Transformation technique problems

Use of Computer Simulation to Solve Differential Equations Pertaining to Diffusion Problems. As shown earlier (Section 4.2.11), differential equations used in the solutions of Fick s second law can often be solved analytically by the use of Laplace transform techniques. However, there are some cases in which the equations can be solved more quickly by using an approximate technique known as the finite-difference method (Feldberg, 1968). 

The problem introduced by this Hamiltonian is studied by means of the unitary - canonical quaziparticle transformation technique in two steps [1],... 

In this section we will consider the case of a multi-level electronic system in interaction with a bosonic bath [288,289], We will use unitary transformation techniques to deal with the problem, but will only focus on the low-bias transport, so that strong non-equilibrium effects can be disregarded. Our interest is to explore how the qualitative low-energy properties of the electronic system are modified by the interaction with the bosonic bath. We will see that the existence of a continuum of vibrational excitations (up to some cut-off frequency) dramatically changes the analytic properties of the electronic Green function and may lead in some limiting cases to a qualitative modification of the low-energy electronic spectrum. As a result, the I-V characteristics at low bias may display metallic behavior (finite current) even if the isolated electronic system does exhibit a band gap. The model to be discussed below... 

To demonstrate Pawlowski s matrix transformation technique, an example will be used in which a forced convection problem, where a fluid with a viscosity p, a density p, a specific heat Cp and a thermal conductivity k, is forced past a surface with a characteristic size D at an average speed u. The temperature difference between the fluid and the surface is described by AT = Tf — Ts and the resulting heat transfer coefficient is defined by h. 

The choice of hydrides should be optimized under a concrete heat problem the heat pump, a refrigerator, a heat transformer. Techniques of a choice of the hydrides can help with it using computer technologies [6],... 

This is a problem which may be solved by the Laplace-transform technique. The solution is given in Ref. I as... 

Various formulations and methodologies have been suggested for describing combined heat and mass transfer problems, such as the integral transform technique, in the development of general solutions. In this chapter, cross phenomena or coupled heat and mass transfer are discussed using the linear nonequilibrium thermodynamics theory. 

Fourier and Laplace transforms are linear transforms and are very often used for analyzing problems in various branches of science and engineering. Since receptivity is studied with respect to onset of instability, it is quite natural that these transform techniques will be the tool of choice for such studies. Fourier transform provides an approach wherein the differential equation of a time dependent system is solved in the transformed plane as. 

Thus, it becomes apparent the output and the impulse response are one-sided in the time domain and this property can be exploited in such studies. Solving linear system problems by Fourier transform is a convenient method. Unfortunately, there are many instances of input/ output functions for which the Fourier transform does not exist. This necessitates developing a general transform procedure that would apply to a wider class of functions than the Fourier transform does. This is the subject area of one-sided Laplace transform that is being discussed here as well. The idea used here is to multiply the function by an exponentially convergent factor and then using Fourier transform technique on this altered function. For causal functions that are zero for t < 0, an appropriate factor turns out to be where a > 0. This is how Laplace transform is constructed and is discussed. However, there is another reason for which we use another variant of Laplace transform, namely the bi-lateral Laplace transform. 

This poses a problem By applying to the system of Eq. (2.27) the Laplace transformation technique illustrated in Chapter I for evaluating the correlation function... 

The stretching and bending modes of zeolite lattices have weak Raman cross sections, which makes measuring high quality Raman spectra difficult. Laser induced fluorescence is also a common problem with dehydrated zeolites, although this can be overcome with the Fourier transform technique. As with the corresponding infrared spectra, the frequencies of the Raman active lattice modes depend on both the local structure and the composition of the zeolite lattice. 

The velocity profile was assumed to be fully-developed. The velocity distribution in a circular microchannel including the slip boundary condition was taken from the literature. However, for the other geometries, they derived the fully-developed velocity profiles from the momentum equation. It is straightforward for flow between parallel plates and flow in an annulus. They applied the integral transform technique to obtain the velocity in a rectangular channel. The problem was simplified by assuming the same amount of slip at all the boundaries. 

The present section reviews the concepts behind the Generalized Integral Transform Technique (GITT) [35-40] as an example of a hybrid method in convective heat transfer applications. The GITT adds to tiie available simulation tools, either as a companion in co-validation tasks, or as an alternative approach for analytically oriented users. We first illustrate the application of this method in the full transformation of a typical convection-diffusion problem, until an ordinary differential system is obtained for the transformed potentials. Then, the more recently introduced strategy of... 

It is clear that the above formulation is linearly constrained in and the solution of (P2) gives us a global optimum in segregated flow for any concave objective function. Moreover, we can reduce this problem to a linear program for both yield and selectivity objective functions by applying suitable transformation techniques (Balakrishna, 1992). Then we can solve the problem by any linear programming algorithm. 

In general, phases (and hence distance Rj) can be determined at present with greater accuracy than amplitudes i.e. co-ordination numbers Aj) the uncertainties in distances, for higher-Z elements are of the order of 0.001-0.002 nm, and in number of atoms in the first co-ordination shell 20%. It has been shown recently that a modification of the usual Fourier transform technique can give accuracies of better than 0.001 nm in distances. Also, the phase problem may be circumvented by considering the beats between two scattering shells. 

One solution to the problem is to increase the ionization probability. This can be done by choosing primary ions with heavy mass, for example, Bi+ or even Ccarbon atoms. The noise level can also be reduced by techniques of digital image processing. For example, a fast Fourier transform technique has been used to remove noise from the image. This technique transforms an image from a space domain to a reciprocal domain by sine and cosine functions. Noise can be readily filtered out in such domain. After a reverse Fourier transform, filtered data produces an image with much less noise. 

Laplace Transform Technique for Parabolic PDEs -Advanced Problems... 

Laplace Transform Technique for Parabolic PDEs - Advanced Problems > u2 =subs( x1 =x,alpha1 =alpha, u2) ... 

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