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Exact Analytical Solution (exact + analytical_solution)

Distribution by Scientific Domains

Engineering	50%
Chemistry	40%

Selected Abstracts

Theoretical and numerical analyses of convective instability in porous media with temperature-dependent viscosity

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 10 2003
Ge Lin
Abstract Exact analytical solutions of the critical Rayleigh numbers have been obtained for a hydrothermal system consisting of a horizontal porous layer with temperature-dependent viscosity. The boundary conditions considered are constant temperature and zero vertical Darcy velocity at both the top and bottom of the layer. Not only can the derived analytical solutions be readily used to examine the effect of the temperature-dependent viscosity on the temperature-gradient driven convective flow, but also they can be used to validate the numerical methods such as the finite-element method and finite-difference method for dealing with the same kind of problem. The related analytical and numerical results demonstrated that the temperature-dependent viscosity destabilizes the temperature-gradient driven convective flow and therefore, may affect the ore body formation and mineralization in the upper crust of the Earth. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Exact analytical solutions to the Kratzer potential by the asymptotic iteration method

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 3 2007
O. Bayrak
Abstract For any n and l values, we present a simple exact analytical solution of the radial Schrödinger equation for the Kratzer potential within the framework of the asymptotic iteration method (AIM). The exact bound-state energy eigenvalues (Enl) and corresponding eigenfunctions (Rnl) are calculated for various values of n and l quantum numbers for CO, NO, O2, and I2 diatomic molecules. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007 [source]

Approximate discharge for constant head test with recharging boundary

GROUND WATER, Issue 3 2005
Philippe Renard
The calculation of the discharge to a constant drawdown well or tunnel in the presence of an infinite linear constant head boundary in an ideal confined aquifer usually relies on the numerical inversion of a Laplace transform solution. Such a solution is used to interpret constant head tests in wells or to roughly estimate ground water inflow into tunnels. In this paper, a simple approximate solution is proposed. Its maximum relative error is on the order of 2% as compared to the exact analytical solution. The approximation is a weighted mean between the early-time and late-time asymptotes. [source]

Stability analysis of the Crank,Nicholson method for variable coefficient diffusion equation

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 1 2007
Charles Tadjeran
Abstract The Crank,Nicholson method is a widely used method to obtain numerical approximations to the diffusion equation due to its accuracy and unconditional stability. When the diffusion coefficient is not a constant, the general approach is to obtain a discretization for the PDE in the same manner as the case for constant coefficients. In this paper, we show that the manner of this discretization may impact the stability of the resulting method and could lead to instability of the numerical solution. It is shown that the classical Crank,Nicholson method will fail to be unconditionally stable if the diffusion coefficient is computed at the time gridpoints instead of at the midpoints of the temporal subinterval. A numerical example is presented and compared with the exact analytical solution to examine its divergence. Copyright © 2006 John Wiley & Sons, Ltd. [source]

Exact analytical solutions to the Kratzer potential by the asymptotic iteration method

Calculation of the instrumental function in X-ray powder diffraction

JOURNAL OF APPLIED CRYSTALLOGRAPHY, Issue 3 2006
A. D. Zuev
A new method for calculating the total instrumental function of a conventional Bragg,Brentano diffractometer has been developed. The method is based on an exact analytical solution, derived from diffraction optics, for the contribution of each incident ray to the intensity registered by a detector of limited size. Because an incident ray is determined by two points (one is related to the source of the X-rays and the other to the sample) the effects of the coupling of specific instrumental functions, for example, equatorial and axial divergence instrumental functions, are treated together automatically. The intensity at any arbitrary point of the total instrumental profile is calculated by integrating the intensities over two simple rectangular regions: possible point positions on the source and possible point positions on the sample. The effects of Soller slits, a monochromator and sample absorption can also be taken into account. The main difference between the proposed method and the convolutive approach (in which the line profile is synthesized by convolving the specific instrumental functions) lies in the fact that the former provides an exact solution for the total instrumental function (exact solutions for specific instrumental functions can be obtained as special cases), whereas the latter is based on the approximations for the specific instrumental functions, and their coupling effects after the convolution are unknown. Unlike the ray-tracing method, in the proposed method the diffracted rays contributing to the registered intensity are considered as combined (part of the diffracted cone) and, correspondingly, the contribution to the instrumental line profile is obtained analytically for this part of the diffracted cone and not for a diffracted unit ray as in ray-tracing simulations. [source]

Diffraction peak profiles from spherical crystallites with lognormal size distribution

JOURNAL OF APPLIED CRYSTALLOGRAPHY, Issue 5 2003
T. Ida
An efficient and accurate method to evaluate the theoretical diffraction peak profiles from spherical crystallites with lognormal size distribution (SLN profile) is presented. Precise results can be obtained typically by an eight-term numerical integral for any values of the parameters, by applying an appropriate substitution of the variable to the integral formula. The calculated SLN profiles have been verified by comparison with those calculated by inverse Fourier transform from the exact analytical solution of the Fourier-transformed SLN profile. It has been found that the shape of the SLN profile strongly depends on the variance of size distribution. When the logarithmic standard deviation , of the size distribution is close to 0.76, the SLN profile becomes close to a Lorentzian profile, and `super-Lorentzian' profiles are predicted for larger values of ,, as has been concluded by Popa & Balzar [J. Appl. Cryst. (2002), 35, 338,346]. The intrinsic diffraction peak profiles of an SiC powder sample obtained by deconvolution of the instrumental function have certainly shown `super-Lorentzian' line profiles, and they are well reproduced by the SLN profile for the value , = 0.93. [source]

Analysis of optical and terahertz multilayer systems using microwave and feedback thoery

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS, Issue 5 2009
Dong-Joon Lee
Abstract The principles of microwave and feedback theory are independently applied to the analysis of both optical and terahertz-regime multilayer systems. An analogy between the two approaches is drawn, and useful recursion relations, along with a signal-flow approach, are presented for both reflection and transmission cases. These relations, in terms of S-parameters, allow an exact analytical solution for even arbitrary, active, stratified structures, not only for any wavelength in the radio-frequency spectrum, but also for optical wavelengths. This approach also provides a bridge between the microwave and optical bands and leads to beneficial design solutions for intermediate bands such as the THz regime. Comparisons with conventional methodologies are provided using practical multilayer simulations. In addition, graphical design techniques from microwave theory are used along with examples for efficient design and understanding. © 2009 Wiley Periodicals, Inc. Microwave Opt Technol Lett 51: 1308,1312, 2009; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.24301 [source]

Modified method of characteristics for solving population balance equations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2003
Laurent Pilon
Abstract This paper presents a new numerical method for solving the population balance equation using the modified method of characteristics. Aggregation and break-up are neglected but the density function variations in the three-dimensional space and its dependence on the external fields are accounted for. The method is an interpretation of the Lagrangian approach. Based on a pre-specified grid, it follows the particles backward in time as opposed to forward in the case of traditional method of characteristics. Unlike the direct marching method, the inverse marching method uses a fixed grid thus, making it compatible with other numerical schemes (e.g. finite-volume, finite elements) that may be used to solve other coupled equations such as the mass, momentum, and energy conservation equations. The numerical solutions are compared with the exact analytical solutions for simple one-dimensional flow cases. Very good agreement between the numerical and the theoretical solutions has been obtained confirming the validity of the numerical procedure and the associated computer program. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Linear waves in a symmetric equatorial channel

THE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 624 2007
C. Erlick
Abstract Using a scaling that allows us to separate the effects of the gravity wave speed from those of boundary location, we reduce the equations for linear waves in a zonal channel on the equatorial beta-plane to a single-parameter eigenvalue problem of the Schrödinger type with parabolic potential. The single parameter can be written , = (,,)2/,1/2, where , = gH(2,R),2, ,, is half the channel width, g is the acceleration due to gravity, H is the typical height of the troposphere or ocean, , is the Earth's rotational frequency, and R is the Earth's radius. The Schrödinger-type equation has exact analytical solutions in the limits , , 0 and , , ,, and one can use these to write an approximate expression for the solution that is accurate everywhere to within 4%. In addition to the simple expression for the eigenvalues, the concise and unified theory also yields explicit expressions for the associated eigenfunctions, which are pure sinusoidal in the , , 0 limit and Gaussian in the , , , limit. Using the same scaling, we derive an eigenvalue formulation for linear waves in an equatorial channel on the sphere with a simple explicit formula for the dispersion relation accurate to O{(,,)2}. From this, we find that the phase velocity of the anti-Kelvin mode on the sphere differs by as much as 10% from , ,1/2. Integrating the linearized shallow-water equations on the sphere, we find that for for larger , and ,,, the phase speeds of all of the negative modes differ substantially from their phase speeds on the beta-plane. Furthermore, the dispersion relations of all of the waves in the equatorial channel on the sphere approach those on the unbounded sphere in a smooth asymptotic fashion, which is not true for the equatorial channel on the beta-plane. Copyright © 2007 Royal Meteorological Society [source]