treatments of cancerous tumors. (Author) 19. Substantive provisions of Numeral-analytical boundary elements method V.F. Orobey 2011-06-01 Full Text Available Substantive propositions of the new method of design calculation, that got the name "Numeral-analytical of boundary elements method", offered by authors, are brought. A method consists of development of the fundamental system of decisions (analytically and Green functions (also analytically for every examined task.For the account of certain border terms, or terms of contact between the separate modules (the separate element of the system is so named the small system of linear algebraic equalizations, that must be decided numeral, is made.Discretisation only of border of the area occupied by an object, sharply diminishes the order of the system of resolvent equalizations; there is possibility of decline of regularity of the decided task. A method is strictly reasonable mathematically, as uses the fundamental decisions of differential equalizations, and, means, within the framework of the accepted hypotheses allows to get the exact meaning of parameters of task (efforts, moving, tensions, currents, frequencies of eigentones, critical forces of loss of stability et cetera into an area.Simplicity of logic of algorithm, good convergence of decision, high stability and small accumulation of errors at numeral operations, are marked also. 20. Analytically-derived sensitivities in one-dimensional models of solute transport in porous media Knopman, D.S. 1987-01-01 Analytically-derived sensitivities are presented for parameters in one-dimensional models of solute transport in porous media. Sensitivities were derived by direct differentiation of closed form solutions for each of the odel, and by a time integral method for two of the models. Models are based on the advection-dispersion equation and include adsorption and first-order chemical decay. Boundary conditions considered are: a constant step input of solute, constant flux input of solute, and exponentially decaying input of solute at the upstream boundary. A zero flux is assumed at the downstream boundary. Initial conditions include a constant and spatially varying distribution of solute. One model simulates the mixing of solute in an observation well from individual layers in a multilayer aquifer system. Computer programs produce output files compatible with graphics software in which sensitivities are plotted as a function of either time or space. (USGS) 1. Analytic solutions of a class of nonlinearly dynamic systems Wang, M-C [System Engineering Institute of Tianjin University, Tianjin, 300072 (China); Zhao, X-S; Liu, X [Tianjin University of Technology and Education, Tianjin, 300222 (China)], E-mail: mchwang123@163.com.cn, E-mail: xszhao@mail.nwpu.edu.cn, E-mail: liuxinhubei@163.com.cn 2008-02-15 In this paper, the homotopy perturbation method (HPM) is applied to solve a coupled system of two nonlinear differential with first-order similar model of Lotka-Volterra and a Bratus equation with a source term. The analytic approximate solutions are derived. Furthermore, the analytic approximate solutions obtained by the HPM with the exact solutions reveals that the present method works efficiently. 2. Approximate analytic solutions to 3D unconfined groundwater flow within regional 2D models Luther, K.; Haitjema, H. M. 2000-04-01 We present methods for finding approximate analytic solutions to three-dimensional (3D) unconfined steady state groundwater flow near partially penetrating and horizontal wells, and for combining those solutions with regional two-dimensional (2D) models. The 3D solutions use distributed singularities (analytic elements) to enforce boundary conditions on the phreatic surface and seepage faces at vertical wells, and to maintain fixed-head boundary conditions, obtained from the 2D model, at the perimeter of the 3D model. The approximate 3D solutions are analytic (continuous and differentiable) everywhere, including on the phreatic surface itself. While continuity of flow is satisfied exactly in the infinite 3D flow domain, water balance errors can occur across the phreatic surface. 3. Problem of the Moving Boundary in Continuous Casting Solved by the Analytic-Numerical Method R. Grzymkowski 2013-01-01 Full Text Available Mathematical modeling of thermal processes combined with the reversible phase transitions of type: solid phase – liquid phase leads to formulation of the parabolic or elliptic moving boundary problem. Solution of such defined problem requires, most often, to use some sophisticated numerical techniques and far advanced mathematical tools. The paper presents an analytic-numerical method, especially attractive from the engineer’s point of view, applied for finding the approximate solutions of the selected class of problems which can be reduced to the one-phase solidification problem of a plate with the unknown a priori, varying in time boundary of the region in which the solution is sought. Proposed method is based on the known formalism of initial expansion of a sought function, describing the field of temperature, into the power series, some coefficients of which are determined with the aid of boundary conditions, and on the approximation of a function defining the freezing front location with the broken line, parameters of which are determined numerically. The method represents a combination of the analytical and numerical techniques and seems to be an effective and relatively easy in using tool for solving problems of considered kind. 4. Stable and chaotic solutions of the complex Ginzburg-Landau equation with periodic boundary conditions Scheuer, Jacob; Malomed, Boris A. 2001-01-01 We study, analytically and numerically, the dynamical behavior of the solutions of the complex Ginzburg-Landau equation with diffraction but without diffusion, which governs the spatial evolution of the field in an active nonlinear laser cavity. Accordingly, the solutions are subject to periodic boundary conditions. The analysis reveals regions of stable stationary solutions in the model’s parameter space, and a wide range of oscillatory and chaotic behaviors. Close to the first bifurca... 5. Analytical Solution for Stellar Density in Globular Clusters M. A. Sharaf; A. M. Sendi 2011-09-01 In this paper, four parameters analytical solution will be established for the stellar density function in globular clusters. The solution could be used for any arbitrary order of outward decrease of the cluster’s density. 6. Analyticity of solutions for randomly forced two-dimensional Navier-Stokes equations A study is made of randomly forced two-dimensional Navier-Stokes equations with periodic boundary conditions. Under the assumption that the random forcing is analytic in the spatial variables and is a white noise in the time, it is proved that a large class of solutions, which contains all stationary solutions with finite energy, admits analytic continuation to a small complex neighbourhood of the torus. Moreover, a lower bound is obtained for the radius of analyticity in terms of the viscosity ν, and it is shown that the Kolmogorov dissipation scale can be asymptotically estimated below by ν2+δ for any δ>0 7. Analytical Solution of Flow and Heat Transfer over a Permeable Stretching Wall in a Porous Medium M. Dayyan; Seyyedi, S. M.; G. G. Domairry; M. Gorji Bandpy 2013-01-01 Boundary layer flow through a porous medium over a stretching porous wall has seen solved with analytical solution. It has been considered two wall boundary conditions which are power-law distribution of either wall temperature or heat flux. These are general enough to cover the isothermal and isoflux cases. In addition to momentum, both first and second laws of thermodynamics analyses of the problem are investigated. The governing equations are transformed into a system of ordinary differen... 8. Analytical chemistry: Sweet solution to sensing Sia, Samuel K.; Chin, Curtis D. 2011-09-01 Glucose meters allow rapid and quantitative measurement of blood sugar levels for diabetes sufferers worldwide. Now a new method allows this proven technology to be used to quantify a much wider range of analytes. 9. Atomically ordered solute segregation behaviour in an oxide grain boundary Feng, Bin; Yokoi, Tatsuya; Kumamoto, Akihito; Yoshiya, Masato; Ikuhara, Yuichi; Shibata, Naoya 2016-01-01 Grain boundary segregation is a critical issue in materials science because it determines the properties of individual grain boundaries and thus governs the macroscopic properties of materials. Recent progress in electron microscopy has greatly improved our understanding of grain boundary segregation phenomena down to atomistic dimensions, but solute segregation is still extremely challenging to experimentally identify at the atomic scale. Here, we report direct observations of atomic-scale yttrium solute segregation behaviours in an yttria-stabilized-zirconia grain boundary using atomic-resolution energy-dispersive X-ray spectroscopy analysis. We found that yttrium solute atoms preferentially segregate to specific atomic sites at the core of the grain boundary, forming a unique chemically-ordered structure across the grain boundary. PMID:27004614 10. Atomically ordered solute segregation behaviour in an oxide grain boundary Feng, Bin; Yokoi, Tatsuya; Kumamoto, Akihito; Yoshiya, Masato; Ikuhara, Yuichi; Shibata, Naoya 2016-03-01 Grain boundary segregation is a critical issue in materials science because it determines the properties of individual grain boundaries and thus governs the macroscopic properties of materials. Recent progress in electron microscopy has greatly improved our understanding of grain boundary segregation phenomena down to atomistic dimensions, but solute segregation is still extremely challenging to experimentally identify at the atomic scale. Here, we report direct observations of atomic-scale yttrium solute segregation behaviours in an yttria-stabilized-zirconia grain boundary using atomic-resolution energy-dispersive X-ray spectroscopy analysis. We found that yttrium solute atoms preferentially segregate to specific atomic sites at the core of the grain boundary, forming a unique chemically-ordered structure across the grain boundary. 11. Approximate analytical solution of MHD flow of an Oldroyd 8-constant fluid in a porous medium Faisal Salah 2014-12-01 Full Text Available The steady flow in an incompressible, magnetohydrodynamic (MHD Oldroyd 8-constant fluid in a porous medium with the motion of an infinite plate is investigated. Using modified Darcy’s law of an Oldroyd 8-constant fluid, the equations governing the flow are modelled. The resulting nonlinear boundary value problem is solved using the homotopy analysis method (HAM. The obtained approximate analytical solutions clearly satisfy the governing nonlinear equations and all the imposed initial and boundary conditions. The convergence of the HAM solutions for different orders of approximation is demonstrated. For the Newtonian case, the approximate analytical solution via HAM is shown to be in close agreement with the exact solution. Finally, the variations of velocity field with respect to the magnetic field, porosity and non-Newtonian fluid parameters are graphically shown and discussed. 12. Analytical model for intergrain expansion and cleavage: random grain boundaries A description of rigid-body grain boundary relaxation and cleavage in tungsten is performed using a pair-wise Morse interatomic potential in real and reciprocal spaces. Cleavage energies and grain boundary dilatation of random grain boundaries were formulated and computed using atomic layer interaction energies. These values were determined using a model for a relaxed random grain boundary that consists of rigid grains on either side of the boundary plane that are allowed to float to reach the equilibrium position. Expressions are given that describe in real space the energy of interatomic interaction on random grain boundaries with twist orientation. It was shown that grain-boundary expansion and cleavage energies of the most widespread random grain boundaries are mainly determined by grain boundary atomic density 13. Analytical solution and computer program (FAST) to estimate fluid fluxes from subsurface temperature profiles Kurylyk, Barret L.; Irvine, Dylan J. 2016-02-01 This study details the derivation and application of a new analytical solution to the one-dimensional, transient conduction-advection equation that is applied to trace vertical subsurface fluid fluxes. The solution employs a flexible initial condition that allows for nonlinear temperature-depth profiles, providing a key improvement over most previous solutions. The boundary condition is composed of any number of superimposed step changes in surface temperature, and thus it accommodates intermittent warming and cooling periods due to long-term changes in climate or land cover. The solution is verified using an established numerical model of coupled groundwater flow and heat transport. A new computer program FAST (Flexible Analytical Solution using Temperature) is also presented to facilitate the inversion of this analytical solution to estimate vertical groundwater flow. The program requires surface temperature history (which can be estimated from historic climate data), subsurface thermal properties, a present-day temperature-depth profile, and reasonable initial conditions. FAST is written in the Python computing language and can be run using a free graphical user interface. Herein, we demonstrate the utility of the analytical solution and FAST using measured subsurface temperature and climate data from the Sendia Plain, Japan. Results from these illustrative examples highlight the influence of the chosen initial and boundary conditions on estimated vertical flow rates. 14. Analytical solutions to dissolved contaminant plume evolution with source depletion during carbon dioxide storage Yang, Yong; Liu, Yongzhong; Yu, Bo; Ding, Tian 2016-06-01 Volatile contaminants may migrate with carbon dioxide (CO2) injection or leakage in subsurface formations, which leads to the risk of the CO2 storage and the ecological environment. This study aims to develop an analytical model that could predict the contaminant migration process induced by CO2 storage. The analytical model with two moving boundaries is obtained through the simplification of the fully coupled model for the CO2-aqueous phase -stagnant phase displacement system. The analytical solutions are confirmed and assessed through the comparison with the numerical simulations of the fully coupled model. Then, some key variables in the analytical solutions, including the critical time, the locations of the dual moving boundaries and the advance velocity, are discussed to present the characteristics of contaminant migration in the multi-phase displacement system. The results show that these key variables are determined by four dimensionless numbers, Pe, RD, Sh and RF, which represent the effects of the convection, the dispersion, the interphase mass transfer and the retention factor of contaminant, respectively. The proposed analytical solutions could be used for tracking the migration of the injected CO2 and the contaminants in subsurface formations, and also provide an analytical tool for other solute transport in multi-phase displacement system. 15. Series Solution for Unsteady Boundary-Layer Flows Due to Impulsively Stretching Plate Seripah Awang Kechil; Ishak Hashim 2007-01-01 @@ The third-order nonlinear partial differential equation modelling the unsteady boundary-layer flows caused by an impulsively stretching flat plate is solved by using the Adomian decomposition method (ADM). The ADM yields analytic solution in the form of a rapidly convergent infinite series with easily computable terms. The series solution using the ADM for the unsteady flows is presented for the first time. 16. Positive Solutions for Boundary Value Problems with Fractional Order Mouffak Benchohra 2013-02-01 Full Text Available In this paper we investigate the existence of at least one, two positive solutions by using the Krasnoselskii fixed-point theorem in cones for nonlinear boundary value problem with fractional order. 17. Positive Solutions for Higher Order Singular -Laplacian Boundary Value Problems Guoliang Shi; Junhong Zhang 2008-05-01 This paper investigates$2m-\\mathrm{th}(m≥ 2)$order singular -Laplacian boundary value problems, and obtains the necessary and sufficient conditions for existence of positive solutions for sublinear 2-th order singular -Laplacian BVPs on closed interval. 18. Fundamental solutions for transversely isotropic magneto-electro-elastic media and boundary integral formulation 丁皓江; 江爱民 2003-01-01 To obtain the fundamental solutions for computation of magneto-electro-elastic media by the boundary element method, the general solutions in the case of distinct eigenvalues are derived and expressed in five harmonic functions from the governing equations and the strict differential operator theorem. On the basis of these general solutions, the fundamental solution of infinite magneto-electro-elastic solid are obtained with the method of trial-and-error. Finally, the boundary integral formulation is derived and the corresponding boundary element method program is implemented to perform two numerical calculations(a column under uni-axial tension, uniform electric displacement or uniform magnetic induction, an annular plate simply-supported on outer and inner surfaces under axial loads). The numerical results agree well with the analytical ones. 19. Analytical solutions of coupled-mode equations for microring resonators ZHAO C Y 2016-06-01 We present a study on analytical solutions of coupled-mode equations for microring resonators with an emphasis on occurrence of all-optical EIT phenomenon, obtained by using a cofactor. As concrete examples, analytical solutions for a$3 \\times 3$linearly distributed coupler and a circularly distributed coupler are obtained. The former corresponds to a non-degenerate eigenvalue problem and the latter corresponds to a degenerate eigenvalue problem. For comparison and without loss of generality, analytical solution for a$4 \\times 4$linearly distributed coupler is also obtained. This paper may be of interest to optical physics and integrated photonics communities. 20. Approximate analytical solution of two-dimensional multigroup P-3 equations Iterative solution of multigroup spherical harmonics equations reduces, in the P-3 approximation and in two-dimensional geometry, to a problem of solving an inhomogeneous system of eight ordinary first order differential equations. With appropriate boundary conditions, these equations have to be solved for each energy group and in each iteration step. The general solution of the corresponding homogeneous system of equations is known in analytical form. The present paper shows how the right-hand side of the system can be approximated in order to derive a particular solution and thus an approximate analytical expression for the general solution of the inhomogeneous system. This combined analytical-numerical approach was shown to have certain advantages compared to the finite-difference method or the Lie-series expansion method, which have been used to solve similar problems. (orig./RW) 1. Approximate analytical solution of two-dimensional multigroup P-3 equations Iterative solution of multigroup spherical harmonics equations reduces, in the P-3 approximation and in two-dimensional geometry, to a problem of solving an inhomogeneous system of eight ordinary first order differential equations. With appropriate boundary conditions, these equations have to be solved for each energy group and in each iteration step. The general solution of the corresponding homogeneous system of equations is known in analytical form. The present paper shows how the right-hand side of the system can be approximated in order to derive a particular solution and thus an approximate analytical expression for the general solution of the inhomogeneous system. This combined analytical-numerical approach was shown to have certain advantages compared to the finite-difference method or the Lie-series expansion method, which have been used to solve similar problems. (author) 2. Analytical Solutions for the Equilibrium states of a Swollen Hydrogel Shell and an Extended Method of Matched Asymptotics Dai, Hui-Hui 2011-01-01 A polymer network can imbibe water, forming an aggregate called hydrogel, and undergo large and inhomogeneous deformation with external mechanical constraint. Due to the large deformation, nonlinearity plays a crucial role, which also causes the mathematical difficulty for obtaining analytical solutions. Based on an existing model for equilibrium states of a swollen hydrogel with a core-shell structure, this paper seeks analytical solutions of the deformations by perturbation methods for three cases, i.e. free-swelling, nearly free-swelling and general inhomogeneous swelling. Particularly for the general inhomogeneous swelling, we introduce an extended method of matched asymptotics to construct the analytical solution of the governing nonlinear second-order variable-coefficient differential equation. The analytical solution captures the boundary layer behavior of the deformation. Also, analytical formulas for the radial and hoop stretches and stresses are obtained at the two boundary surfaces of the shell, ma... 3. Analytical solutions of the extended Boussinesq equation The extended Boussinesq equation for the description of the Fermi-Pasta-Ulam problem has been studied and analyzed with the Painleve test. It has been shown that the equation does not pass the Painleve test, but the necessary condition for the existence of meromorphic solutions is satisfied 4. Analyticity of solutions of the Korteweg-de Vries equation Tarama, Shigeo 2004-01-01 We consider the analytic smoothing effect for the KdV equation. That is to say, if the initial data given at$t = 0$decays very rapidly, the solution to the Cauchy problem becomes analytic with respect to the space variable for$t > 0. In this paper we show this effect by using the inverse scattering method which transforms the KdV equation to a linear dispersive equation whose analytic smoothing effect is shown through the properties of the Airy function. 5. Analytical r-mode solution with gravitational radiation reaction force Dias, O J C; S\\'a, Paulo M. 2005-01-01 We present and discuss the analytical r-mode solution to the linearized hydrodynamic equations of a slowly rotating, Newtonian, barotropic, non-magnetized, perfect-fluid star in which the gravitational radiation reaction force is present. 6. False Vacuum Transitions - Analytical Solutions and Decay Rate Values Correa, R A C; da Rocha, Roldao 2015-01-01 In this work we show a class of oscillating configurations for the evolution of the domain walls in Euclidean space. The solutions are obtained analytically. We also find the decay rate of the false vacuum. 7. Series solutions of boundary-layer flows in porous media with lateral mass flux Awang Kechil, Seripah [Universiti Tekonologi MARA, Department of Mathematics, Shah Alam Selangor (Malaysia); Hashim, Ishak [Universiti Kebangsaan Malaysia, School of Mathematical Sciences, UKM Bangi Selangor (Malaysia) 2008-08-15 Approximate analytical solutions for free convection boundary layers on a heated vertical plate with lateral mass flux embedded in a saturated porous medium are presented using the modified Adomian decomposition method and Pade technique. Several values of the wall temperature exponent for illustrating the effects of suction/injection parameter on the flow and heat transfer are considered. This study also includes the influence of the exponent on an impermeable surface. The results obtained are comparable to the exact analytical solutions and elucidate reliability and efficiency of the technique. (orig.) 8. New software solutions for analytical spectroscopists Davies, Antony N. 1999-05-01 Analytical spectroscopists must be computer literate to effectively carry out the tasks assigned to them. This has often been resisted within organizations with insufficient funds to equip their staff properly, a lack of desire to deliver the essential training and a basic resistance amongst staff to learn the new techniques required for computer assisted analysis. In the past these problems were compounded by seriously flawed software which was being sold for spectroscopic applications. Owing to the limited market for such complex products the analytical spectroscopist often was faced with buying incomplete and unstable tools if the price was to remain reasonable. Long product lead times meant spectrometer manufacturers often ended up offering systems running under outdated and sometimes obscure operating systems. Not only did this mean special staff training for each instrument where the knowledge gained on one system could not be transferred to the neighbouring system but these spectrometers were often only capable of running in a stand-alone mode, cut-off from the rest of the laboratory environment. Fortunately a number of developments in recent years have substantially changed this depressing picture. A true multi-tasking operating system with a simple graphical user interface, Microsoft Windows NT4, has now been widely introduced into the spectroscopic computing environment which has provided a desktop operating system which has proved to be more stable and robust as well as requiring better programming techniques of software vendors. The opening up of the Internet has provided an easy way to access new tools for data handling and has forced a substantial re-think about results delivery (for example Chemical MIME types, IUPAC spectroscopic data exchange standards). Improved computing power and cheaper hardware now allows large spectroscopic data sets to be handled without too many problems. This includes the ability to carry out chemometric operations in 9. Analytic solutions for marginal deformations in open superstring field theory We extend the calculable analytic approach to marginal deformations recently developed in open bosonic string field theory to open superstring field theory formulated by Berkovits. We construct analytic solutions to all orders in the deformation parameter when operator products made of the marginal operator and the associated superconformal primary field are regular. (orig.) 10. A hybrid ICT-solution for smart meter data analytics Liu, Xiufeng; Nielsen, Per Sieverts 2016-01-01 analytics. The proposed solution offers an information integration pipeline for ingesting data from smart meters, a scalable platform for processing and mining big data sets, and a web portal for visualizing analytics results. The implemented system has a hybrid architecture of using Spark or Hive for big... 11. Analytical solutions for the Rabi model Yu, Lixian; Liang, Qifeng; Chen, Gang; Jia, Suotang 2012-01-01 The Rabi model that describes the fundamental interaction between a two-level system with a quantized harmonic oscillator is one of the simplest and most ubiquitous models in modern physics. However, this model has not been solved exactly because it is hard to find a second conserved quantity besides the energy. Here we present a unitary transformation to map this unsolvable Rabi model into a solvable Jaynes-Cummings-like model by choosing a proper variation parameter. As a result, the analytical energy spectrums and wavefunctions including both the ground and the excited states can be obtained easily. Moreover, these explicit results agree well with the direct numerical simulations in a wide range of the experimental parameters. In addition, based on our obtained energy spectrums, the recent experimental observation of Bloch-Siegert in the circuit quantum electrodynamics with the ultrastrong coupling can be explained perfectly. Our results have the potential application in the solid-state quantum information... 12. Solution of Exterior Acoustic Problems by the Boundary Element Method. Kirkup, Stephen Martin Available from UMI in association with The British Library. The boundary element method is described and investigated, especially in respect of its application to exterior two -dimensional Laplace problems. Both empirical and algebraic analyses (including the effects of approximation of the boundary and boundary functions and the precision of the evaluation of the discrete forms) are developed. Methods for the automatic evaluation of the discrete forms of the Laplace and Helmholtz integral operators are reviewed and extended. Boundary element methods for the solution of exterior Helmholtz problems with general (but most importantly Neumann) boundary conditions are reviewed and some are explicitly stated using a new notation. Boundary element methods based on the boundary integral equations introduced by Brakhage & Werner/ Leis/ Panich/ Kussmaul (indirect) and Burton & Miller (direct) are given prime consideration and implemented for three -dimensional problems. The influence of the choice of weighting parameter on the performance of the methods is explored and further guidance is given. The application of boundary element methods and methods based on the Rayleigh integral to acoustic radiation problems are considered. Methods for speeding up their solution via the boundary element method are developed. Library subroutines for the solution of acoustic radiation problems are described and demonstrated. Computational techniques for the problem of predicting the noise produced by a running engine are reviewed and appraised. The application of the boundary element method to low-noise engine design and in the design of noise shields is considered. The boundary element method is applied to the Ricardo crankcase simulation rig, which is an engine -like structure. A comparison of predicted and measured sound power spectra is given. 13. Analytical Solution for the Current Distribution in Multistrand Superconducting Cables Bottura, L; Fabbri, M G 2002-01-01 Current distribution in multistrand superconducting cables can be a major concern for stability in superconducting magnets and for field quality in particle accelerator magnets. In this paper we describe multistrand superconducting cables by means of a distributed parameters circuit model. We derive a system of partial differential equations governing current distribution in the cable and we give the analytical solution of the general system. We then specialize the general solution to the particular case of uniform cable properties. In the particular case of a two-strand cable, we show that the analytical solution presented here is identical to the one already available in the literature. For a cable made of N equal strands we give a closed form solution that to our knowledge was never presented before. We finally validate the analytical solution by comparison to numerical results in the case of a step-like spatial distribution of the magnetic field over a short Rutherford cable, both in transient and steady ... 14. Analytical solutions to SSC coil end design As part of the SCC magnet effort, Fermilab will build and test a series of one meter model SSC magnets. The coils in these magnets will be constructed with several different end configurations. These end designs must satisfy both mechanical and magnetic criteria. Only the mechanical problem will be addressed. Solutions will attempt to minimize stresses and provide internal support for the cable. Different end designs will be compared in an attempt to determine which is most appropriate for the SSC dipole. The mathematics required to create each end configuration will be described. The computer aided design, programming and machine technology needed to make the parts will be reviewed. 2 refs., 10 figs 15. On analytic continuability of the missing Cauchy datum for Helmholtz boundary problems Karamehmedovic, Mirza 2015-01-01 We relate the domains of analytic continuation of Dirichlet and Neumann boundary data for Helmholtz problems in two or more independent variables. The domains are related à priori, locally and explicitly in terms of complex polyrectangular neighbourhoods of planar pieces of the boundary. To this end we identify and characterise a special subspace of the standard pseudodifferential operators with real-analytic symbols. The result is applicable in the estimation of the domain of analytic contin... 16. Speciation—targets, analytical solutions and markets Łobiński, Ryszard 1998-02-01 An analysis of speciation-relevant issues leads to the conclusion that, despite the rapidly increasing number of reports, the field has reached a level of virtual stagnation in terms of research originality and market perspectives. A breakthrough is in sight but requires an advanced interdisciplinary collaboration of chemists-analysts with clinicians, ecotoxicologists and nutricionists aimed at the definition of metal (metalloid)-dependent problems relevant to human health. The feedback from analytical chemists will be stimulated by a wider availability of efficient HPLC (CZE)-inductively coupled plasma mass spectrometry (ICP MS) interfaces, chromatographic software for ICP AES and MS and sensitive on-line methods for compound identification (electrospray MS/MS). The maturity of purge and trap thermal desorption techniques and capillary GC chromatography is likely to be reflected by an increasing number of commercial dedicated systems for small molecules containing Hg, Pb, Sn and metalloids. The pre-requisite of success for such systems is the integration of a sample preparation step (based on focused low-power microwave technology) into the marketed set-up. 17. Analytic solution of simplified Cardan's shaft model Zajíček M. 2014-12-01 Full Text Available Torsional oscillations and stability assessment of the homokinetic Cardan shaft with a small misalignment angle is described in this paper. The simplified mathematical model of this system leads to the linearized equation of the Mathieu's type. This equation with and without a stationary damping parameter is considered. The solution of the original differential equation is identical with those one of the Fredholm’s integral equation with degenerated kernel assembled by means of a periodic Green's function. The conditions of solvability of such problem enable the identification of the borders between stability and instability regions. These results are presented in the form of stability charts and they are verified using the Floquet theory. The correctness of oscillation results for the system with periodic stiffness is then validated by means of the Runge-Kutta integration method. 18. Analytical determination of distillation boundaries for ternary azeotropic systems Marcilla Gomis, Antonio; Reyes Labarta, Juan Antonio; Velasco, Raúl; Serrano Cayuelas, María Dolores; Olaya López, María del Mar 2009-01-01 A new straight forward algorithm to calculate distillation boundaries in ternary azeotropic systems has been developed. The proposed method allows, using cubic splines, the calculation of distillation trajectories and the calculation of that corresponding to the searched distillation boundaries. The algorithm is applied to 4 ternary liquid-vapour systems to test its validity. 19. Analytical Solution of the Time Fractional Fokker-Planck Equation Sutradhar T. 2014-05-01 Full Text Available A nonperturbative approximate analytic solution is derived for the time fractional Fokker-Planck (F-P equation by using Adomian’s Decomposition Method (ADM. The solution is expressed in terms of Mittag- Leffler function. The present method performs extremely well in terms of accuracy, efficiency and simplicity. 20. AN ANALYTICAL SOLUTION FOR CALCULATING THE INITIATION OF SEDIMENT MOTION Thomas LUCKNER; Ulrich ZANKE 2007-01-01 This paper presents an analytical solution for calculating the initiation of sediment motion and the risk of river bed movement. It thus deals with a fundamental problem in sediment transport, for which no complete analytical solution has yet been found. The analytical solution presented here is based on forces acting on a single grain in state of initiation of sediment motion. The previous procedures for calculating the initiation of sediment motion are complemented by an innovative combination of optical surface measurement technology for determining geometrical parameters and their statistical derivation as well as a novel approach for determining the turbulence effects of velocity fluctuations. This two aspects and the comparison of the solution functions presented here with the well known data and functions of different authors mainly differ the presented solution model for calculating the initiation of sediment motion from previous approaches. The defined values of required geometrical parameters are based on hydraulically laboratory tests with spheres. With this limitations the derivated solution functions permit the calculation of the effective critical transport parameters of a single grain, the calculation of averaged critical parameters for describing the state of initiation of sediment motion on the river bed, the calculation of the probability density of the effective critical velocity as well as the calculation of the risk of river bed movement. The main advantage of the presented model is the closed analytical solution from the equilibrium of forces on a single grain to the solution functions describing the initiation of sediment motion. 1. Constructing analytic approximate solutions to the Lane–Emden equation We derive analytic approximations to the solutions of the Lane–Emden equation, a basic equation in Astrophysics that describes the Newtonian equilibrium structure of a self-gravitating polytropic fluid sphere. After recalling some basic results, we focus on the construction of rational approximations, discussing the limitations of previous attempts, and providing new accurate approximate solutions. - Highlights: • We make a critical survey of the literature concerning the Lane–Emden equation. • We discuss problems in the construction of accurate rational approximate solutions. • We derive new analytic approximations of interest for star and cluster dynamics 2. Periodic solutions to nonlinear equations with oblique boundary conditions Allergretto, Walter; Papini, Duccio 2012-01-01 We study the existence of positive periodic solutions to nonlinear elliptic and parabolic equations with oblique and dynamical boundary conditions and non-local terms. The results are obtained through fixed point theory, topological degree methods and properties of related linear elliptic problems with natural boundary conditions and possibly non-symmetric principal part. As immediate consequences, we also obtain estimates on the principal eigenvalue for non-symmetric elliptic ... 3. An analytical solution for improved HIFU SAR estimation Accurate determination of the specific absorption rates (SARs) present during high intensity focused ultrasound (HIFU) experiments and treatments provides a solid physical basis for scientific comparison of results among HIFU studies and is necessary to validate and improve SAR predictive software, which will improve patient treatment planning, control and evaluation. This study develops and tests an analytical solution that significantly improves the accuracy of SAR values obtained from HIFU temperature data. SAR estimates are obtained by fitting the analytical temperature solution for a one-dimensional radial Gaussian heating pattern to the temperature versus time data following a step in applied power and evaluating the initial slope of the analytical solution. The analytical method is evaluated in multiple parametric simulations for which it consistently (except at high perfusions) yields maximum errors of less than 10% at the center of the focal zone compared with errors up to 90% and 55% for the commonly used linear method and an exponential method, respectively. For high perfusion, an extension of the analytical method estimates SAR with less than 10% error. The analytical method is validated experimentally by showing that the temperature elevations predicted using the analytical method's SAR values determined for the entire 3D focal region agree well with the experimental temperature elevations in a HIFU-heated tissue-mimicking phantom. (paper) 4. Irreversible and reversible reactive chromatography: analytical solutions and moment analysis for rectangular pulse injections. Bibi, Sameena; Qamar, Shamsul; Seidel-Morgenstern, Andreas 2015-03-13 This work is concerned with the analysis of models for linear reactive chromatography describing irreversible A→B and reversible A↔B reactions. In contrast to previously published results rectangular reactant pulses are injected into initially empty or pre-equilibrated columns assuming both Dirichlet and Danckwerts boundary conditions. The models consist of two partial differential equations, accounting for convection, longitudinal dispersion and first order chemical reactions. Due to the effect of involved mechanisms on solute transport, analytical and numerical solutions of the models could be helpful to understand, design and optimize chromatographic reactors. The Laplace transformation is applied to solve the model equations analytically for linear adsorption isotherms. Statistical temporal moments are derived from solutions in the Laplace domain. Analytical results are compared with numerical predictions generated using a high-resolution finite volume scheme for two sets of boundary conditions. Several case studies are carried out to analyze reactive liquid chromatographic processes for a wide range of mass transfer and reaction kinetics. Good agreements in the results validate the correctness of the analytical solutions and accuracy of the proposed numerical algorithm. PMID:25670415 5. Analytical Solution of Smoluchowski Equation in Harmonic Oscillator Potential SUN Xiao-Jun; LU Xiao-Xia; YAN Yu-Liang; DUAN Jun-Feng; ZHANG Jing-Shang 2005-01-01 Non-equilibrium fission has been described by diffusion model. In order to describe the diffusion process analytically, the analytical solution of Smoluchowski equation in harmonic oscillator potential is obtained. This analytical solution is able to describe the probability distribution and the diffusive current with the variable x and t. The results indicate that the probability distribution and the diffusive current are relevant to the initial distribution shape, initial position, and the nuclear temperature T; the time to reach the quasi-stationary state is proportional to friction coefficient β, but is independent of the initial distribution status and the nuclear temperature T. The prerequisites of negative diffusive current are justified. This method provides an approach to describe the diffusion process for fissile process in complicated potentials analytically. 6. Analytical solutions of the simplified Mathieu’s equation Nicolae MARCOV 2016-03-01 Full Text Available Consider a second order differential linear periodic equation. The periodic coefficient is an approximation of the Mathieu’s coefficient. This equation is recast as a first-order homogeneous system. For this system we obtain analytical solutions in an explicit form. The first solution is a periodic function. The second solution is a sum of two functions, the first is a continuous periodic function, but the second is an oscillating function with monotone linear increasing amplitude. We give a formula to directly compute the slope of this increase, without knowing the second numeric solution. The periodic term of the second solution may be computed directly. The coefficients of fundamental matrix of the system are analytical functions. 7. Numerical solutions of telegraph equations with the Dirichlet boundary condition Ashyralyev, Allaberen; Turkcan, Kadriye Tuba; Koksal, Mehmet Emir 2016-08-01 In this study, the Cauchy problem for telegraph equations in a Hilbert space is considered. Stability estimates for the solution of this problem are presented. The third order of accuracy difference scheme is constructed for approximate solutions of the problem. Stability estimates for the solution of this difference scheme are established. As a test problem to support theoretical results, one-dimensional telegraph equation with the Dirichlet boundary condition is considered. Numerical solutions of this equation are obtained by first, second and third order of accuracy difference schemes. 8. An analytical model for radioactive pollutant release simulation in the atmospheric boundary layer Weymar, Guilherme J.; Vilhena, Marco T.; Bodmann, Bardo E.J., E-mail: guicefetrs@gmail.com, E-mail: mtmbvilhena@gmail.com, E-mail: bejbodmann@gmail.com [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica; Buske, Daniela; Quadros, Regis, E-mail: danielabuske@gmail.com, E-mail: quadros99@gmail.com [Universidade Federal de Pelotas (UFPel), Capao do Leao, RS (Brazil). Programa de Pos-Graduacao em Modelagem Matematica 2013-07-01 Simulations of emission of radioactive substances in the atmosphere from the Brazilian nuclear power plant Angra 1 are a necessary tool for control and elaboration of emergency plans as a preventive action for possible accidents. In the present work we present an analytical solution for radioactive pollutant dispersion in the atmosphere, solving the time-dependent three-dimensional advection-diffusion equation. The experiment here used as a reference in the simulations consisted of the controlled releases of radioactive tritiated water vapor from the meteorological tower close to the power plant at Itaorna Beach. The wind profile was determined using experimental meteorological data and the micrometeorological parameters were calculated from empirical equations obtained in the literature. We report on a novel analytical formulation for the concentration of products of a radioactive chain released in the atmospheric boundary layer and solve the set of coupled equations for each chain radionuclide by the GILTT solution, assuming the decay of all progenitors radionuclide for each equation as source term. Further we report on numerical simulations, as an explicit but fictitious example and consider three radionuclides in the radioactive chain of Uranium 235. (author) 9. Analytical solution for a coaxial plasma gun: Weak coupling limit The analytical solution of the system of coupled ODE's which describes the time evolution of an ideal (i.e., zero resistance) coaxial plasma gun operating in the snowplow mode is obtained in the weak coupling limit, i.e, when the gun is fully influenced by the driving (RLC) circuit in which it resides but the circuit is negligibly influenced by the gun. Criteria for the validity of this limit are derived and numerical examples are presented. Although others have obtained approximate, asymptotic and numerical solutions of the equations, the present analytical results seem not to have appeared previously in the literature 10. A Hybrid Analytical-Numerical Solution to the Laminar Flow inside Biconical Ducts Thiago Antonini Alves 2015-10-01 Full Text Available In this work was presented a hybrid analytical-numerical solution to hydrodynamic problem of fully developed Newtonian laminar flow inside biconical ducts employing the Generalized Integral Transform Technique (GITT. In order to facilitate the analytical treatment and the application of the boundary conditions, a Conformal Transform was used to change the domain into a more suitable coordinate system. Thereafter, the GITT was applied on the momentum equation to obtain the velocity field. Numerical results were obtained for quantities of practical interest, such as maximum and minimum velocity, Fanning friction factor, Poiseuille number, Hagenbach factor and hydrodynamic entry length. 11. Positive Solutions for Nonlinear Differential Equations with Periodic Boundary Condition Shengjun Li 2012-01-01 Full Text Available We study the existence of positive solutions for second-order nonlinear differential equations with nonseparated boundary conditions. Our nonlinearity may be singular in its dependent variable. The proof of the main result relies on a nonlinear alternative principle of Leray-Schauder. Recent results in the literature are generalized and significantly improved. 12. Analytical and grid-free solutions to the Lighthill-Whitham-Richards traffic flow model Mazaré, Pierre Emmanuel 2011-12-01 In this article, we propose a computational method for solving the Lighthill-Whitham-Richards (LWR) partial differential equation (PDE) semi-analytically for arbitrary piecewise-constant initial and boundary conditions, and for arbitrary concave fundamental diagrams. With these assumptions, we show that the solution to the LWR PDE at any location and time can be computed exactly and semi-analytically for a very low computational cost using the cumulative number of vehicles formulation of the problem. We implement the proposed computational method on a representative traffic flow scenario to illustrate the exactness of the analytical solution. We also show that the proposed scheme can handle more complex scenarios including traffic lights or moving bottlenecks. The computational cost of the method is very favorable, and is compared with existing algorithms. A toolbox implementation available for public download is briefly described, and posted at http://traffic.berkeley.edu/project/downloads/lwrsolver. © 2011 Elsevier Ltd. 13. Comparison of Web Analytics : Hosted Solutions vs Server-side Analytics Mutai, Dominic 2015-01-01 The ratability of websites allows the aggregation of detailed data about the behavior and characteristics of website visitors. This thesis examines the value of different web metrics based on the analytics tools used and the behavior of website visitors. The objective is to test and identify key metrics and discuss how they compare between hosted solutions and server-side analytics. The value of the web metrics is evaluated by examining the relationships of the metrics to website conversions.... 14. Analytical solution to one-dimensional consolidation in unsaturated soils under sinusoidal cyclic loading 冯君; 巫锡勇; 朱宝龙; 杨期祥 2015-01-01 An analytical solution was presented to the unsaturated soil with a finite thickness under confinement in the lateral direction and sinusoidal cyclic loading in the vertical direction based on Fredlund’s one-dimensional consolidation equation for unsaturated soil. The transfer relationship between the state vectors at the top surface and any depth was gained by applying the Laplace transform and Cayley−Hamilton mathematical methods to the governing equations of water and air, Darcy’s law and Fick’s law. The excess pore-air and pore-water pressures and settlement in the Laplace-transformed domain were obtained by using the Laplace transform with the initial and boundary conditions. The analytical solutions of the excess pore-air and pore-water pressures at any depth and settlement were obtained in the time domain by performing the inverse Laplace transforms. A typical example illustrates the consolidation characteristics of unsaturated soil under sinusoidal loading from analytical results. Finally, comparisons between the analytical solutions and results of the numerical method indicate that the analytical solution is correct. 15. Analytic solution for the propagation velocity in superconducting composities The propagation velocity of normal zones in composite superconductors has been calculated analytically for the case of constant thermophysical properties, including the effects of current sharing. The solution is compared with that of a more elementary theory in which current sharing is neglected, i.e., in which there is a sharp transition from the superconducting to the normal state. The solution is also compared with experiment. This comparison demonstrates the important influence of transient heat transfer on the propagation velocity 16. Efficient analytical solutions for heated, pressurized multi-layered cylinders 2013-01-01 Two independent sets of analytical solutions, one based on matrix inversion and one based on iteration, are derived for the displacement field and corresponding stress state in multi-layer cylinders subjected to pressure and thermal loading. Solutions are developed for cylinders that are axially free with no friction between layers (plane stress), for cylinders that are fully restrained axially (plane strain) and for axially loaded and spring-mounted cylinders, assuming that the combined two-... 17. On analytic continuability of the missing Cauchy datum for Helmholtz boundary problems Karamehmedovic, Mirza 2015-01-01 We relate the domains of analytic continuation of Dirichlet and Neumann boundary data for Helmholtz problems in two or more independent variables. The domains are related à priori, locally and explicitly in terms of complex polyrectangular neighbourhoods of planar pieces of the boundary. To this... 18. Horses for courses: analytical tools to explore planetary boundaries van Vuuren, D. P.; Lucas, P. L.; Häyhä, T.; Cornell, S. E.; Stafford-Smith, M. 2015-09-01 There is a need for further integrated research on developing a set of sustainable development objectives, based on the proposed framework of planetary boundaries indicators. The relevant research questions are divided in this paper into four key categories, related to the underlying processes and selection of key indicators, understanding the impacts of different exposure levels and influence of connections between different types of impacts, a better understanding of different response strategies and the available options to implement changes. Clearly, different categories of scientific disciplines and associated models exist that can contribute to the necessary analysis, noting that the distinctions between them are fuzzy. In the paper, we both indicate how different models relate to the four categories of questions but also how further insights can be obtained by connecting the different disciplines (without necessarily fully integrating them). Research on integration can support planetary boundary quantification in a credible way, linking human drivers and social and biophysical impacts. 19. Analytical Loss Factors in Approximation of the Leontovich Boundary Conditions Baturin, S S 2014-01-01 Recently the new method of the Cherenkov fields and loss factors of a point-like electron bunch passing through longitudinally homogeneous structures lined with arbitrary slowdown layers was proposed. It was shown that the Cherenkov loss factor of the short bunch does not depend on the waveguide system material and is a constant for any given transverse dimensions and cross-section shapes of the waveguides. It was shown that with the proposed approach one can use a relatively simple method for the calculation of the total loss factor using an integral relation based on the cylindrical slowdown waveguide model. With this paper, we demonstrate that the same integral relation that we call relativistic Gauss theorem can be applied in case impedance boundary conditions (IBC) also known as Leontovich boundary conditions. 20. General analytical shakedown solution for structures with kinematic hardening materials Guo, Baofeng; Zou, Zongyuan; Jin, Miao 2016-04-01 The effect of kinematic hardening behavior on the shakedown behaviors of structure has been investigated by performing shakedown analysis for some specific problems. The results obtained only show that the shakedown limit loads of structures with kinematic hardening model are larger than or equal to those with perfectly plastic model of the same initial yield stress. To further investigate the rules governing the different shakedown behaviors of kinematic hardening structures, the extended shakedown theorem for limited kinematic hardening is applied, the shakedown condition is then proposed, and a general analytical solution for the structural shakedown limit load is thus derived. The analytical shakedown limit loads for fully reversed cyclic loading and non-fully reversed cyclic loading are then given based on the general solution. The resulting analytical solution is applied to some specific problems: a hollow specimen subjected to tension and torsion, a flanged pipe subjected to pressure and axial force and a square plate with small central hole subjected to biaxial tension. The results obtained are compared with those in literatures, they are consistent with each other. Based on the resulting general analytical solution, rules governing the general effects of kinematic hardening behavior on the shakedown behavior of structure are clearly. 1. Analytic Solutions for Tachyon Condensation with General Projectors Okawa, Y; Zwiebach, B; Okawa, Yuji; Rastelli, Leonardo; Zwiebach, Barton 2006-01-01 The tachyon vacuum solution of Schnabl is based on the wedge states, which close under the star product and interpolate between the identity state and the sliver projector. We use reparameterizations to solve the long-standing problem of finding an analogous family of states for arbitrary projectors and to construct analytic solutions based on them. The solutions simplify for special projectors and allow explicit calculations in the level expansion. We test the solutions in detail for a one-parameter family of special projectors that includes the sliver and the butterfly. Reparameterizations further allow a one-parameter deformation of the solution for a given projector, and in a certain limit the solution takes the form of an operator insertion on the projector. We discuss implications of our work for vacuum string field theory. 2. Analytic solutions for tachyon condensation with general projectors Okawa, Y. [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany); Rastelli, L. [C.N. Yang Institute for Theoretical Physics, Stony Brook, NY (United States); Zwiebach, B. [Massachusetts Inst. of Tech., Cambridge, MA (United States). Center for Theoretical Physics 2006-11-15 The tachyon vacuum solution of Schnabl is based on the wedge states, which close under the star product and interpolate between the identity state and the sliver projector. We use reparameterizations to solve the long-standing problem of finding an analogous family of states for arbitrary projectors and to construct analytic solutions based on them. The solutions simplify for special projectors and allow explicit calculations in the level expansion. We test the solutions in detail for a one-parameter family of special projectors that includes the sliver and the butterfly. Reparameterizations further allow a one-parameter deformation of the solution for a given projector, and in a certain limit the solution takes the form of an operator insertion on the projector. We discuss implications of our work for vacuum string field theory. (orig.) 3. Modelling stellar jets with magnetospheres using as initial states analytical MHD solutions Todorov, P; Cayatte, V; Sauty, C; Lima, J J G; Tsinganos, K 2016-01-01 In this paper we focus on the construction of stellar outflow models emerging from a polar coronal hole-type region surrounded by a magnetosphere in the equatorial regions during phases of quiescent accretion. The models are based on initial analytical solutions. We adopt a meridionally self-similar solution of the time-independent and axisymmetric MHD equations which describes effectively a jet originating from the corona of a star. We modify appropriately this solution in order to incorporate a physically consistent stellar magnetosphere. We find that the closed fieldline region may exhibit different behaviour depending on the associated boundary conditions and the distribution of the heat flux. However, the stellar jet in all final equilibrium states is very similar to the analytical one prescribed in the initial conditions. When the initial net heat flux is maintained, the magnetosphere takes the form of a dynamical helmet streamer with a quasi steady state slow magnetospheric wind. With no heat flux, a s... 4. MATHEMATIC MODEL AND ANALYTIC SOLUTION FOR CYLINDER SUBJECT TO UNEVEN PRESSURES LIU Wen 2006-01-01 According to the inverse solution of elasticity mechanics, a stress function is constructed which meets the space biharmonic equation, this stress functions is about cubic function pressure on the inner and outer surfaces of cylinder. When borderline condition that is predigested according to the Saint-Venant's theory is joined, an equation suit is constructed which meets both the biharmonic equations and the boundary conditions. Furthermore, its analytic solution is deduced with Matlab.When this theory is applied to hydraulic bulging rollers, the experimental results inosculate with the theoretic calculation. Simultaneously, the limit along the axis invariable direction is given and the model building of hollow cylinder and for the analytic solution of hollow cylinder with randomly uneven pressure. 5. On numerical-analytic techniques for boundary value problems Rontó, András; Rontó, M.; Shchobak, N. 2012-01-01 Roč. 12, č. 3 (2012), s. 5-10. ISSN 1335-8243 Institutional support: RVO:67985840 Keywords : numerical-analytic method * periodic successive approximations * Lyapunov-Schmidt method Subject RIV: BA - General Mathematics http://www.degruyter.com/view/j/aeei.2012.12.issue-3/v10198-012-0035-1/v10198-012-0035-1.xml?format=INT 6. A three-dimensional analytical solution for radioactive contaminant dispersion in the atmosphere In this work, we report an analytical solution for steady-state three-dimensional advection-diffusion equation for simulation of radioactive pollutant in atmosphere considering a vertically inhomogeneous Planetary Boundary Layer. The main idea relies in solution of the steady-state three-dimensional advection-diffusion equation by the combined ADMM and GILTT techniques. We also report numerical simulation assuming power wind profile and we compare with the ones achieved by the GILTT method with Gaussian in y-direction as well experimental data. (author) 7. An analytical solution for the electrical properties of a Radio Frequency Quadrupole (RFQ) with simple vanes Lancaster, H. 1982-02-01 Although the SUPERFISH program is used for calculating the design parameters of an radio frequency quadrupole (RFQ) structure with complex vanes, an analytical solution for electrical properties of an RFQ with simple vanes provides insight into the parametric behavior of these more complicated resonators. The fields in an inclined plane wave guide with proper boundary conditions match those in one quadrant of an RFQ. The principle of duality is used to exploit the solutions to a radial transmission line in solving the field equations. Calculated are the frequency equation, frequency sensitivity factors (5), electric field (E), magnetic field (H), stored energy (U), power dissipation (P), and quality factor (Q). 8. Semi-Analytical Solution of Functionally Graded Circular Short Hollow Cylinder Subject to Transient Thermal Loading Eskandari Jam Jafar 2014-12-01 Full Text Available In this paper, by using a semi-analytical solution based on multi-layered approach, the authors present the solutions of temperature, displacements, and transient thermal stresses in functionally graded circular hollow cylinders subjected to transient thermal boundary conditions. The cylinder has finite length and is subjected to axisymmetric thermal loads. It is assumed that the functionally graded circular hollow cylinder is composed of N fictitious layers and the properties of each layer are assumed to be homogeneous and isotropic. Time variations of the temperature, displacements, and stresses are obtained by employing series solving method for ordinary differential equation, Laplace transform techniques and a numerical Laplace inversion. 9. Mathematical Model of Suspension Filtering and Its Analytical Solution Normahmad Ravshanov 2013-01-01 Full Text Available The work develops mathematical model and computing algorithm to analyze, project and identify the basic parameters of filter units operation and their variation range. On their basis, numerical analytic solution of the problem of ionized liquid solutions filtering was obtained. Computing experiments, resulting in graphic format were presented. Analysis of calculation results enables to determine the optimum modes of filter units operation, used in liquid ionized solutions filtration technology, in food preparation, in drug production and for drinking water purification. Selection of the most suitable parameters contributes to the improvement of economic and technologic efficiency of production and filter units operability. 10. The big bang and inflation united by an analytic solution Exact analytic solutions for a class of scalar-tensor gravity theories with a hyperbolic scalar potential are presented. Using an exact solution we have successfully constructed a model of inflation that produces the spectral index, the running of the spectral index, and the amplitude of scalar perturbations within the constraints given by the WMAP 7 years data. The model simultaneously describes the big bang and inflation connected by a specific time delay between them so that these two events are regarded as dependent on each other. In solving the Friedmann equations, we have utilized an essential Weyl symmetry of our theory in 3+1 dimensions which is a predicted remaining symmetry of 2T-physics field theory in 4+2 dimensions. This led to a new method of obtaining analytic solutions in the 1T field theory which could in principle be used to solve more complicated theories with more scalar fields. Some additional distinguishing properties of the solution includes the fact that there are early periods of time when the slow-roll approximation is not valid. Furthermore, the inflaton does not decrease monotonically with time; rather, it oscillates around the potential minimum while settling down, unlike the slow-roll approximation. While the model we used for illustration purposes is realistic in most respects, it lacks a mechanism for stopping inflation. The technique of obtaining analytic solutions opens a new window for studying inflation, and other applications, more precisely than using approximations. 11. Exact Analytical Solution of Alfven Waves in Nonuniform Plasmas Full text: The propagation of Alfven waves in non-uniform plasmas is described through linear second-order differential equations, governing the total pressure and radial plasma velocity. In general, these two differential equations only admit numerical solutions, whose behavior is very much complicated especially near resonance surfaces which encompass essential degeneracies. It is well-known that most existing analytical methods, including the famous Wentzel-Karmers-Brillouin (WKB) approximation fail near such singularities. In this paper, a power analytical method, which is recently developed and named the Differential Transfer Matrix Method (DTMM), is applied to find a rigorously exact solution to the problem of interest. We also present an approximate solution based on the Airy functions. (author) 12. Analytical Determination of the Boundaries of Transition Natural Zones (Ecotones Rulev Aleksandr Sergeevich 2015-04-01 Full Text Available The morphological units that are part of the catena, are recognized in accordance with the response to the geomorphological and soil processes. The spatial relationship is the main unit between them. In this regard, the landscape patterns acquire a cascade type, and their main link becomes the zonal catena, which has specific stable features, reflecting the dependence of the complex of natural conditions and processes of latitude. However, clear-cut boundaries do not exist – they have spatial and temporal displacement, associated with the cyclical nature of the global climatic processes. The landscapes in these transition zones (ecotones a priori can be considered unstable. The detection of ecotones boundaries provides the opportunity to divide natural zones to potentially stable and potentially unstable parts for planning measures on preventing the degradation of landscapes localized in them. The latitude of the ecotones localization can be determined through the connection of the radiation heat flux on land (R with the normalized geographical latitude of the subboreal belt (x, which is described by the equation of the energy balance, expressed in the logistic function R = А / [1 + 0,72 exp(4,25 – Bx] + C. 13. Horses for courses: analytical tools to explore planetary boundaries van Vuuren, Detlef P.; Lucas, Paul L.; Häyhä, Tiina; Cornell, Sarah E.; Stafford-Smith, Mark 2016-03-01 There is a need for more integrated research on sustainable development and global environmental change. In this paper, we focus on the planetary boundaries framework to provide a systematic categorization of key research questions in relation to avoiding severe global environmental degradation. The four categories of key questions are those that relate to (1) the underlying processes and selection of key indicators for planetary boundaries, (2) understanding the impacts of environmental pressure and connections between different types of impacts, (3) better understanding of different response strategies to avoid further degradation, and (4) the available instruments to implement such strategies. Clearly, different categories of scientific disciplines and associated model types exist that can accommodate answering these questions. We identify the strength and weaknesses of different research areas in relation to the question categories, focusing specifically on different types of models. We discuss that more interdisciplinary research is need to increase our understanding by better linking human drivers and social and biophysical impacts. This requires better collaboration between relevant disciplines (associated with the model types), either by exchanging information or by fully linking or integrating them. As fully integrated models can become too complex, the appropriate type of model (the racehorse) should be applied for answering the target research question (the race course). 14. A semi-analytical solution for slug tests in an unconfined aquifer considering unsaturated flow Sun, Hongbing 2016-01-01 A semi-analytical solution considering the vertical unsaturated flow is developed for groundwater flow in response to a slug test in an unconfined aquifer in Laplace space. The new solution incorporates the effects of partial penetrating, anisotropy, vertical unsaturated flow, and a moving water table boundary. Compared to the Kansas Geological Survey (KGS) model, the new solution can significantly improve the fittings of the modeled to the measured hydraulic heads at the late stage of slug tests in an unconfined aquifer, particularly when the slug well has a partially submerged screen and moisture drainage above the water table is significant. The radial hydraulic conductivities estimated with the new solution are comparable to those from the KGS, Bouwer and Rice, and Hvorslev methods. In addition, the new solution also can be used to examine the vertical conductivity, specific storage, specific yield, and the moisture retention parameters in an unconfined aquifer based on slug test data. 15. Analytical and Numerical Modeling of Plasma Flow Around a Circular Boundary Scott, Roger B. 2016-05-01 The flow of plasma around a circular boundary is a topic of great importance for solar physics, in which individual flux tubes are often treated as discrete elements that interact with their environment through the forces exerted across their surfaces. This scenario has been suggested by McKenzie (2000) and later refined by Savage et al. (2012) as an explanation for the dark lanes that commonly form in supra-arcade fan structures, i.e. SADs. Here we present a rigorous model in which a retracting flux tube serves as an interior boundary for the surrounding magnetic field, with the associated plasma forced to flow around it. Through a zero-β analytical solution we show that under certain circumstances these flows lead to shocks that can extend far out into the unreconnected field, altering the plasma properties in the affected region. We then employ a numerical simulation that relaxes the low-β assumption and we find that the collimated shocks described in our previous study are recovered for plasma β in the range of 0 energy of the fluid exceeds the internal energy of both the fluid and the magnetic field, thereby leading to the formation of unconfined, fast-mode magnetohydrodynamic shocks, even in the limit of small β. 16. Analytical Solution of Flow and Heat Transfer over a Permeable Stretching Wall in a Porous Medium M. Dayyan 2013-01-01 Full Text Available Boundary layer flow through a porous medium over a stretching porous wall has seen solved with analytical solution. It has been considered two wall boundary conditions which are power-law distribution of either wall temperature or heat flux. These are general enough to cover the isothermal and isoflux cases. In addition to momentum, both first and second laws of thermodynamics analyses of the problem are investigated. The governing equations are transformed into a system of ordinary differential equations. The transformed ordinary equations are solved analytically using homotopy analysis method. A comprehensive parametric study is presented, and it is shown that the rate of heat transfer increases with Reynolds number, Prandtl number, and suction to the surface. 17. Solution of Boundary-Value Problems using Kantorovich Method Gusev, A. A.; Hai, L. L.; Chuluunbaatar, O.; Vinitsky, S. I.; Derbov, V. L. 2016-02-01 We propose a computational scheme for solving the eigenvalue problem for an elliptic differential equation in a two-dimensional domain with Dirichlet boundary conditions. The solution is sought in the form of Kantorovich expansion over the basis functions of one of the independent variables with the second variable treated as a parameter. The basis functions are calculated as solutions of the parametric eigenvalue problem for an ordinary second-order differential equation. As a result, the initial problem is reduced to a boundary-value problem for a set of self-adjoint second-order differential equations for functions of the second independent variable. The discrete formulation of the problem is implemented using the finite element method with Hermite interpolation polynomials. The effciency of the calculation scheme is shown by benchmark calculations for a square membrane with a degenerate spectrum. 18. Solution of Boundary-Value Problems using Kantorovich Method Gusev A.A. 2016-01-01 Full Text Available We propose a computational scheme for solving the eigenvalue problem for an elliptic differential equation in a two-dimensional domain with Dirichlet boundary conditions. The solution is sought in the form of Kantorovich expansion over the basis functions of one of the independent variables with the second variable treated as a parameter. The basis functions are calculated as solutions of the parametric eigenvalue problem for an ordinary second-order differential equation. As a result, the initial problem is reduced to a boundary-value problem for a set of self-adjoint second-order differential equations for functions of the second independent variable. The discrete formulation of the problem is implemented using the finite element method with Hermite interpolation polynomials. The effciency of the calculation scheme is shown by benchmark calculations for a square membrane with a degenerate spectrum. 19. Solute interaction in grain boundary segregation and cohesion Lejček, Pavel Zurich : Trans Tech Publications, 2014 - (Šandera, P.), s. 389-392 ISBN 978-3-03785-934-6. ISSN 1013-9826. - (Key Engineering Materials. 592-593). [International Conference on Materials Structure and Micromechanics of Fracture /7./. Brno (CZ), 01.07.2013-03.07.2013] R&D Projects: GA ČR GAP108/12/0144 Institutional support: RVO:68378271 Keywords : interfacial segregation * grain boundary embrittlemenmt * solute interaction modeling Subject RIV: BM - Solid Matter Physics ; Magnetism 20. Mathematic Model and Analytic Solution for a Cylinder Subject to Exponential Function LIU Wen; SHAN Rui 2009-01-01 Hollow cylinders are widely used in spacecraft, rockets, weapons, metallurgy, materials, and mechanical manufacturing industries, and so on, hydraulic bulging roll cylinder and hydraulic press work all belong to hollow cylinders. However, up till now, the solution of the cylinder subjected to the pressures in the three-dimensional space is still at the stage of the analytical solution to the normal pressure or the approximate solution to the variable pressure by numerical method. The analytical solution to the variable pressure of the cylinder has not yet made any breakthrough in theory and can not meet accurate theoretical analysis and calculation requirements of the cylindrical in Engineering. In view of their importance, the precision calculation and theoretical analysis are required to investigate on engineering. A stress function which meets both the biharmonic equations and boundary conditions is constructed in the three-dimensional space. Furthermore, the analytic solution of a hollow cylinder subjected to exponential function distributed variable pressure on its inner and outer surfaces is deduced. By controlling the pressure subject to exponential function distributed variable pressure in the hydraulic bulging roller without any rolling load, using a static tester to record the strain supported hydraulic bulging roll, and comparing with the theoretical calculation, the experimental test result has a higher degree of agreement with the theoretical calculation. Simultaneously, the famous Lamè solution can be deduced when given the unlimited length of cylinder along the axis. The analytic solution paves the way for the mathematic building and solution of hollow cylinder with randomly uneven pressure. 1. Analytical solutions for flow of horizontal well in compressible, three-dimensional unconsolidated formations A novel mathematical model for single-phase fluid flow from unconsolidated formations to a horizontal well with the consideration of stress-sensitive permeability is presented. The model assumes the formation permeability is an exponential function of the pore pressure. Using a perturbation technique, the model is solved for either constant pressure or constant flux or infinite lateral boundary conditions with closed top and bottom boundaries. Through Laplace transformation, finite Fourier transformation and numerical inversion methods, the solutions are obtained and the pressure response curves are analyzed. The agreement between the analytical solutions in this paper and the numerical results from commercial software (Saphir) is excellent, which manifests the accuracy of the results derived in this paper. (paper) 2. Phononic heat transport in the transient regime: An analytic solution Tuovinen, Riku; Säkkinen, Niko; Karlsson, Daniel; Stefanucci, Gianluca; van Leeuwen, Robert 2016-06-01 We investigate the time-resolved quantum transport properties of phonons in arbitrary harmonic systems connected to phonon baths at different temperatures. We obtain a closed analytic expression of the time-dependent one-particle reduced density matrix by explicitly solving the equations of motion for the nonequilibrium Green's function. This is achieved through a well-controlled approximation of the frequency-dependent bath self-energy. Our result allows for exploring transient oscillations and relaxation times of local heat currents, and correctly reduces to an earlier known result in the steady-state limit. We apply the formalism to atomic chains, and benchmark the validity of the approximation against full numerical solutions of the bosonic Kadanoff-Baym equations for the Green's function. We find good agreement between the analytic and numerical solutions for weak contacts and baths with a wide energy dispersion. We further analyze relaxation times from low to high temperature gradients. 3. Analytical representation of a black hole puncture solution The 'moving-puncture' technique has led to dramatic advancements in the numerical simulations of binary black holes. Hannam et al. have recently demonstrated that, for suitable gauge conditions commonly employed in moving-puncture simulations, the evolution of a single black hole leads to a well-known, time-independent, maximal slicing of Schwarzschild spacetime. They construct the corresponding solution in isotropic coordinates numerically and demonstrate its usefulness, for example, for testing and calibrating numerical codes that employ moving-puncture techniques. In this brief report we point out that this solution can also be constructed analytically, making it even more useful as a test case for numerical codes 4. Barrierless Electronic Relaxation in Solution: An Analytically Solvable Model Chakraborty, Aniruddha 2013-01-01 We propose an analytical method for understanding the problem of electronic relaxation in solution, modeled by a particle undergoing diffusive motion under the influence of two potentials. The coupling between the two potentials is assumed to be represented by a Dirac Delta function. The diffusive motion in this paper is described by the Smoluchowskii equation. Our solution requires the knowledge of the Laplace transform of the Green's function for the motion in both the uncoupled potentials. Our model is more general than all the earlier models, because we are the first one to consider the effect of ground state potential energy surface explicitly. 5. An Analytical Method of Auxiliary Sources Solution for Plane Wave Scattering by Impedance Cylinders Larsen, Niels Vesterdal; Breinbjerg, Olav Analytical Method of Auxiliary Sources solutions for plane wave scattering by circular impedance cylinders are derived by transformation of the exact eigenfunction series solutions employing the Hankel function wave transformation. The analytical Method of Auxiliary Sources solution thus obtained... 6. Analytical Analysis and Numerical Solution of Two Flavours Skyrmion Hadi, Miftachul; Hermawanto, Denny 2010-01-01 Two flavours Skyrmion will be analyzed analytically, in case of static and rotational Skyrme equations. Numerical solution of a nonlinear scalar field equation, i.e. the Skyrme equation, will be worked with finite difference method. This article is a more comprehensive version of \\textit{SU(2) Skyrme Model for Hadron} which have been published at Journal of Theoretical and Computational Studies, Volume \\textbf{3} (2004) 0407. 7. Analytic solution of certain second-order functional differential equation Theeradach Kaewong 2006-09-01 Full Text Available We consider the existence of analytic solutions of a certain class of iterative second-order functional differential equation of the form xÃ¢Â€Â³(x[r](z=c0z2+c1(x(z2+(c2x[2](z2+Ã¢Â‹Â¯+cm(x[m](z2, m,rÃ¢Â‰Â¥0. 8. Semi-analytical solution for soliton propagation in colloidal suspension Senthilkumar Selvaraj 2013-04-01 Full Text Available We consider the propagation of soliton in colloidal nano-suspension. We derive the semi analytical solution for soliton propagation in colloidal nano-suspensions for both one and two spatial dimensions using variational method. This Variational method uses both Averaged Lagrangian and suitable trial functions. Finally we analyse about Rayleigh scattering loss in the soliton propagation through the colloidal nano-suspensions. 9. Approximate analytical solutions of the baby Skyrme model Ioannidou, T. A.; Kopeliovich, V. B.; Zakrzewski, W. J. 2002-01-01 In present paper we show that many properties of the baby skyrmions, which have been determined numerically, can be understood in terms of an analytic approximation. In particular, we show that this approximation captures properties of the multiskyrmion solutions (derived numerically) such as their stability towards decay into various channels, and that it is more accurate for the "new baby Skyrme model" which describes anisotropic physical systems in terms of multiskyrmion fields with axial ... 10. Analytical Solutions of the Balance Equation for the Scalar Variance in One-Dimensional Turbulent Flows under Stationary Conditions Andrea Amicarelli 2015-01-01 Full Text Available This study presents 1D analytical solutions for the ensemble variance of reactive scalars in one-dimensional turbulent flows, in case of stationary conditions, homogeneous mean scalar gradient and turbulence, Dirichlet boundary conditions, and first order kinetics reactions. Simplified solutions and sensitivity analysis are also discussed. These solutions represent both analytical tools for preliminary estimations of the concentration variance and upwind spatial reconstruction schemes for CFD (Computational Fluid Dynamics—RANS (Reynolds-Averaged Navier-Stokes codes, which estimate the turbulent fluctuations of reactive scalars. 11. A new analytical solution to axisymmetric Blot's consolidation of a finite soil layer 2008-01-01 A new analytical method is presented to study the axisymmetric Blot's consolidation of a finite soil layer. Starting from the governing equations of axisymmetric Blot's consolidation, and based on the property of Laplace transform, the relation of basic variables for a point of a finite soil layer is established between the ground surface (z= 0) and the depth z in the Laplace and Hankel transform domains. Combined with the boundary conditions of the finite soil layer, the analytical solution of any point in the transform domain can be obtained. The actual solution in the physical domain can be obtained by inverse Laplace and Hankel transforms. A numerical analysis for the axisymmetric consolidation of a finite soil layer is carried out. 12. Development of CAD implementing the algorithm of boundary elements’ numerical analytical method Yulia V. Korniyenko 2015-03-01 Full Text Available Up to recent days the algorithms for numerical-analytical boundary elements method had been implemented with programs written in MATLAB environment language. Each program had a local character, i.e. used to solve a particular problem: calculation of beam, frame, arch, etc. Constructing matrices in these programs was carried out “manually” therefore being time-consuming. The research was purposed onto a reasoned choice of programming language for new CAD development, allows to implement algorithm of numerical analytical boundary elements method and to create visualization tools for initial objects and calculation results. Research conducted shows that among wide variety of programming languages the most efficient one for CAD development, employing the numerical analytical boundary elements method algorithm, is the Java language. This language provides tools not only for development of calculating CAD part, but also to build the graphic interface for geometrical models construction and calculated results interpretation. 13. Numerical solutions of multi-dimensional solidification/melting problems by the dual reciprocity boundary element method This paper presents an effective and simple procedure for the simulation of the motion of the solid-liquid interfacial boundary and the transient temperature field during phase change process. To accomplish this purpose, an iterative implicit solution algorithm has been developed by employing the dual reciprocity boundary element method. The dual reciprocity boundary element approach provided in this paper is much simpler than the usual boundary element method applying a reciprocity principle and an available technique for dealing with domain integral of boundary element formulation simultaneously. The effectiveness of the present analysis method have been illustrated through comparisons of the calculation results of an example with its semi-analytical or other numerical solutions where available 14. Numerical solutions of multi-dimensional solidification/melting problems by the dual reciprocity boundary element method Jo, Jong Chull; Shin, Won Ky [Korea Institute of Nuclear Safety, Taejon (Korea, Republic of) 1997-12-31 This paper presents an effective and simple procedure for the simulation of the motion of the solid-liquid interfacial boundary and the transient temperature field during phase change process. To accomplish this purpose, an iterative implicit solution algorithm has been developed by employing the dual reciprocity boundary element method. The dual reciprocity boundary element approach provided in this paper is much simpler than the usual boundary element method applying a reciprocity principle and an available technique for dealing with domain integral of boundary element formulation simultaneously. The effectiveness of the present analysis method have been illustrated through comparisons of the calculation results of an example with its semi-analytical or other numerical solutions where available. 22 refs., 3 figs. (Author) 15. Analytical model for temperature distribution of an end diode-pumped laser slab with Robin boundary conditions by Green’s function method Moghtader Dindarlu, M. H.; Kavosh Tehrani, M.; Saghafifar, H.; Maleki, A. 2016-05-01 In this paper, an analytical model is introduced for temperature distribution of an end diode-pumped laser slab by Green’s function method. To solve the heat equation, Robin boundary conditions are considered because four lateral faces of the slab are cooled by water. An analytical model is extracted for single and dual end-pumping configuration. For an example, the 2D and 3D temperature distributions are plotted and our analytical model is validated by numerical solution based on the finite element method (FEM). The results show that our model has very good agreement with numerical solution. Furthermore, dependence of the temperature distribution on absorbed pump power is shown. 16. Analytical solution for multilayer plates using general layerwise plate theory Vuksanović Đorđe M. 2005-01-01 Full Text Available This paper deals with closed-form solution for static analysis of simply supported composite plate, based on generalized laminate plate theory (GLPT. The mathematical model assumes piece-wise linear variation of in-plane displacement components and a constant transverse displacement through the thickness. It also include discrete transverse shear effect into the assumed displacement field, thus providing accurate prediction of transverse shear stresses. Namely, transverse stresses satisfy Hook's law, 3D equilibrium equations and traction free boundary conditions. With assumed displacement field, linear strain-displacement relation, and constitutive equations of the lamina, equilibrium equations are derived using principle of virtual displacements. Navier-type closed form solution of GLPT, is derived for simply supported plate, made of orthotropic laminae, loaded by harmonic and uniform distribution of transverse pressure. Results are compared with 3D elasticity solutions and excellent agreement is found. 17. Monotone positive solution for three-point boundary value problem SUN Yong-ping 2008-01-01 In this paper, the existence of monotone positive solution for the following secondorder three-point boundary value problem is studied:x"(t)+f(t,x(t))=0,0solution are obtained and the bounds of solution are given.At last,an example is given to illustrate the result of the paper. 18. Analyticity of solutions for quasilinear wave equations and other quasilinear systems Kuksin, Sergei; Nadirashvili, Nikolai 2012-01-01 We prove the persistence of analyticity for classical solution of the Cauchy problem for quasilinear wave equations with analytic data. Our results show that the analyticity of solutions, stated by the Cauchy-Kowalewski and Ovsiannikov-Nirenberg theorems, lasts till a classical solution exists. Moreover, they show that if the equation and the Cauchy data are analytic only in a part of space-variables, then a classical solution also is analytic in these variables. The approach applies to other... 19. Analytic solution of pseudocolloid migration in fractured rock A form of colloid migration that can enhance or retard the migration of a dissolved contaminant in ground water is the sorption of the contaminant on the moving colloidal particulate to form pseudocolloids. In this paper we develop analytical solutions for the interactive migration of radioactive species dissolved in ground water and sorbed as pseudocolloids. The solute and pseudocolloids are assumed to undergo advection and dispersion in a one-dimensional flow field in planar fractures in porous rock. Interaction between pseudocolloid and dissolved species is described by equilibrium sorption. Sorbed species on the pseudocolloids undergo radioactive decay, and pseudocolloids can sorb on fracture surfaces and sediments. Filtration is neglected. The solute can decay and sorb on pseudocolloids, on the fracture surfaces, and on sediments and can diffuse into the porous rock matrix. 1 fig 20. Analytical solutions for multidimensional transport of a four-member radionuclide decay chain in ground water Analytical solutions based on the Laplace and Fourier transformation techniques are presented for the two- and three-dimensional space-time-dependent convective-dispersive transport of a four-member radionuclide decay chain in homogeneous porous media. The longitudinal dispersion-free solution is also reported. The computation was executed using the MASCOT model on a VAX/VMS-Version 4.1. The solutions are designed for an unbounded medium flow field assumed to be semi-infinite in the direction normal to the source, and infinite orthogonal to the source, with a variety of boundary conditions (single or multiple finite line source or a Gaussian distributed source in the two-dimensional case; single or multiple patch source or bivariate normally distributed source in the three-dimensional case). Radionuclide release modes of the constant and nuclide-dependent type are taken into account. An optimization of the convergence of the integration required by these solutions is achieved after operating a transformation of the infinite interval into the sum of two finite ones. The efficiency of two quadrature formulas (Gauss-Legendre and a fourth-order Newton-Cotes based on an iterative approach) was investigated. Solution accuracy was verified against available one- and two-dimensional analytical solutions. 15 refs 1. JOVIAN STRATOSPHERE AS A CHEMICAL TRANSPORT SYSTEM: BENCHMARK ANALYTICAL SOLUTIONS We systematically investigated the solvable analytical benchmark cases in both one- and two-dimensional (1D and 2D) chemical-advective-diffusive systems. We use the stratosphere of Jupiter as an example but the results can be applied to other planetary atmospheres and exoplanetary atmospheres. In the 1D system, we show that CH4 and C2H6 are mainly in diffusive equilibrium, and the C2H2 profile can be approximated by modified Bessel functions. In the 2D system in the meridional plane, analytical solutions for two typical circulation patterns are derived. Simple tracer transport modeling demonstrates that the distribution of a short-lived species (such as C2H2) is dominated by the local chemical sources and sinks, while that of a long-lived species (such as C2H6) is significantly influenced by the circulation pattern. We find that an equator-to-pole circulation could qualitatively explain the Cassini observations, but a pure diffusive transport process could not. For slowly rotating planets like the close-in extrasolar planets, the interaction between the advection by the zonal wind and chemistry might cause a phase lag between the final tracer distribution and the original source distribution. The numerical simulation results from the 2D Caltech/JPL chemistry-transport model agree well with the analytical solutions for various cases. 2. JOVIAN STRATOSPHERE AS A CHEMICAL TRANSPORT SYSTEM: BENCHMARK ANALYTICAL SOLUTIONS Zhang Xi; Shia Runlie; Yung, Yuk L., E-mail: xiz@gps.caltech.edu [Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125 (United States) 2013-04-20 We systematically investigated the solvable analytical benchmark cases in both one- and two-dimensional (1D and 2D) chemical-advective-diffusive systems. We use the stratosphere of Jupiter as an example but the results can be applied to other planetary atmospheres and exoplanetary atmospheres. In the 1D system, we show that CH{sub 4} and C{sub 2}H{sub 6} are mainly in diffusive equilibrium, and the C{sub 2}H{sub 2} profile can be approximated by modified Bessel functions. In the 2D system in the meridional plane, analytical solutions for two typical circulation patterns are derived. Simple tracer transport modeling demonstrates that the distribution of a short-lived species (such as C{sub 2}H{sub 2}) is dominated by the local chemical sources and sinks, while that of a long-lived species (such as C{sub 2}H{sub 6}) is significantly influenced by the circulation pattern. We find that an equator-to-pole circulation could qualitatively explain the Cassini observations, but a pure diffusive transport process could not. For slowly rotating planets like the close-in extrasolar planets, the interaction between the advection by the zonal wind and chemistry might cause a phase lag between the final tracer distribution and the original source distribution. The numerical simulation results from the 2D Caltech/JPL chemistry-transport model agree well with the analytical solutions for various cases. 3. Electromagnetic wave theory for boundary-value problems an advanced course on analytical methods Eom, Hyo J 2004-01-01 Electromagnetic wave theory is based on Maxwell's equations, and electromagnetic boundary-value problems must be solved to understand electromagnetic scattering, propagation, and radiation. Electromagnetic theory finds practical applications in wireless telecommunications and microwave engineering. This book is written as a text for a two-semester graduate course on electromagnetic wave theory. As such, Electromagnetic Wave Theory for Boundary-Value Problems is intended to help students enhance analytic skills by solving pertinent boundary-value problems. In particular, the techniques of Fourier transform, mode matching, and residue calculus are utilized to solve some canonical scattering and radiation problems. 4. Approximate analytical solutions to the condensation-coagulation equation of aerosols Smith, Naftali R.; Shaviv, Nir J.; Svensmark, Henrik 2016-01-01 We present analytical solutions to the steady state nucleation-condensation-coagulation equation of aerosols in the atmosphere. These solutions are appropriate under different limits but more general than previously derived analytical solutions. For example, we provide an analytic solution to the... 5. Analytical solution to a fracture problem in a tough layered structure Hamamoto, Yukari; Okumura, Ko 2008-08-01 Nacre causes the shining beauty of pearl due to its remarkable layered structure, which is also strong. We reconsider a simplified layered model of nacre proposed previously [Okumura and de Gennes, Eur. Phys. J. E 4, 121 (2001)] and obtain an analytical solution to a fundamental crack problem. The result asserts that the fracture toughness is enhanced due to a large displacement around the crack tip (even if the crack-tip stress is not reduced). The derivation offers ideas for solving a number of boundary problems for partial differential equations important in many fields. 6. An analytical solution for the magneto-electro-elastic bimorph beam forced vibrations problem Based on the Timoshenko beam theory and on the assumption that the electric and magnetic fields can be treated as steady, since elastic waves propagate very slowly with respect to electromagnetic ones, a general analytical solution for the transient analysis of a magneto-electro-elastic bimorph beam is obtained. General magneto-electric boundary conditions can be applied on the top and bottom surfaces of the beam, allowing us to study the response of the bilayer structure to electromagnetic stimuli. The model reveals that the magneto-electric loads enter the solution as an equivalent external bending moment per unit length and as time-dependent mechanical boundary conditions through the definition of the bending moment. Moreover, the influences of the electro-mechanic, magneto-mechanic and electromagnetic coupling on the stiffness of the bimorph stem from the computation of the beam equivalent stiffness constants. Free and forced vibration analyses of both multiphase and laminated magneto-electro-elastic composite beams are carried out to check the effectiveness and reliability of the proposed analytic solution 7. An analytical solution for the magneto-electro-elastic bimorph beam forced vibrations problem Milazzo, A.; Orlando, C.; Alaimo, A. 2009-08-01 Based on the Timoshenko beam theory and on the assumption that the electric and magnetic fields can be treated as steady, since elastic waves propagate very slowly with respect to electromagnetic ones, a general analytical solution for the transient analysis of a magneto-electro-elastic bimorph beam is obtained. General magneto-electric boundary conditions can be applied on the top and bottom surfaces of the beam, allowing us to study the response of the bilayer structure to electromagnetic stimuli. The model reveals that the magneto-electric loads enter the solution as an equivalent external bending moment per unit length and as time-dependent mechanical boundary conditions through the definition of the bending moment. Moreover, the influences of the electro-mechanic, magneto-mechanic and electromagnetic coupling on the stiffness of the bimorph stem from the computation of the beam equivalent stiffness constants. Free and forced vibration analyses of both multiphase and laminated magneto-electro-elastic composite beams are carried out to check the effectiveness and reliability of the proposed analytic solution. 8. Analytical solutions of transport problems in anisotropic media Recently, the problem of neutron transport in anisotropic media has received new attention in connection with safety studies of water reactors and design of gas-cooled systems. In situations presenting large voided regions, as the axial streaming is dominating with respect to the transverse one, the average properties of the homogenized material should physically account for such macroscopic anisotropy. Hence, it is suggested that cell calculations produce anisotropic average cross sections, e.g., axial (σA) and transverse (σT) values. Since material anisotropy is due to leakage, as a first-step approximation, the medium can be considered isotropic with respect to scattering phenomena. Transport codes are currently being adapted to include anisotropic cross sections. An important aspect of code development is the validation of algorithms by analytical benchmarks. For that purpose, the present work is devoted to the fully analytical solution of transport problems in slab geometry 9. Analytical solutions for finite cylindrical dynamic cavity expansion in compressible elastic-plastic materials 甄明; 蒋志刚; 宋殿义; 刘飞 2014-01-01 Analytical solutions for the dynamic cylindrical cavity expansion in a com-pressible elastic-plastic cylinder with a finite radius are developed by taking into account of the effect of lateral free boundary, which are different from the traditional cavity expan-sion models for targets with infinite dimensions. The finite cylindrical cavity expansion process begins with an elastic-plastic stage followed by a plastic stage. The elastic-plastic stage ends and the plastic stage starts when the plastic wave front reaches the lateral free boundary. Approximate solutions of radial stress on cavity wall are derived by using the Von-Mise yield criterion and Forrestal’s similarity transformation method. The effects of the lateral free boundary and finite radius on the radial stress on the cavity wall are discussed, and comparisons are also conducted with the finite cylindrical cavity expansion in incompressible elastic-plastic materials. Numerical results show that the lateral free boundary has significant influence on the cavity expansion process and the radial stress on the cavity wall of metal cylinder with a finite radius. 10. Cooling and warming laws: an exact analytical solution This paper deals with temperature variations over time of objects placed in a constant-temperature environment in the presence of thermal radiation. After a historical introduction, the paper discusses cooling and warming laws, by taking into account first solely object-environment energy exchange by thermal radiation, and then adding object-environment heat exchange by convection. These processes are usually evaluated by approximating the law of exchange of thermal radiation by a linear relationship between power exchange and temperature difference. In contrast, in this paper an exact analytical solution considering Stefan's fourth power law is provided, under some general hypotheses, for both cases. A comparison with exponential approximations and with a historical law proposed by Dulong and Petit in 1817 is presented. Data of an experiment are used to test the analytical solution: the test has allowed evaluating the heat transfer coefficient h of the experiment and has shown that our solution provides a better fit with the measured values than any exponential function. The topic is developed in a way which can be suitable both for undergraduate students and for general physicists. 11. Analytic theory of curvature effects for wave problems with general boundary conditions Willatzen, Morten; Gravesen, Jens; Voon, L. C. Lew Yan 2010-01-01 A formalism based on a combination of differential geometry and perturbation theory is used to obtain analytic expressions for confined eigenmode changes due to general curvature effects. In cases of circular-shaped and helix-shaped structures, where alternative analytic solutions can be found, the... 12. Numerical and analytical solutions for problems relevant for quantum computers Quantum computers are one of the next technological steps in modern computer science. Some of the relevant questions that arise when it comes to the implementation of quantum operations (as building blocks in a quantum algorithm) or the simulation of quantum systems are studied. Numerical results are gathered for variety of systems, e.g. NMR systems, Josephson junctions and others. To study quantum operations (e.g. the quantum fourier transform, swap operations or multiply-controlled NOT operations) on systems containing many qubits, a parallel C++ code was developed and optimised. In addition to performing high quality operations, a closer look was given to the minimal times required to implement certain quantum operations. These times represent an interesting quantity for the experimenter as well as for the mathematician. The former tries to fight dissipative effects with fast implementations, while the latter draws conclusions in the form of analytical solutions. Dissipative effects can even be included in the optimisation. The resulting solutions are relaxation and time optimised. For systems containing 3 linearly coupled spin (1)/(2) qubits, analytical solutions are known for several problems, e.g. indirect Ising couplings and trilinear operations. A further study was made to investigate whether there exists a sufficient set of criteria to identify systems with dynamics which are invertible under local operations. Finally, a full quantum algorithm to distinguish between two knots was implemented on a spin(1)/(2) system. All operations for this experiment were calculated analytically. The experimental results coincide with the theoretical expectations. (orig.) 13. Analytical Solution of the Bosonic Three-Body Problem We revisit the problem of three identical bosons in free space, which exhibits a universal hierarchy of bound states (Efimov trimers). Modeling a narrow Feshbach resonance within a two-channel description, we map the integral equation for the three-body scattering amplitude to a one-dimensional Schroedinger-type single-particle equation, where an analytical solution of exponential accuracy is obtained. We give exact results for the trimer binding energies, the three-body parameter, the threshold to the three-atom continuum, and the recombination rate 14. Mathematical Model of Suspension Filtration and Its Analytical Solution Normahmad Ravshanov 2013-01-01 Full Text Available The work develops advanced mathematical model and computing algorithm to analyze, predict and identify the basic parameters of filter units and their variation ranges. Numerical analytic solution of liquid ionized mixtures filtration was got on their basis. Computing experiments results are presented in graphics form. Calculation results analysis enables to determine the optimum performance of filter units, used for liquid ionized mixtures filtration, food preparation, drug production and water purification. Selection of the most suitable parameters contributes to the improvement of economic and technological efficiency of production and filter units working efficiency. 15. Analytical Solution of The Two-Qubit Quantum Rabi Model Abo-Kahla, Doaa A M; Abdel-Aty, Mahmoud 2015-01-01 In this paper, an analytical solution of the two-qubit Rabi model for the general case is presented. Furthermore, a comparison between the information entropies and the Von Neumann entropy(\\rho_{A})$is given for some special values of the qubit-photon coupling constants in case of the detuning parameters. It is demonstrated that oscillations of the occupation probabilities$\\rho_{11}, \\rho_{22}, \\rho_{33}$and$\\rho_{44}$are equivalent to the case of the spontaneous emission. The occupation probability$\\rho_{11}$reaches the case of sudden death, when the detuning parameters$\\Delta_{2}equals zero. 16. Regression techniques and analytical solutions to demonstrate intrinsic bioremediation It is now generally recognized that a major factor responsible for the attenuation and mass reduction of benzene, toluene, ethylbenzene, and xylenes (BTEX) in groundwater plumes is hydrocarbon biodegradation by indigenous microorganisms in aquifer material. Their objective is to apply well-known regression techniques and analytical solutions to estimate the contribution of advection, dispersion, sorption, and biodecay to the overall attenuation of petroleum hydrocarbons. These calculations yield an apparent biodecay rate based on field data. This biodecay rate is a significant portion of the overall attenuation in stable, dissolved hydrocarbon plumes 17. Approximate analytical solutions of the baby Skyrme model Ioannidou, T A; Zakrzewski, W J 2002-01-01 In present paper we show that many properties of the baby skyrmions, which have been determined numerically, can be understood in terms of an analytic approximation. In particular, we show that this approximation captures properties of the multiskyrmion solutions (derived numerically) such as their stability towards decay into various channels, and that it is more accurate for the "new baby Skyrme model" which describes anisotropic physical systems in terms of multiskyrmion fields with axial symmetry. Some universal characteristics of configurations of this kind are demonstrated, which do not depend on their topological number. 18. An analytical model for solute transport in an infiltration tracer test in soil with a shallow groundwater table Liang, Ching-Ping; Hsu, Shao-Yiu; Chen, Jui-Sheng 2016-09-01 It is recommended that an in-situ infiltration tracer test is considered for simultaneously determining the longitudinal and transverse dispersion coefficients in soil. Analytical solutions have been derived for two-dimensional advective-dispersive transport in a radial geometry in the literature which can be used for interpreting the result of such a tracer test. However, these solutions were developed for a transport domain with an unbounded-radial extent and an infinite thickness of vadose zone which might not be realistically manifested in the actual solute transport during a field infiltration tracer test. Especially, the assumption of infinite thickness of vadose zone should be invalid for infiltration tracer tests conducted in soil with a shallow groundwater table. This paper describes an analytical model for interpreting the results of an infiltration tracer test based on improving the transport domain with a bounded-radial extent and a finite thickness of vadose zone. The analytical model is obtained with the successive application of appropriate integral transforms and their corresponding inverse transforms. A comparison of the newly derived analytical solution against the previous analytical solutions in which two distinct sets of radial extent and thickness of vadose zone are considered is conducted to determine the influence of the radial and exit boundary conditions on the solute transport. The results shows that both the radial and exit boundary conditions substantially affect the trailing segment of the breakthrough curves for a soil medium with large dispersion coefficients. Previous solutions derived for a transport domain with an unbounded-radial and an infinite thickness of vadose zone boundary conditions give lower concentration predictions compared with the proposed solution at late times. Moreover, the differences between two solutions are amplified when the observation positions are near the groundwater table. In addition, we compare our 19. RESEARCH OF INTERACTIONS BETWEEN SHEAR FRACTURES ON THE BASIS OF APPROXIMATE ANALYTICAL ELASTIC SOLUTIONS Anastasia S. Lermontova 2015-09-01 Full Text Available The article describes a method yielding approximate analytical solutions under the theory of elasticity for a set of interacting arbitrarily spaced shear fractures. Accurate analytical solutions of this problem are now available only for the simplest individual cases, such as a single fracture or two collinear fractures. A large amount of computation is required to yield a numerical solution for a case considering arbitrary numbers and locations of fractures, while this problem has important practical applications, such as assessment of the state of stress in seismically active regions, forecasts of secondary destruction impacts near systems of large faults, studies of reservoir properties of the territories comprising oil and gas provinces.In this study, an approximate estimation is obtained with the following simplification assumptions: (1 functions showing shear of fractures’ borders are determined similar to the shear function for a single fracture, and (2 boundary conditions for the fractures are specified in the integrated form as mean values along each fracture. Upon simplification, the solution is obtained through the system of linear algebraic equations for unknown values of tangential stress drop. With this approach, the accuracy of approximate solutions is consistent with the accuracy of the available data on real fractures.The reviewed examples of estimations show that the resultant stress field is dependent on the number, size and location of fractures and the sequence of displacements of the fractures’ borders. 20. Analytical dynamic solution of a flexible cable-suspended manipulator Bamdad, Mahdi 2013-12-01 Cable-suspended manipulators are used in large scale applications with, heavy in weight and long in span cables. It seems impractical to maintain cable assumptions of smaller robots for large scale manipulators. The interactions among the cables, platforms and actuators can fully evaluate the coupled dynamic analysis. The structural flexibility of the cables becomes more pronounced in large manipulators. In this paper, an analytic solution is provided to solve cable vibration. Also, a closed form solution can be adopted to improve the dynamic response to flexibility. The output is provided by the optimal torque generation subject to the actuator limitations in a mechatronic sense. Finally, the performance of the proposed algorithm is examined through simulations. 1. A general solution for vertical-drain consolidation with impeded drainage boundaries 付崔伟; 雷国辉 2016-01-01 An analytical solution is derived from the generalized governing equations of equal-strain consolidation with vertical drains under multi-ramp surcharge preloading. The hydraulic boundary conditions at both top and bottom of the consolidating soil are modelled as impeded drainage. The impeded drainage is described by using the third type boundary condition with a characteristic factor of drainage efficiency. Fully drained and undrained boundary conditions can also be modelled by applying an infinite and a zero characteristic factor, respectively. Simultaneous radial and vertical flow conditions are considered, together with the effects of drain resistance and smear. An increase in total stress due to multi-ramp loading is reasonably modelled as a function of both time and depth. A solution to calculate excess pore-water pressure at any arbitrary point in soil is derived, and the overall average degree of consolidation is obtained. It shows that the proposed solution can be used to analyze not only vertical-drain consolidation but also one-dimensional consolidation under either one-way or two-way vertical drainage conditions. The characteristic factors of drainage efficiency of top and bottom boundaries have a potentially important influence on consolidation. The boundary may be considered fully drained when the characteristic factor is greater than 100 and fully undrained when the characteristic factor is less than 0.1. The stress distribution along depth induced by the surcharge loading has a limited effect on the overall average degree of consolidation. However, it has a significant effect on the dissipation of excess pore-water pressure. 2. A Three-Dimensional Analytical Solution for the Study of Air Pollutant Dispersion in a Finite Layer Marie, Ema'a. Ema'a. Jean; Hubert, Ben-Bolie Germain; Patrice, Ele Abiama; Zarma, Ali; Pierre, Owono Ateba 2015-05-01 We present a closed-form analytical solution for the advection-diffusion equation, where the planetary boundary layer is divided into subdomains, where in each subdomain averaged values of eddy diffusivity and wind speed are assumed. The solution procedure combines an appropriate auxiliary eigenvalue problem with mathematical induction. A transcendental equation giving the eigenvalues for any numbers of subdomains is also developed. Convergence of the solution is numerically verified. The solution is used to evaluate the model against the Copenhagen experiment and computed results are in agreement with experimental ones. 3. A closed-form exact solution for the value of American put and its optimal exercise boundary (Invited Paper) Zhu, Song-Ping 2005-05-01 Searching for a closed-form exact solution for American put options under the Black-Scholes framework has been a long standing problem in the past; many researchers believe that it is impossible to find such a solution. In this paper, a closed-form exact solution, in the form of a Taylor's series expansion, of the well-known Black-Scholes equation is presented for the first time. As a result of this analytic solution, the optimal exercise boundary, which is the main difficulty of the problem, is found as an explicit function of the risk-free interest rate, the volatility and the time to expiration. 4. Theoretical progress in non-equilibrium grain-boundary segregation(II):Micro-mechanism of grain boundary anelastic relaxation and its analytical formula 2009-01-01 Finding the internal-friction peak of grain boundary anelastic relaxation was one of the important breakthroughs in the study of internal friction in the last century.But the micro-mechanism of grain boundary anelastic relaxations is still obscure.Based on the observations of the grain boundary seg-regation or depletion of solute induced by an applied stress,the following micro-mechanism was suggested:grain-boundaries will work as sources to emit vacancies when a compressive stress is exerted on them and as sinks to absorb vacancies when a tensile stress is exerted,inducing grain-boundary depletion or segregation of solute,respectively.The equations of vacancy and solute con-centrations at grain boundaries were established under the equilibrium of grain-boundary anelastic relaxation.With these the kinetic equations were established for grain boundary segregation and depletion during the grain boundary relaxation progress. 5. New chemical evolution analytical solutions including environment effects Spitoni, E 2015-01-01 In the last years, more and more interest has been devoted to analytical solutions, including inflow and outflow, to study the metallicity enrichment in galaxies. In this framework, we assume a star formation rate which follows a linear Schmidt law, and we present new analytical solutions for the evolution of the metallicity (Z) in galaxies. In particular, we take into account environmental effects including primordial and enriched gas infall, outflow, different star formation efficiencies, and galactic fountains. The enriched infall is included to take into account galaxy-galaxy interactions. Our main results can be summarized as: i) when a linear Schmidt law of star formation is assumed, the resulting time evolution of the metallicity Z is the same either for a closed-box model or for an outflow model. ii) The mass-metallicity relation for galaxies which suffer a chemically enriched infall, originating from another evolved galaxy with no pre-enriched gas, is shifted down in parallel at lower Z values, if co... 6. Analytic solutions of tunneling time through smooth barriers Xiao, Zhi; Huang, Hai 2016-03-01 In the discussion of temporary behaviors of quantum tunneling, people usually like to focus their attention on rectangular barrier with steep edges, or to deal with smooth barrier with semi-classical or even numerical calculations. Very few discussions on analytic solutions of tunneling through smooth barrier appear in the literature. In this paper, we provide two such examples, a semi-infinite long barrier V ( x ) = /A 2 [ 1 + tanh ( x / a ) ] and a finite barrier V(x) = A sech2(x/a). To each barrier, we calculate the associated phase time and dwell time after obtaining the analytic solution. The results show that, different from rectangular barrier, phase time or dwell time does increase with the length parameter a controlling the effective extension of the barrier. More interestingly, for the finite barrier, phase time or dwell time exhibits a peak in k-space. A detailed analysis shows that this interesting behavior can be attributed to the strange tunneling probability Ts(k), i.e., Ts(k) displays a unit step function-like profile Θ(k - k0), especially when a is large, say, a ≫ 1/κ, 1/k. And k 0 ≡ √{ m A } / ħ is exactly where the peak appears in phase or dwell time k-spectrum. Thus only those particles with k in a very narrow interval around k0 are capable to dwell in the central region of the barrier sufficiently long. 7. Decision exploration lab: a visual analytics solution for decision management. Broeksema, Bertjan; Baudel, Thomas; Telea, Arthur G; Crisafulli, Paolo 2013-12-01 We present a visual analytics solution designed to address prevalent issues in the area of Operational Decision Management (ODM). In ODM, which has its roots in Artificial Intelligence (Expert Systems) and Management Science, it is increasingly important to align business decisions with business goals. In our work, we consider decision models (executable models of the business domain) as ontologies that describe the business domain, and production rules that describe the business logic of decisions to be made over this ontology. Executing a decision model produces an accumulation of decisions made over time for individual cases. We are interested, first, to get insight in the decision logic and the accumulated facts by themselves. Secondly and more importantly, we want to see how the accumulated facts reveal potential divergences between the reality as captured by the decision model, and the reality as captured by the executed decisions. We illustrate the motivation, added value for visual analytics, and our proposed solution and tooling through a business case from the car insurance industry. PMID:24051763 8. Creation of the CMB blackbody spectrum: precise analytic solutions Khatri, Rishi 2012-01-01 The blackbody spectrum of CMB was created behind the blackbody surface at redshiftsz\\gtrsim 2\\times 10^6$. At earlier times, the Universe was dense and hot enough that complete thermal equilibrium between baryonic matter (electrons and ions) and photons could be established. Any perturbation away from the blackbody spectrum was suppressed exponentially. New physics, for example annihilation and decay of dark matter, can add energy and photons to CMB at redshifts$z\\gtrsim 10^5$and result in a non-zero chemical potential ($\\mu$) of CMB. Precise evolution of the CMB spectrum around the critical redshift of$z\\gtrsim 2\\times 10^6$is required in order to calculate the$\\mu$-type spectral distortion. Although numerical calculation of important processes involved (double Compton process, comptonization and bremsstrahlung) is not difficult, analytic solutions are much faster and easier to calculate and provide valuable physical insights. We provide precise (better than 1%) analytic solutions for the decay of$\\m...

9. Grain-boundary migration in nonstoichiometric solid solutions of magnesium aluminate spinel 2; Effects of grain-boundary nonstoichiometry

Chiang, Y.M. (Massachusetts Inst. of Tech., Cambridge, MA (USA). Dept. of Materials Science and Engineering); Kingery, W.D. (Arizona Univ., Tucson, AZ (USA))

1990-05-01

The grain-boundary chemistry of magnesium aluminate spinel solid solutions MgO {center dot} nAl{sub 2}O{sub 3} has been investigated in order to understand the mechanism of grain-boundary migration. It is found that although segregation of impurity Ca and Si is common, much larger deviations in grain-boundary stoichiometry are present. There is an excess of Al and O relative to Mg at grain boundaries in all compositions. Grain-boundary migration appears to be rate-limited by solute drag from intrinsic defects accommodating lattice nonstoichiometry, rather than by extrinsic solutes, consistent with the observed impurity tolerance of grain-boundary mobility. Different rate-limiting defects are proposed for magnesia-rich and alumina-rich spinels.

10. Analytical Solution of Projectile Motion with Quadratic Resistance and Generalisations

Ray, Shouryya

2013-01-01

The paper considers the motion of a body under the influence of gravity and drag of the surrounding fluid. Depending on the fluid mechanical regime, the drag force can exhibit a linear, quadratic or even more general dependence on the velocity of the body relative to the fluid. The case of quadratic drag is substantially more complex than the linear case, as it nonlinearly couples both components of the momentum equation, and no explicit analytic solution is known for a general trajectory. After a detailed account of the literature, the paper provides such a solution in form of a series expansion. This result is discussed in detail and related to other approaches previously proposed. In particular, it is shown to yield certain approximate solutions proposed in the literature as limiting cases. The solution technique employs a strategy to reduce systems of ordinary differential equations with a triangular dependence of the right-hand side on the vector of unknowns to a single equation in an auxiliary variable....

11. Bondi-Hoyle-Lyttleton accretion flow revisited: Analytic solution

Matsuda, Takuya; Isaka, Hiromu; Ohsugi, Yukimasa

2015-11-01

The time-steady equation for a 1D wind accretion flow, i.e. the Bondi-Hoyle-Lyttleton (BHL) equation, is investigated analytically. The BHL equation is well known to have infinitely many solutions. Traditionally, the accretion radius has been assumed to be 2textit {GM}/v_{infty }2, but its mathematical foundation has not been clarified because of the non-uniqueness of the solution. Here, we assume that the solution curves possess physically nice characteristics, i.e. velocity and line mass-density increase monotonically with radial distance. This condition restricts the accretion radius to the range left (0.71 - 1.0right ) × 2textit {GM}/v_{infty }2. Further assumptions, specifically, that the solution curves for velocity and line mass-density are convex upward, restrict the accretion radius to (0.84 - 0.94) × 2textit {GM}/v_{infty }2, and 0.90 × 2textit {GM}/v_{infty }2, respectively. Therefore, we conclude that the accretion radius is almost uniquely determined to be 0.90 × 2textit {GM}/v_{infty }2.

12. Exact analytical solution of shear-induced flexural vibration of functionally graded piezoelectric beam

Sharma, Pankaj; Parashar, Sandeep Kumar

2016-05-01

The priority of this paper is to obtain the exact analytical solution for free flexural vibration of FGPM beam actuated using the d15 effect. In piezoelectric actuators, the potential use of d15 effect has been of particular interest for engineering applications since shear piezoelectric coefficient d15 is much higher than the other piezoelectric coupling constants d31 and d33. The applications of shear actuators are to induce and control the flexural vibrations of beams and plates. In this study, a modified Timoshenko beam theory is used where electric potential is assumed to vary sinusoidaly along the thickness direction. The material properties are assumed to be graded across the thickness in accordance with power law distribution. Hamilton`s principle is employed to obtain the equations of motion along with the associated boundary conditions for FGPM beams. Exact analytical solution is derived thus obtained equations of motion. Results for clamped-clamped and clamped-free boundary conditions are presented. The presented result and method shell serve as benchmark for comparing the results obtained from the other approximate methods.

13. Analytic electrostatic solution of an axisymmetric accelerator gap

Numerous computer codes calculate beam dynamics of particles traversing an accelerating gap. In order to carry out these calculations the electric field of a gap must be determined. The electric field is obtained from derivatives of the scalar potential which solves Laplace's equation and satisfies the appropriate boundary conditions. An integral approach for the solution of Laplace's equation is used in this work since the objective is to determine the potential and fields without solving on a traditional spatial grid. The motivation is to quickly obtain forces for particle transport, and eliminate the need to keep track of a large number of grid point fields. The problem then becomes one of how to evaluate the appropriate integral. In this work the integral solution has been converted to a finite sum of easily computed functions. Representing the integral solution in this manner provides a readily calculable formulation and avoids a number of difficulties inherent in dealing with an integral that can be weakly convergent in some regimes, and is, in general, highly oscillatory

14. Exact analytical solution to three-dimensional phase change heat transfer problems in biological tissues subject to freezing

Fang-fang LI; Jing LIU; Kai YUE

2009-01-01

Analytically solving a three-dimensional (3-D) bioheat transfer problem with phase change during a freezing process is extremely difficult but theoretically important. The moving heat source model and the Green function method are introduced to deal with the cryopreservation process of in vitro biomaterials. Exact solutions for the 3-D temper-ature transients of tissues under various boundary conditions, such as totally convective cooling, totally fixed temperature cooling and a hybrid between them on tissue surfaces, are obtained. Furthermore, the cryosurgical process in living tissues subject to freezing by a single or multiple cryoprobes is also analytically solved. A closed-form analytical solution to the bioheat phase change process is derived by considering contributions from blood perfusion heat transfer, metabolic heat generation, and heat sink of a cryoprobe. The present method is expected to have significant value for analytically solving complex bioheat transfer problems with phase change.

15. A non-grey analytical model for irradiated atmospheres. II: Analytical vs. numerical solutions

Parmentier, Vivien; Fortney, Jonathan J; Marley, Mark S

2013-01-01

The recent discovery and characterization of the diversity of the atmospheres of exoplanets and brown dwarfs calls for the development of fast and accurate analytical models. In this paper we first quantify the accuracy of the analytical solution derived in paper I for an irradiated, non-grey atmosphere by comparing it to a state-of-the-art radiative transfer model. Then, using a grid of numerical models, we calibrate the different coefficients of our analytical model for irradiated solar-composition atmospheres of giant exoplanets and brown dwarfs. We show that the so-called Eddington approximation used to solve the angular dependency of the radiation field leads to relative errors of up to 5% on the temperature profile. For grey or semi-grey atmospheres we show that the presence of a convective zone has a limited effect on the radiative atmosphere above it and leads to modifications of the radiative temperature profile of order 2%. However, for realistic non-grey planetary atmospheres, the presence of a con...

16. Exact analytical solutions to the master equation of quantum Brownian motion for a general environment

Fleming, C H; Hu, B L

2010-01-01

We revisit the model of a quantum Brownian oscillator linearly coupled to an environment of quantum oscillators at finite temperature. By introducing a compact and particularly well-suited formulation, we give a rather quick and direct derivation of the master equation and its solutions for general spectral functions and arbitrary temperatures. The flexibility of our approach allows for an immediate generalization to cases with an external force and with an arbitrary number of Brownian oscillators. More importantly, we point out an important mathematical subtlety concerning boundary-value problems for integro-differential equations which led to incorrect master equation coefficients and impacts on the description of nonlocal dissipation effects in all earlier derivations. Furthermore, we provide explicit, exact analytical results for the master equation coefficients and its solutions in a wide variety of cases, including ohmic, sub-ohmic and supra-ohmic environments with a finite cut-off.

17. An analytic solution of steady Stokes flow on a rotating polar cap

An analytic solution of two-dimensional, steady, linear, viscous flow on a polar cap-the polar region of a sphere that lies above (or below) a given plane normal to the rotation axis-rotating about its center is obtained. Inflow and outflow on the boundary of the polar cap drive the fluid motion. The solution of the stream function is expressed as the Fourier series in longitudes and the associated Legendre functions of complex degrees in cosines of colatitudes. Fluid particles move almost along lines of constant latitude, some circulate cyclonically and others anticyclonically, in the geostrophic balance everywhere except near the north pole where the flow is relatively slow and the viscous force dominates over the Coriolis force. Our results support the approximation analysis and laboratory experiment studied by Imawaki and Takano (1974 Deep-Sea Res. 21 69-77).

18. Unified semi-analytical wall boundary conditions applied to 2-D incompressible SPH

Leroy, A.; Violeau, D.; Ferrand, M.; Kassiotis, C.

2014-03-01

This work aims at improving the 2-D incompressible SPH model (ISPH) by adapting it to the unified semi-analytical wall boundary conditions proposed by Ferrand et al. [10]. The ISPH algorithm considered is as proposed by Lind et al. [25], based on the projection method with a divergence-free velocity field and using a stabilising procedure based on particle shifting. However, we consider an extension of this model to Reynolds-Averaged Navier-Stokes equations based on the k-ɛ turbulent closure model, as done in [10]. The discrete SPH operators are modified by the new description of the wall boundary conditions. In particular, a boundary term appears in the Laplacian operator, which makes it possible to accurately impose a von Neumann pressure wall boundary condition that corresponds to impermeability. The shifting and free-surface detection algorithms have also been adapted to the new boundary conditions. Moreover, a new way to compute the wall renormalisation factor in the frame of the unified semi-analytical boundary conditions is proposed in order to decrease the computational time. We present several verifications to the present approach, including a lid-driven cavity, a water column collapsing on a wedge and a periodic schematic fish-pass. Our results are compared to Finite Volumes methods, using Volume of Fluids in the case of free-surface flows. We briefly investigate the convergence of the method and prove its ability to model complex free-surface and turbulent flows. The results are generally improved when compared to a weakly compressible SPH model with the same boundary conditions, especially in terms of pressure prediction.

19. Local existence of solution to free boundary value problem for compressible Navier-Stokes equations

Liu, Jian

2015-01-01

This paper is concerned with the free boundary value problem for multi-dimensional Navier-Stokes equations with density-dependent viscosity where the flow density vanishes continuously across the free boundary. A local (in time) existence of weak solution is established, in particular, the density is positive and the solution is regular away from the free boundary.

20. Analytic solution of Hubbell's model of local community dynamics

McKane, A; Sole, R; Kane, Alan Mc; Alonso, David; Sole, Ricard

2003-01-01

Recent theoretical approaches to community structure and dynamics reveal that many large-scale features of community structure (such as species-rank distributions and species-area relations) can be explained by a so-called neutral model. Using this approach, species are taken to be equivalent and trophic relations are not taken into account explicitly. Here we provide a general analytic solution to the local community model of Hubbell's neutral theory of biodiversity by recasting it as an urn model i.e.a Markovian description of states and their transitions. Both stationary and time-dependent distributions are analysed. The stationary distribution -- also called the zero-sum multinomial -- is given in closed form. An approximate form for the time-dependence is obtained by using an expansion of the master equation. The temporal evolution of the approximate distribution is shown to be a good representation for the true temporal evolution for a large range of parameter values.

1. Segregation and clustering of solutes at grain boundaries in Mg–rare earth solid solutions

The present study validates the previously reported investigations about segregation of rare-earth (RE) elements at grain boundaries in Mg–RE alloys and ultimately provides a direct visualization of the distribution of the solute atoms in the structure of a Mg–Gd alloy. It is demonstrated that Gd forms a solid solution within the Mg matrix in addition to substantial segregation at high-angle grain boundaries in the form of 1–2 nm clusters, with a postulated face-centered cubic Gd structure. The results suggest significant implications for the texture development during alloy processing and recrystallization, and thus for the mechanical behavior and properties of Mg–RE alloys

2. Measurement of Actinides in Molybdenum-99 Solution Analytical Procedure

Soderquist, Chuck Z. [Pacific Northwest National Lab. (PNNL), Richland, WA (United States); Weaver, Jamie L. [Pacific Northwest National Lab. (PNNL), Richland, WA (United States)

2015-11-01

This document is a companion report to a previous report, PNNL 24519, Measurement of Actinides in Molybdenum-99 Solution, A Brief Review of the Literature, August 2015. In this companion report, we report a fast, accurate, newly developed analytical method for measurement of trace alpha-emitting actinide elements in commercial high-activity molybdenum-99 solution. Molybdenum-99 is widely used to produce 99mTc for medical imaging. Because it is used as a radiopharmaceutical, its purity must be proven to be extremely high, particularly for the alpha emitting actinides. The sample of 99Mo solution is measured into a vessel (such as a polyethylene centrifuge tube) and acidified with dilute nitric acid. A gadolinium carrier is added (50 µg). Tracers and spikes are added as necessary. Then the solution is made strongly basic with ammonium hydroxide, which causes the gadolinium carrier to precipitate as hydrous Gd(OH)3. The precipitate of Gd(OH)3 carries all of the actinide elements. The suspension of gadolinium hydroxide is then passed through a membrane filter to make a counting mount suitable for direct alpha spectrometry. The high-activity 99Mo and 99mTc pass through the membrane filter and are separated from the alpha emitters. The gadolinium hydroxide, carrying any trace actinide elements that might be present in the sample, forms a thin, uniform cake on the surface of the membrane filter. The filter cake is first washed with dilute ammonium hydroxide to push the last traces of molybdate through, then with water. The filter is then mounted on a stainless steel counting disk. Finally, the alpha emitting actinide elements are measured by alpha spectrometry.

3. Solution of a Problem Linear Plane Elasticity with Mixed Boundary Conditions by the Method of Boundary Integrals

Nahed S. Hussein

2014-01-01

Full Text Available A numerical boundary integral scheme is proposed for the solution to the system of eld equations of plane. The stresses are prescribed on one-half of the circle, while the displacements are given. The considered problem with mixed boundary conditions in the circle is replaced by two problems with homogeneous boundary conditions, one of each type, having a common solution. The equations are reduced to a system of boundary integral equations, which is then discretized in the usual way, and the problem at this stage is reduced to the solution to a rectangular linear system of algebraic equations. The unknowns in this system of equations are the boundary values of four harmonic functions which define the full elastic solution and the unknown boundary values of stresses or displacements on proper parts of the boundary. On the basis of the obtained results, it is inferred that a stress component has a singularity at each of the two separation points, thought to be of logarithmic type. The results are discussed and boundary plots are given. We have also calculated the unknown functions in the bulk directly from the given boundary conditions using the boundary collocation method. The obtained results in the bulk are discussed and three-dimensional plots are given. A tentative form for the singular solution is proposed and the corresponding singular stresses and displacements are plotted in the bulk. The form of the singular tangential stress is seen to be compatible with the boundary values obtained earlier. The efficiency of the used numerical schemes is discussed.

4. Mathematical model of complex technical asymmetric system based on numerical-analytical boundary elements method

Dina V. Lazareva

2015-06-01

Full Text Available A new mathematical model of asymmetric support structure frame type is built on the basis of numerical-analytical boundary elements method (BEM. To describe the design scheme used is the graph theory. Building the model taken into account is the effect of frame members restrained torsion, which presence is due to the fact that these elements are thin-walled. The built model represents a real object as a two-axle semi-trailer platform. To implement the BEM algorithm obtained are analytical expressions of the fundamental functions and vector load components. The effected calculations are based on the semi-trailer two different models, using finite elements and boundary elements methods. The analysis showed that the error between the results obtained on the basis of two numerical methods and experimental data is about 4%, that indicates the adequacy of the proposed mathematical model.

5. Investigation of wall bounded flows using SPH and the unified semi-analytical wall boundary conditions

Mayrhofer, Arno; Violeau, Damien; Ferrand, Martin

2013-01-01

The semi-analytical wall boundary conditions present a mathematically rigorous framework to prescribe the influence of solid walls in SPH for fluid flows. In this paper they are investigated with respect to the skew-adjoint property which implies exact energy conservation. It will be shown that this property holds only in the limit of the continuous SPH approximation, whereas in the discrete SPH formulation it is only approximately true, leading to numerical noise. This noise, interpreted as form of "turbulence", is treated using an additional volume diffusion term in the continuity equation which we show is equivalent to an approximate Riemann solver. Subsequently two extensions to the boundary conditions are presented. The first dealing with a variable driving force when imposing a volume flux in a periodic flow and the second showing a generalization of the wall boundary condition to Robin type and arbitrary-order interpolation. Two modifications for free-surface flows are presented for the volume diffusio...

6. General analytical solutions for DC/AC circuit network analysis

Rubido, Nicolás; Baptista, Murilo S

2014-01-01

In this work, we present novel general analytical solutions for the currents that are developed in the edges of network-like circuits when some nodes of the network act as sources/sinks of DC or AC current. We assume that Ohm's law is valid at every edge and that charge at every node is conserved (with the exception of the source/sink nodes). The resistive, capacitive, and/or inductive properties of the lines in the circuit define a complex network structure with given impedances for each edge. Our solution for the currents at each edge is derived in terms of the eigenvalues and eigenvectors of the Laplacian matrix of the network defined from the impedances. This derivation also allows us to compute the equivalent impedance between any two nodes of the circuit and relate it to currents in a closed circuit which has a single voltage generator instead of many input/output source/sink nodes. Contrary to solving Kirchhoff's equations, our derivation allows to easily calculate the redistribution of currents that o...

7. A Priori Estimates for Solutions of Boundary Value Problems for Fractional-Order Equations

Alikhanov, A A

2011-01-01

We consider boundary value problems of the first and third kind for the diffusionwave equation. By using the method of energy inequalities, we find a priori estimates for the solutions of these boundary value problems.

8. LIMIT BEHAVIOUR OF SOLUTIONS TO EQUIVALUED SURFACE BOUNDARY VALUE PROBLEM FOR PARABOLIC EQUATIONS

LI Fengquan

2002-01-01

In this paper, we discuss the limit behaviour of solutions to equivalued surface boundary value problem for parabolic equations when the equivalued surface boundary shrinks to a point and the space dimension of the domain is two or more.

9. Analytical solutions of the planar cyclic voltammetry process for two soluble species with equal diffusivities and fast electron transfer using the method of eigenfunction expansions

Adib Samin; Erik Lahti; Jinsuo Zhang

2015-01-01

Cyclic voltammetry is a powerful tool that is used for characterizing electrochemical processes. Models of cyclic voltammetry take into account the mass transport of species and the kinetics at the electrode surface. Analytical solutions of these models are not well-known due to the complexity of the boundary conditions. In this study we present closed form analytical solutions of the planar voltammetry model for two soluble species with fast electron transfer and equal diffusivities using th...

10. POLYNOMIAL SOLUTIONS TO PIEZOELECTRIC BEAMS(Ⅱ)--ANALYTICAL SOLUTIONS TO TYPICAL PROBLEMS

DING Hao-jiang; JIANG Ai-min

2005-01-01

For the orthotropic piezoelectric plane problem, a series of piezoelectric beams is solved and the corresponding analytical solutions are obtained with the trialand-error method on the basis of the general solution in the case of three distinct eigenvalues, in which all displacements, electrical potential, stresses and electrical displacements are expressed by three displacement functions in terms of harmonic polynomials. These problems are cantilever beam with cross force and point charge at free end, cantilever beam and simply-supported beam subjected to uniform loads on the upper and lower surfaces, and cantilever beam subjected to linear electrical potential.

11. Concerning an analytical solution of some families of Kepler’s transcendental equation

Slavica M. Perovich

2016-03-01

Full Text Available The problem of finding an analytical solution of some families of Kepler transcendental equation is studied in some detail, by the Special Trans Functions Theory – STFT. Thus, the STFT mathematical approach in the form of STFT iterative methods with a novel analytical solutions are presented. Structure of the STFT solutions, numerical results and graphical simulations confirm the validity of the basic principle of the STFT. In addition, the obtained analytical results are compared with the calculated values of other analytical methods for alternative proving its significance. Undoubtedly, the proposed novel analytical approach implies qualitative improvement in comparison with conventional numerical and analytical methods.

12. An approximate analytical solution of free convection problem for vertical isothermal plate via transverse coordinate Taylor expansion

Leble, Sergey

2013-01-01

The model under consideration is based on approximate analytical solution of two dimensional stationary Navier-Stokes and Fourier-Kirchhoff equations. Approximations are based on the typical for natural convection assumptions: the fluid noncompressibility and Bousinesq approximation. We also assume that ortogonal to the plate component (x) of velocity is neglectible small. The solution of the boundary problem is represented as a Taylor Series in $x$ coordinate for velocity and temperature which introduces functions of vertical coordinate (y), as coefficients of the expansion. The correspondent boundary problem formulation depends on parameters specific for the problem: Grashoff number, the plate height (L) and gravity constant. The main result of the paper is the set of equations for the coefficient functions for example choice of expansion terms number. The nonzero velocity at the starting point of a flow appears in such approach as a development of convecntional boundary layer theory formulation.

13. Food Adulteration: From Vulnerability Assessment to New Analytical Solutions.

Cavin, Christophe; Cottenet, Geoffrey; Blancpain, Carine; Bessaire, Thomas; Frank, Nancy; Zbinden, Pascal

2016-01-01

Crises related to the presence of melamine in milk or horse meat in beef have been a wake-up call to the whole food industry showing that adulteration of food raw materials is a complex issue. By analysing the situation, it became clear that the risk-based approach applied to ensure the safety related to chemical contaminants in food is not adequate for food fraud. Therefore, a specific approach has been developed to evaluate adulteration vulnerabilities within the food chain. Vulnerabilities will require the development of new analytical solutions. Fingerprinting methodologies can be very powerful in determining the status of a raw material without knowing the identity of each constituent. Milk adulterated by addition of adulterants with very different chemical properties could be detected rapidly by Fourier-transformed mid-infrared spectroscopy (FT-mid-IR) fingerprinting technology. In parallel, a fast and simple multi-analytes liquid-chromatography tandem mass-spectrometry (LC/MS-MS) method has been developed to detect either high levels of nitrogen-rich compounds resulting from adulteration or low levels due to accidental contamination either in milk or in other sensitive food matrices. To verify meat species authenticity, DNA-based methods are preferred for both raw ingredients and processed food. DNA macro-array, and more specifically the Meat LCD Array have showed efficient and reliable meat identification, allowing the simultaneous detection of 32 meat species. While the Meat LCD Array is still a targeted approach, DNA sequencing is a significant step towards an untargeted one. PMID:27198809

14. Analytical Solution and Physics of a Propellant Damping Device

Yang, H. Q.; Peugeot, John

2011-01-01

NASA design teams have been investigating options for "detuning" Ares I to prevent oscillations originating in the vehicle solid-rocket main stage from synching up with the natural resonance of the rest of the vehicle. An experimental work started at NASA MSFC center in 2008 using a damping device showed great promise in damping the vibration level of an 8 resonant tank. However, the mechanisms of the vibration damping were not well understood and there were many unknowns such as the physics, scalability, technology readiness level (TRL), and applicability for the Ares I vehicle. The objectives of this study are to understand the physics of intriguing slosh damping observed in the experiments, to further validate a Computational Fluid Dynamics (CFD) software in propellant sloshing against experiments with water, and to study the applicability and efficiency of the slosh damper to a full scale propellant tank and to cryogenic fluids. First a 2D fluid-structure interaction model is built to model the system resonance of liquid sloshing and structure vibration. A damper is then added into the above model to simulate experimentally observed system damping phenomena. Qualitative agreement is found. An analytical solution is then derived from the Newtonian dynamics for the thrust oscillation damper frequency, and a slave mass concept is introduced in deriving the damper and tank interaction dynamics. The paper will elucidate the fundamental physics behind the LOX damper success from the derivation of the above analytical equation of the lumped Newtonian dynamics. Discussion of simulation results using high fidelity multi-phase, multi-physics, fully coupled CFD structure interaction model will show why the LOX damper is unique and superior compared to other proposed mitigation techniques.

15. An analytical study of the instability of a superelastic shape memory alloy cylinder subject to practical boundary conditions

In this paper, we study phase transitions in a slender circular cylinder composed of a compressible hyperelastic material with a non-convex strain energy function. We aim to construct asymptotic solutions based on an axisymmetrical three-dimensional setting and use the results to describe the key features observed in the experiments by others. The problem of the solution bifurcations of the governing nonlinear partial differential equations (PDEs) is solved through a novel approach involving coupled series–asymptotic expansions. We derive the normal form equation of the original complicated system of nonlinear PDEs. By writing the normal form equation into a first-order dynamical system and with a phase-plane analysis, we deduce the global bifurcation properties and solve the boundary-value problem analytically. The asymptotic solutions in terms of integrals are obtained. The engineering stress–strain curve plotted from the asymptotic solutions can capture some key features of the curve measured in the experiments. It appears that the asymptotic solutions obtained shed certain light on the instability phenomena associated with phase transitions in a cylinder. Also, an important feature of this work is that we consider the clamped end conditions, which are more practical but rarely used in the literature for phase transition problems

16. The BV solution of the parabolic equation with degeneracy on the boundary

Zhan Huashui

2016-12-01

Full Text Available Consider a parabolic equation which is degenerate on the boundary. By the degeneracy, to assure the well-posedness of the solutions, only a partial boundary condition is generally necessary. When 1 ≤ α < p – 1, the existence of the local BV solution is proved. By choosing some kinds of test functions, the stability of the solutions based on a partial boundary condition is established.

17. Embedded Analytical Solutions Improve Accuracy in Convolution-Based Particle Tracking Models using Python

Starn, J. J.

2013-12-01

Particle tracking often is used to generate particle-age distributions that are used as impulse-response functions in convolution. A typical application is to produce groundwater solute breakthrough curves (BTC) at endpoint receptors such as pumping wells or streams. The commonly used semi-analytical particle-tracking algorithm based on the assumption of linear velocity gradients between opposing cell faces is computationally very fast when used in combination with finite-difference models. However, large gradients near pumping wells in regional-scale groundwater-flow models often are not well represented because of cell-size limitations. This leads to inaccurate velocity fields, especially at weak sinks. Accurate analytical solutions for velocity near a pumping well are available, and various boundary conditions can be imposed using image-well theory. Python can be used to embed these solutions into existing semi-analytical particle-tracking codes, thereby maintaining the integrity and quality-assurance of the existing code. Python (and associated scientific computational packages NumPy, SciPy, and Matplotlib) is an effective tool because of its wide ranging capability. Python text processing allows complex and database-like manipulation of model input and output files, including binary and HDF5 files. High-level functions in the language include ODE solvers to solve first-order particle-location ODEs, Gaussian kernel density estimation to compute smooth particle-age distributions, and convolution. The highly vectorized nature of NumPy arrays and functions minimizes the need for computationally expensive loops. A modular Python code base has been developed to compute BTCs using embedded analytical solutions at pumping wells based on an existing well-documented finite-difference groundwater-flow simulation code (MODFLOW) and a semi-analytical particle-tracking code (MODPATH). The Python code base is tested by comparing BTCs with highly discretized synthetic steady

18. A study on evaluation and analytical methods for groundwater flow with considering sea/fresh-water boundary. 1

Sea/fresh-water boundary caused by density and concentration balance of sea-water and fresh-water is an important item for groundwater flow evaluation in deep underground near the coast. Also, in order to evaluate groundwater quality, it is important to understand the characteristics of sea/fresh-water boundary, for example boundary shape, salt distribution. In order to establish the evaluation and analytical methods for groundwater flow with considering sea/fresh-water boundary, we investigated the following items in this study. (1) Literature survey and data collection. (2) Investigation of analytical methods. (3) Planning of further study. (author). 78 refs

19. A study on evaluation and analytical methods for groundwater flow with considering sea/fresh-water boundary. 1

Anezaki, S. [Taisei Corp., Tokyo (Japan)

1998-03-01

Sea/fresh-water boundary caused by density and concentration balance of sea-water and fresh-water is an important item for groundwater flow evaluation in deep underground near the coast. Also, in order to evaluate groundwater quality, it is important to understand the characteristics of sea/fresh-water boundary, for example boundary shape, salt distribution. In order to establish the evaluation and analytical methods for groundwater flow with considering sea/fresh-water boundary, we investigated the following items in this study. (1) Literature survey and data collection. (2) Investigation of analytical methods. (3) Planning of further study. (author). 78 refs.

20. Grain boundary structure and solute segregation in titanium-doped sapphire bicrystals

Taylor, Seth T.

2002-05-17

Solute segregation to ceramic grain boundaries governs material processing and microstructure evolution, and can strongly influence material properties critical to engineering performance. Understanding the evolution and implications of grain boundary chemistry is a vital component in the greater effort to engineer ceramics with controlled microstructures. This study examines solute segregation to engineered grain boundaries in titanium-doped sapphire (Al2O3) bicrystals, and explores relationships between grain boundary structure and chemistry at the nanometer scale using spectroscopic and imaging techniques in the transmission electron microscope (TEM). Results demonstrate dramatic changes in solute segregation stemming from small fluctuations in grain boundary plane and structure. Titanium and silicon solute species exhibit strong tendencies to segregate to non-basal and basal grain boundary planes, respectively. Evidence suggests that grain boundary faceting occurs in low-angle twis t boundaries to accommodate nonequilibrium solute segregation related to slow specimen cooling rates, while faceting of tilt grain boundaries often occurs to expose special planes of the coincidence site lattice (CSL). Moreover, quantitative analysis of grain boundary chemistry indicates preferential segregation of charged defects to grain boundary dislocations. These results offer direct proof that static dislocations in ionic materials can assume a net charge, and emphasize the importance of interactions between charged point, line, and planar defects in ionic materials. Efforts to understand grain boundary chemistry in terms of space charge theory, elastic misfit and nonequilibrium segregation are discussed for the Al2O3 system.

1. New analytic solutions for modeling vertical gravity gradient anomalies

Kim, Seung-Sep; Wessel, Paul

2016-05-01

Modern processing of satellite altimetry for use in marine gravimetry involves computing the along-track slopes of observed sea-surface heights, projecting them into east-west and north-south deflection of the vertical grids, and using Laplace's equation to algebraically obtain a grid of the vertical gravity gradient (VGG). The VGG grid is then integrated via overlapping, flat Earth Fourier transforms to yield a free-air anomaly grid. Because of this integration and associated edge effects, the VGG grid retains more short-wavelength information (e.g., fracture zone and seamount signatures) that is of particular importance for plate tectonic investigations. While modeling of gravity anomalies over arbitrary bodies has long been a standard undertaking, similar modeling of VGG anomalies over oceanic features is not commonplace yet. Here we derive analytic solutions for VGG anomalies over simple bodies and arbitrary 2-D and 3-D sources. We demonstrate their usability in determining mass excess and deficiency across the Mendocino fracture zone (a 2-D feature) and find the best bulk density estimate for Jasper seamount (a 3-D feature). The methodologies used herein are implemented in the Generic Mapping Tools, available from gmt.soest.hawaii.edu.

2. Analytical solutions for peak and residual uplift resistance of pipelines

Nixon, J.F. [Nixon Geotech Ltd., Calgary, AB (Canada); Oswell, J.M. [Naviq Consulting Inc., Calgary, AB (Canada)

2010-07-01

Frost heave can occur on cold pipelines that traverse unfrozen, non permafrost terrain. The stresses experienced by the pipeline are partly a function of the strength of the soil on the non heaving side of the frozen-unfrozen interface. This paper proposed three analytical solutions to estimate the soil uplift resistance by considering the pipeline and soil to act similar to a strip footing, a punching shear failure, and by considering the formation of horizontal crack emanating from the spring line of the pipe. Peak uplift resistance and residual uplift resistance were discussed. Results for full scale pipe and for laboratory scale model pipes were presented, with particular reference to cover depth, temperature and crack width; and limits to residual uplift resistance. It was concluded that the peak uplift resistance and the residual uplift resistance are generally independent and controlled by different factors. The peak resistance is related directly to pipe diameter, and less strongly dependent on springline depth. It is also strongly dependent on soil temperature. However, the residual uplift resistance is strongly dependent on burial depth, weakly dependent on pipe displacement rate and also on soil temperature. 15 refs., 19 figs.

3. Approximate analytic solutions to the NPDD: Short exposure approximations

Close, Ciara E.; Sheridan, John T.

2014-04-01

There have been many attempts to accurately describe the photochemical processes that take places in photopolymer materials. As the models have become more accurate, solving them has become more numerically intensive and more 'opaque'. Recent models incorporate the major photochemical reactions taking place as well as the diffusion effects resulting from the photo-polymerisation process, and have accurately described these processes in a number of different materials. It is our aim to develop accessible mathematical expressions which provide physical insights and simple quantitative predictions of practical value to material designers and users. In this paper, starting with the Non-Local Photo-Polymerisation Driven Diffusion (NPDD) model coupled integro-differential equations, we first simplify these equations and validate the accuracy of the resulting approximate model. This new set of governing equations are then used to produce accurate analytic solutions (polynomials) describing the evolution of the monomer and polymer concentrations, and the grating refractive index modulation, in the case of short low intensity sinusoidal exposures. The physical significance of the results and their consequences for holographic data storage (HDS) are then discussed.

4. New possibilities of accurate particle characterisation by applying direct boundary models to analytical centrifugation

Walter, Johannes; Thajudeen, Thaseem; Süß, Sebastian; Segets, Doris; Peukert, Wolfgang

2015-04-01

Analytical centrifugation (AC) is a powerful technique for the characterisation of nanoparticles in colloidal systems. As a direct and absolute technique it requires no calibration or measurements of standards. Moreover, it offers simple experimental design and handling, high sample throughput as well as moderate investment costs. However, the full potential of AC for nanoparticle size analysis requires the development of powerful data analysis techniques. In this study we show how the application of direct boundary models to AC data opens up new possibilities in particle characterisation. An accurate analysis method, successfully applied to sedimentation data obtained by analytical ultracentrifugation (AUC) in the past, was used for the first time in analysing AC data. Unlike traditional data evaluation routines for AC using a designated number of radial positions or scans, direct boundary models consider the complete sedimentation boundary, which results in significantly better statistics. We demonstrate that meniscus fitting, as well as the correction of radius and time invariant noise significantly improves the signal-to-noise ratio and prevents the occurrence of false positives due to optical artefacts. Moreover, hydrodynamic non-ideality can be assessed by the residuals obtained from the analysis. The sedimentation coefficient distributions obtained by AC are in excellent agreement with the results from AUC. Brownian dynamics simulations were used to generate numerical sedimentation data to study the influence of diffusion on the obtained distributions. Our approach is further validated using polystyrene and silica nanoparticles. In particular, we demonstrate the strength of AC for analysing multimodal distributions by means of gold nanoparticles.

5. An analytical solution of non-Fourier Chen-Holmes bioheat transfer equation

GOU Chenhua; CAI Ruixian

2005-01-01

An algebraically explicit analytical solution with heat wave effect is derived for the non-Fourier bioheat transfer Chen-Holmes model. Besides its important theoretical meaning (for example, to expand the understanding of heat wave phenomena in living tissues), this analytical solution is also valuable as the benchmark solution to check the numerical calculation and to develop various numerical computational approaches.

6. Analytical solutions of heat transfer for laminar flow in rectangular channels

Rybiński Witold

2014-12-01

Full Text Available The paper presents two analytical solutions namely for Fanning friction factor and for Nusselt number of fully developed laminar fluid flow in straight mini channels with rectangular cross-section. This type of channels is common in mini- and microchannel heat exchangers. Analytical formulae, both for velocity and temperature profiles, were obtained in the explicit form of two terms. The first term is an asymptotic solution of laminar flow between parallel plates. The second one is a rapidly convergent series. This series becomes zero as the cross-section aspect ratio goes to infinity. This clear mathematical form is also inherited by the formulae for friction factor and Nusselt number. As the boundary conditions for velocity and temperature profiles no-slip and peripherally constant temperature with axially constant heat flux were assumed (H1 type. The velocity profile is assumed to be independent of the temperature profile. The assumption of constant temperature at the channel’s perimeter is related to the asymptotic case of channel’s wall thermal resistance: infinite in the axial direction and zero in the peripheral one. It represents typical conditions in a minichannel heat exchanger made of metal.

7. Semi-analytical solutions for the effect of well shut down on rock stability

Han, G.; Ioannidis, M.; Dusseault, M.B. [Waterloo Univ., ON (Canada)

2002-06-01

This paper presents three newly developed models to describe the effect of well shut down (or sharp change of production rate) on rock stress distributions. The methods are particularly useful in poorly consolidated rock around a wellbore which may become unstable after the process of well shut down and restart. Analytical solutions for quasi-static pressure recovery processes in a bounded oil reservoir are combined with a poro-elastic geomechanics model in which pressure fluctuations inside the wellbore provide a boundary condition to the formation outside the wellbore. Analytical solutions explain the direct relationships between fluid properties, rock properties and production parameters. Stress fluctuations are examined in the context of rock stability changes resulting from dynamic loading. Model calculations show that the fluctuations of effective stresses and shear stress could reach several hundred kPa due to pressure waves created by the water hammer effect inside a wellbore. The models can be used to quantify the effects of pressure oscillation, resulting from operation at the surface, on the stability of underground rock. It is noted that more research is needed to obtain accurate information on the dynamic response of unconsolidated sandstones to rapidly oscillating pressures before this method can be widely used. The model can be used to evaluate risks such as rock instability. It can also be used to choose which wells may start sanding if they are shut down or started up abruptly. 13 refs., 1 tab., 7 figs., 1 append.

8. Boundary Integral Solutions to Three-Dimensional Unconfined Darcy's Flow

Lennon, Gerard P.; Liu, Philip L.-F.; Liggett, James A.

1980-08-01

The boundary integral equation method (BIEM) is used to solve three-dimensional potential flow problems in porous media. The problems considered here are time dependent and have a nonlinear boundary condition on the free surface. The entire boundary, including the moving free surface, discretized into linear finite elements for the purpose of evaluating the boundary integrals. The technique allows transient, three-dimensional problems to be solved with reasonable computational costs. Numerical examples include recharge through rectangular and circular areas and seepage flow from a surface pond. The examples are used to illustrate the method and show the nonlinear effects.

9. Local solution for a class of mixed boundary value problems

A local method is developed for solving locally partial differential equations with mixed boundary conditions. The method is based on a heuristic idea, properties of diffusion processes, stopping times and the Ito formula for semimartingales. According to the heuristic idea, the diffusion process used for solving locally a partial differential with mixed boundary conditions is stopped when it reaches a Neumann boundary and then restarted inside the domain of definition of this equation at a point depending on the Neumann conditions. The proposed method is illustrated and its accuracy assessed by two simple numerical examples solving locally mixed boundary value problems in one and two space dimensions

10. An analytical solution for one-dimensional contaminant diffusion through multi-layered system and its applications

Chen, Yunmin; Xie, Haijian; Ke, Han; Chen, Renpeng

2009-09-01

An analytical solution for one-dimensional contaminant diffusion through multi-layered media is derived regarding the change of the concentration of contaminants at the top boundary with time. The model accounts for the arbitrary initial conditions and the conditions of zero concentration and zero mass flux on the bottom boundary. The average degree of diffusion of the layered system is introduced on the basis of the solution. The results obtained by the presented analytical solutions agree well with those obtained by the numerical methods presented in the literature papers. The application of the analytical solution to the problem of landfill liner design is illustrated by considering a composite liner consisting of geomembrane and compacted clay liner. The results show that the 100-year mass flux of benzene at the bottom of the composite liner is 45 times higher than that of acetone for the same composite liner. The half-life of the contaminant has a great influence on the solute flux of benzene diffused into the underlying aquifer. Results also indicates that an additional 2.9-5.0 m of the conventional (untreated) compacted clay liner under the geomembrane is required to achieve the same level of protection as provided by 0.60 m of the Hexadecyltrimethylammonium (HDTMA)-treated compacted clay liners in conjunction with the geomembrane. Applications of the solution are also presented in the context of a contaminated two-layered media to demonstrate that different boundary and initial conditions can greatly affect the decontamination rate of the problem. The method is relatively simple to apply and can be used for performing equivalency analysis of landfill liners, preliminary design of groundwater remediation system, evaluating experimental results, and verifying more complex numerical models.