The interaction effect between two neutron absorbers using Wallace's method

2000 ◽  
Vol 78 (12) ◽  
pp. 1035-1054 ◽  
Author(s):  
M MR Williams
Keyword(s):  
Integral Equation ◽  
Transport Equation ◽  
Absorption Rate ◽  
Exact Result ◽  
Integral Form ◽  
High Accuracy ◽  
Approximation Procedure ◽  
Plate Separation ◽  
Source Sink ◽  
Constant Source

The dependence of Galanin's constant on fuel-plate separation is studied. Galanin's constant arises in the source–sink method of lattice calculations and normally its value for an isolated fuel plate is used. Our calculations therefore enable the accuracy of that assumption to be assessed. To illustrate the problem, two absorbing and scattering plates, embedded in an infinite moderator and separated by a distance b are considered, with a source in the moderator region. The calculations are carried out via the integral form of the transport equation using the little-known method of Wallace. This approach enables the integral equation to be cast into a particularly useful form for evaluation of the flux and absorption rate in the plates. A variational method and a numerical solution provide results that show that there are significant deviations from the isolated plate approximation. Tables and figures are provided to illustrate these conclusions. In addition to the main result, we take the opportunity to show how Wallace's method can be used to simplify the transport equation for an infinitely reflected slab reactor. We also obtain results for the flux in a half-space in which there is a constant source, using a novel approximation procedure. Comparison with the exact result shows high accuracy.

scholarly journals Entire Solutions of an Integral Equation in R5

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Xin Feng ◽  
Xingwang Xu
Keyword(s):  
Integral Equation ◽  
Closed Form ◽  
Integral Form ◽  
Entire Solution ◽  
Entire Solutions ◽  
Positive Entire Solution ◽  
Moving Sphere Method

We will study the entire positive C0 solution of the geometrically and analytically interesting integral equation: u(x)=1/C5∫R5‍|x-y|u-q(y)dy with 0<q in R5. We will show that only when q=11, there are positive entire solutions which are given by the closed form u(x)=c(1+|x|2)1/2 up to dilation and translation. The paper consists of two parts. The first part is devoted to showing that q must be equal to 11 if there exists a positive entire solution to the integral equation. The tool to reach this conclusion is the well-known Pohozev identity. The amazing cancelation occurred in Pohozev’s identity helps us to conclude the claim. It is this exponent which makes the moving sphere method work. In the second part, as normal, we adopt the moving sphere method based on the integral form to solve the integral equation.

scholarly journals Splash formation at the nose of a smoothly curved body in a stream

1999 ◽  
Vol 40 (4) ◽  
pp. 421-436 ◽  
Author(s):  
E. O. Tuck ◽  
S. T. Simakov
Keyword(s):  
Integral Equation ◽  
Free Surface ◽  
Velocity Potential ◽  
Integral Form ◽  
The Body ◽  
Front Face ◽  
Separation Condition ◽  
Polygonal Approximation ◽  
Side Condition ◽  
Two Dimensional Flow

AbstractIn two-dimensional flow past a body close to a free surface, the upwardly diverted portion may separate to form a splash. We model the nose of such a body by a semi-infinite obstacle of finite draft with a smoothly curved front face. This problem leads to a nonlinear integral equation with a side condition, a separation condition and an integral constraint requiring the far-upstream free surface to be asymptotically plane. The integral equation, called Villat's equation, connects a natural parametrisation of the curved front face with the parametrisation by the velocity potential near the body. The side condition fixes the position of the separation point, whereas the separation condition, known as the Brillouin-Villat condition, imposes a continuity relation to be satisfied at separation. For the described flow we derive the Brillouin-Villat condition in integral form and give a numerical solution to the problem using a polygonal approximation to the front face.

scholarly journals Improvements in Simulations for Radiotherapy Wedge Filter dose and AAA-Convolution Factor Algorithms

2019 ◽  
pp. 194-219
Author(s):  
Francisco Casesnoves
Keyword(s):  
Integral Equation ◽  
Gaussian Model ◽  
Spherical Geometry ◽  
Numerical Data ◽  
Integral Form ◽  
Dose Delivery ◽  
Equation Resolution ◽  
Exponential Factor ◽  
Multi Leaf Collimator ◽  
Series Development

Analytical-convoluted and numerical Gaussian models have been used in recent decades for radiotherapy treatment planning software/calculations, to perform accurately radiation dose delivery –numerical, analytical, or numerical-analytical. The objective of this evoluted-contribution was to obtain an exact dose delivery, 3D analytical-integral-equation solution, for the triple Gaussian model of wedge filters, since previous/initial 2D approximations of other authors, although correct, were not completely exact. Additionally, to set conceptual and mathematical-geometrical differences between the beam modification created by Multi-Leaf Collimator and Wedge Filters, either standard or Conformal. Ever the precision, from mathematical theory algorithms to real laboratory measurements, a series of simulations are presented. The generic triple Gaussian model of Ulmer and Harder sets an Attenuation Exponential Factor, AEF, well approximated in 2 variables, namely, u and z. This evoluted contribution of the research contribution was specially focused on numerical methods and approximation analysis of the integral equation resolution –with extent details about numerical data, Appendix 3. In this paper we set a detailed spatial-spherical geometry discussion/proof towards the determination of a 3D integral form of the delivery dose in water. In other words, with an AEF for magnitude-values of variables u,v, and z. Simulations, based on these new determinations were shown with sharp presentation of the numerical-computational software and functional programming series development. Computing encode techniques are explained with some practical examples for numerical radiotherapy calculus.

Application of Hybrid Subgroup Decomposition Method to Gas Cooled Thermal Reactor Problem

2014 ◽  
Author(s):  
Saam Yasseri ◽  
Farzad Rahnema
Keyword(s):  
Transport Equation ◽  
Computational Efficiency ◽  
Decomposition Method ◽  
High Accuracy ◽  
Numerical Results ◽  
New Method ◽  
Thermal Reactor ◽  
Problem Characteristic ◽  
Group Transport ◽  
Subgroup Decomposition

In this paper, a newly developed hybrid subgroup decomposition method is tested in a 1D problem characteristic of gas cooled thermal reactors (GCR). The new method couples an efficient coarse-group eigenvalue calculation with a set of fine-group transport source iterations to unfold the fine-group flux. It is shown that the new method reproduces the fine-group transport solution by iteratively solving the coarse-group quasi transport equation. The numerical results demonstrate that the new method applied to 1D GCR problem is capable of achieving high accuracy while gaining computational efficiency up to 5 times compared to direct fine-group transport calculations.

An Explicit Second-Order Numerical Scheme to Solve Decoupled Forward Backward Stochastic Equations

2014 ◽  
Vol 4 (4) ◽  
pp. 368-385 ◽  
Author(s):  
Yu Fu ◽  
Weidong Zhao
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Experiments ◽  
Numerical Scheme ◽  
High Accuracy ◽  
Stochastic Equations ◽  
Explicit Scheme ◽  
Second Order ◽  
General Error

AbstractAn explicit numerical scheme is proposed for solving decoupled forward backward stochastic differential equations (FBSDE) represented in integral equation form. A general error inequality is derived for this numerical scheme, which also implies its stability. Error estimates are given based on this inequality, showing that the explicit scheme can be second-order. Some numerical experiments are carried out to illustrate the high accuracy of the proposed scheme.

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