scholarly journals Towards a Direct Numerical Solution of Schrödinger’s Equation for (e, 2e) Reactions

1999 ◽  
Vol 52 (3) ◽  
pp. 621 ◽  
Author(s):  
S. Jones ◽  
A. T. Stelbovics
Keyword(s):  
Differential Equation ◽  
Scattering Amplitudes ◽  
Cross Sections ◽  
Asymptotic Boundary ◽  
Collision Problem ◽  
Model Collision ◽  
Direct Numerical Solution ◽  
Particular Solution ◽  
The Finite Difference Method ◽  
Atomic Centre

The finite-difference method for electron{hydrogen scattering is presented in a simple, easily understood form for a model collision problem in which all angular momentum is neglected. The model Schrödinger equation is integrated outwards from the atomic centre on a grid of fixed spacing h. The number of difference equations is reduced each step outwards using an algorithm due to Poet, resulting in a propagating solution of the partial-differential equation. By imposing correct asymptotic boundary conditions on this general, propagating solution, the particular solution that physically corresponds to scattering is obtained along with the scattering amplitudes. Previous works using finite differences (and finite elements) have extracted scattering amplitudes only for low-level transitions (elastic scattering and n = 2 excitation). If we are to eventually extract ionisation amplitudes, however, the numerical method must remain stable for higher-level transitions. Here we report converged cross sections for transitions up to n = 8, as a first step towards obtaining ionisation (e; 2e) results.

scholarly journals Boundedness of Solutions to Differential Equations of Fourth Order with Oscillatory Restoring and Forcing Terms

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Cemil Tunç ◽  
Muzaffer Ateş
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Fourth Order ◽  
Cauchy Formula ◽  
Boundedness Of Solutions ◽  
Constant Coefficients ◽  
Particular Solution

This paper deals with the boundedness of solutions to a nonlinear differential equation of fourth order. Using the Cauchy formula for the particular solution of nonhomogeneous differential equations with constant coefficients, we prove that the solution and its derivatives up to order three are bounded.

Assessment of the numerical and experimental performance of screw tidal turbines

2018 ◽  
Vol 232 (7) ◽  
pp. 912-925 ◽  
Author(s):  
Hamid Rahmani ◽  
Mojtaba Biglari ◽  
Mohammad Sadegh Valipour ◽  
Kamran Lari
Keyword(s):  
Output Power ◽  
Water Flow ◽  
Cross Sections ◽  
Turbine Blades ◽  
Transient State ◽  
Experimental Results ◽  
Ansys Fluent ◽  
Tidal Turbines ◽  
Collision Problem ◽  
Mesh Independence

This study was aimed at the numerical and experimental modeling of water flow during collision between water and vertical screw turbine blades with different cross sections (i.e. Darrieus, spoon, and airfoil). ANSYS Fluent was used to model water flow under tidal currents in a flume, and mesh independence was ensured after the selection of appropriate geometry. The collision problem was then solved in the transient state, and results on the momentum and power generated by different inlet velocities and different blade cross sections were analyzed. The findings showed that torque and turbine power increased with increasing inlet velocity. Subsequently, a turbine was experimentally created, with cross sections drawn in the numerical model and tested under the same conditions as that imposed on the model. Installing a multimeter on the turbine enabled the generation of turbine power in different dimensions. The resultant power increased with rising turbine dimensions. After obtaining the numerical and experimental results, the value of the output power of the turbine was validated. The validation indicated a 7% difference in output power between the numerical and experimental results, indicating acceptable accuracy.

scholarly journals An application of Ritt’s low power theorem

1977 ◽  
Vol 68 ◽  
pp. 17-19 ◽  
Author(s):  
Michihiko Matsuda
Keyword(s):  
Differential Equation ◽  
General Solution ◽  
Low Power ◽  
Singular Solution ◽  
Algebraic Differential Equation ◽  
Classical Concept ◽  
First Order ◽  
Particular Solutions ◽  
Rigorous Definition ◽  
Particular Solution

AbstractConsider an algebraic differential equation F = 0 of the first order. A rigorous definition will be given to the classical concept of “particular solutions” of F = 0. By Ritt’s low power theorem we shall prove that a singular solution of F = 0 belongs to the general solution of F if and only if it is a particular solution of F = 0.

ON THE FACTORIZATION OF THE SCATTERING OF W BOSONS

2013 ◽  
Vol 28 (12) ◽  
pp. 1330008 ◽  
Author(s):  
ROBERTO FRANCESCHINI
Keyword(s):  
Scattering Amplitudes ◽  
Cross Sections ◽  
W Bosons ◽  
Numerical Calculations ◽  
Physical Origin ◽  
Exact Computation ◽  
Scattering Cross ◽  
Scattering Cross Sections

Recent results on the application of the method of the effective quanta to the case of WW scattering are reviewed. In particular, it is reviewed how effective W bosons can be used to compute scattering amplitudes in a generalized Effective W Approximation (gEWA). The physical origin and the expected size of the discrepancy between the exact computation and the gEWA computation are discussed. Particular attention is devoted to the peculiarities of the formulation of the gEWA for processes that involve longitudinal W bosons. These formal results are checked through explicit numerical calculations of both scattering amplitudes and scattering cross-sections. Furthermore, the use of the gEWA for the interpretation and for the presentation of the measurement of WW scattering at the LHC is briefly discussed.

scholarly journals Exact Longitudinal Vibration Characteristics of Rods with Variable Cross-Sections

2011 ◽  
Vol 18 (4) ◽  
pp. 555-562 ◽  
Author(s):  
Bulent Yardimoglu ◽  
Levent Aydin
Keyword(s):  
Differential Equation ◽  
Cross Section ◽  
Cross Sections ◽  
Longitudinal Vibration ◽  
Transformation Method ◽  
Variable Cross ◽  
Cross Sectional ◽  
Cross Section Area ◽  
Section Area ◽  
Sinusoidal Functions

Longitudinal natural vibration frequencies of rods (or bars) with variable cross-sections are obtained from the exact solutions of differential equation of motion based on transformation method. For the rods having cross-section variations as power of the sinusoidal functions ofax+b, the differential equation is reduced to associated Legendre equation by using the appropriate transformations. Frequency equations of rods with certain cross-section area variations are found from the general solution of this equation for different boundary conditions. The present solutions are benchmarked by the solutions available in the literature for the special case of present cross-sectional variations. Moreover, the effects of cross-sectional area variations of rods on natural characteristics are studied with numerical examples.

Electron energy distributions in uniform electric fields and the Townsend ionization coefficient

1958 ◽  
Vol 244 (1237) ◽  
pp. 166-185 ◽  
Keyword(s):  
Differential Equation ◽  
Cross Section ◽  
Electric Fields ◽  
Cross Sections ◽  
Collision Cross Section ◽  
Present Theory ◽  
Uniform Electric Field ◽  
Ionization Coefficient ◽  
Special Cases ◽  
Wide Range

The second-order differential equation which expresses the equilibrium condition of an electron swarm in a uniform electric field in a gas, the electrons suffering both elastic and inelastic collisions with the gas molecules, is solved by the Jeffreys or W.K.B. method of approximation. The distribution function F(ε) of electrons of energy ε is obtained immediately in a general form involving the elastic and inelastic collision cross-sections and without any restriction on the range of E/p (electric strength/gas pressure) save that introduced in the original differential equation. In almost all applications the approximation is likely to be of high accuracy, and easy to use. Several of the earlier derivations of F(ε) are obtained as special cases. Using the function F(ε) an attempt is made to relate the Townsend ionization coefficient a to the properties of the gas in a more general manner than hitherto, using realistic functions for the collision cross-section. It is finally expressed by the equation α/ p = A exp ( — Bp/E ) in which A and B are functions involving the properties of the gas and the ratio E/p . The important coefficient B is directly related to the form and magnitude of the total inelastic cross-section below the ionization potential and can be evaluated for a particular gas once the cross-section is known experimentally. The present theory shows clearly the influence of E/p on both A and B, a matter which has not been satisfactorily discussed previously. The theory is illustrated by calculations of F (ε) and a/p for hydrogen over a range of E/p from 10 to 1000. The agreement between the calculated results and recent reliable observations of α/ p is surprisingly good considering the nature of the calculations and the wide range of E/p .

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