Cranking method

In order to be able to determine re we must consider rotations. A straightforward procedure is a generalization of that discussed in the previous section, and it is called the cranking method. The method has been used extensively in nuclear physics (Schaaser and Brink, 1984). It consists in evaluating the expectation value of the modified Hamiltonian... 

The full calculational result [BEN83] for 15 Er is compared with experiment in fig. 3. The calculations are based on the Nilsson-Strutinsky cranking method where we follow individual configurations as functions of spin [BEN85 ] The energy of each state is minimised with respect to deformation, e, y, and 4. The calculated bands denoted by 1, 2 and 3 are the (kn/2 11/2 11/2) configurations of fig. 2. Compared... 

Errors are proportional to At for small At. When the trapezoid rule is used with the finite difference method for solving partial differential equations, it is called the Crank-Nicolson method. The implicit methods are stable for any step size but do require the solution of a set of nonlinear equations, which must be solved iteratively. The set of equations can be solved using the successive substitution method or Newton-Raphson method. See Ref. 36 for an application to dynamic distillation problems. 

The implicit Crank-Nicholson integration method was used to solve the equation. Radial temperature and concentrations were calculated using the Thomas algorithm (Lapidus 1962, Carnahan et al,1969). This program allowed the use of either ideal or non-ideal gas laws. For cases using real gas assumptions, heat capacity and heat of reactions were made temperature dependent. 

The Crank-Nicholson method is a special case of the formula... 

The Crank-Nicolson method is popular as a time-step scheme for CFD problems, as it is stable and computationally less expensive than the implicit Euler scheme. 

This then provides a physical derivation of the finite-difference technique and shows how the solution to the differential equations can be propagated forward in time from a knowledge of the concentration profile at a series of mesh points. Algebraic derivations of the finite-difference equations can be found in most textbooks on numerical analysis. There are a variety of finite-difference approximations ranging from the fully explicit method (illustrated above) via Crank-Nicolson and other weighted implicit forward. schemes to the fully implicit backward method, which can be u.sed to solve the equations. The methods tend to increase in stability and accuracy in the order given. The difference scheme for the cylindrical geometry appropriate for a root is... 

Let us now turn to the implicit Crank-Nicholson method and form the matrix A as... 

When faster reactions are dealt with, it may be profitable to remove the At/Ay2 < 0.5 condition and use an implicit method such as the Crank-Nicholson method.15 17 The finite difference approximation is then applied at the value of t corresponding to the middle of the j to j + 1 interval, leading to... 

Linearizing the kinetic term as before, a set of three unknown linear equations is obtained, which is completed by the finite difference expression of the initial and boundary conditions. Inversion of the ensuing matrix allows the calculation of C at each node of the calculation grid and finally, of the current flowing through the electrode, or of the corresponding dimensionless function, by means of its finite difference expression. Calculation inside thin reaction layers may thus be more efficiently carried out than with explicit methods. The combination of the Crank-Nicholson... 

Transient is a C-program for solving systems of generally non-linear, parabolic partial differential equations in two variables (that is, space and time), in particular, reaction-diffusion equations within the generalized Crank-Nicolson Finite Difference Method. 

As initial distribution corresponds to the linear mode (2.11) of the given waveguide, the deviation of T z) with respeet to unity may he eonsidered as a measure of the error in this method. The results presented in Fig.2 allow one to analyze the accuracy of the method depending on the type of finite-difference scheme (Crank-Nicholson" or Douglas" schemes have been applied) and on the method of simulation of conditions at the interface between the core and the cladding for both (2D-FT) and 2D problems. 

Crank-Nicolson implicit method This method is a little more complicated but it offers high precision and unconditional stability. Let... 

Only one other general solution exists. Two methods may be used to solve a partial differential equation such as the diffusion equation, or wave equation separation of variables or Laplace transformation (Carslaw and Jaeger [26] Crank [27]). The Laplace transformation route is often easier, especially if the inversion of the Laplace transform can be found in standard tables [28]. The Laplace transform of a function of time, (t), is defined as... 

So far, relatively little attention has been given to the variational method of solving diffusion problems. Nevertheless, it is a technique which may become of more interest as the nature of problems becomes more complex. Indeed, the variational method is the basis of the finite element method of numerical calculations and so is, in many ways, an equal alternative to the more familiar Crank—Nicholson approach [505a, 505b]. The author hopes that the comments made in this chapter will indicate how useful and versatile this approach can be. 

The equations were transformed into dimensionless form and solved by numerical methods. Solutions of the diffusion equations (7 or 13) were obtained by the Crank-Nicholson method (9) while Equation 2 was solved by a forward finite difference scheme. The theoretical breakthrough curves were obtained in terms of the following dimensionless variables... 

The volume of gas inside the cylinder is a function of the piston position. The piston position is a function of the crank-shaft position. The crank-shaft position can be measured by a magnetic pickup attached to the crank shaft. Anyone who has had an automobile spark-plug firing timing adjusted electronically is familiar with this method of determining piston position. 

In Section 4.2.3 we described application of the method of superposition to infinite and semi-infinite systems. The method can also be applied, in principle, to finite systems, but it often becomes unwieldy (see Crank s discussion of the reflection method [2]). 

We determined the phase behavior of the HDPE/styrene/CC>2 using the method described by Berens et al. (1992), modeling mass uptake data as Fickian dilfusion into a planar sheet (Crank, 1975). Ethylbenzene was used as the penetrant to model styrene. 

Equation (5.62) for the current-potential response in CV has been deduced by assuming that the diffusion coefficients of species O and R fulfill the condition Do = >r = D. If this assumption cannot be fulfilled, this equation is not valid since in this case the surface concentrations are not constant and it has not been possible to obtain an explicit solution. Under these conditions, the CV curves corresponding to Nemstian processes have to be obtained by using numerical procedures to solve the diffusion differential equations (finite differences, Crank-Nicholson methods, etc. see Appendix I and ([28])3. 

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