Fourier-Bessel serie

A numerical algorithm for the solution of the system of Eqs. (15), (19) and (51) consists of the expansion of the two-particle functions into a Fourier-Bessel series. We omit all the details of the numerical method they can be found in Refs. 55-58, 85, 86. In Fig. 3 we show a comparison of the total... 

In the numerical solution the matrix structure is evaluated from Eqs. (44)-(46). Then Eqs. (47)-(49) with corresponding closure approximations are solved. Details of the solution have been presented in Refs. 32 and 33. Briefly, the numerical algorithm uses an expansion of the two-particle functions into a Fourier-Bessel series. The three-fold integrations are then reduced to sums of one-dimensional integrations. In the case of hard-sphere potentials, the BGY equation contains the delta function due to the derivative of the pair interactions. Therefore, the integrals in Eqs. (48) and (49) are onefold and contain the contact values of the functions... 

The final step is to choose the A so that w(r, 7) satisfies the initial condition 777(7, 0) = -(1 - r2)/4. The general Sturm-Louiville theory16 guarantees that the eigenfunctions (3-106) form a complete set of orthogonal functions. Thus it is possible to express the smooth initial condition (1 -r2) by means of the Fourier-Bessel series (3-109) with 7 = 0, that is,... 

Within the interval r < R the charge distribution is expanded into a Fourier-Bessel series and beyond the cutoff-radius R, the charge distribution is assumed to be zero ... 

Here n = cq/cq is the ratio of the wave speeds in the two media. They also provide more general expressions for the case of sound beams (rather than plane waves) that are approximated by a one-term Fourier-Bessel series, having both a radial and an axial wave number. The radiation pressure difference (32) at the interface is said to explain the acoustic fountain effect where the directed sound beam creates a liquid jet. 

Using the valence profiles of the 10 measured directions per sample it is now possible to reconstruct as a first step the Ml three-dimensional momentum space density. According to the Fourier Bessel method [8] one starts with the calculation of the Fourier transform of the Compton profiles which is the reciprocal form factor B(z) in the direction of the scattering vector q. The Ml B(r) function is then expanded in terms of cubic lattice harmonics up to the 12th order, which is to take into account the first 6 terms in the series expansion. These expansion coefficients can be determined by a least square fit to the 10 experimental B(z) curves. Then the inverse Fourier transform of the expanded B(r) function corresponds to a series expansion of the momentum density, whose coefficients can be calculated from the coefficients of the B(r) expansion. 

In this section no attempt has been made to discuss the very difficult problem of the convergence of Fourier-Bcssel series. For a very full discussion of this topic the reader is referred to Chapter XVIII of G. N. Watson s A Trealise on the Theory of Bessel Functions, 2nd. edit., (Cambridge University Press, 1044). 

Thus, the form factor F q) is the Fourier-Bessel (or Hankel) transform (of order zero) of the charge density distribution p r) [33]. With the short-range series expansion for jo x) one obtains easily the expansion... 

The solutions to Mathieu s equation comprise an orthogonal set and possess the curious property that the coefficients of their Fourier series expansions are identical in magnitude, with alternating signs, to corresponding coefficients of their Bessel series expansions [2, 3]. Floquefs theorem asserts that any solution of equation (Eq. 20.7) is of the form... 

The time dependence of the light intensity from the PMF is considerably more complicated than that from the RAF, since the basis functions are sin (A sin (cot)) and cos (A sin (cot)), rather than sin(cot) and cos(cot) as for the RAF. However, these functions can be expressed in terms of a Fourier-Bessel infinite series ... 

Analytical solution is possible only for first or zero order. Otherwise a numerical solution by finite differences, method of lines or finite elements is required. The analytical solution proceeds by the method of separation of variables which converts the PDE into one ODE with variables separable and the other a Bessel equation. The final solution is an infinite series whose development is quite elaborate and should be sought in books on Fourier series or partial differential equations. 

Now let s look briefly at just enough of the mathematics of fiber diffraction to explain the origin of the X patterns. Whereas each reflection in the diffraction pattern of a crystal is described by a Fourier series of sine and cosine waves, each layer line in the diffraction pattern of a noncrystalline fiber is described by one or more Bessel functions, graphs that look like sine or cosine waves that damp out as they travel away from the origin (Fig. 9.3). Bessel functions appear when you apply the Fourier transform to helical objects. A Bessel function is of the form... 

The Bessel functions tp(r,9, ip) that vanish at r = a for l = 0 define a Fourier series [54]. The ground-state series is the Fourier transform of sin(ka)/ka, which is the box function... 

An extensive study of analytical techniques used in conduction heat transfer requires a background in the theory of orthogonal functions. Fourier series are one example of orthogonal functions, as are Bessel functions and other special functions applicable to different geometries and boundary conditions. The interested reader may consult one or more of the conduction heat-transfer texts listed in the references for further information on the subject. 

To determine the frequenqf spectrum of a transmitted FM waveform, it is necessary to compute a Fourier series or Fourier expansion to show the actual signal components involved. This work is difficult for a waveform of this type, as the integrals that must be performed in the Fourier expansion or Fourier series are difficult to solve. The actual result is that the integral produces a particular class of solution that is identified as the Bessel function. 

At fixed 5, the terms in (5.180) represent the modes of the f-shape of the potential. The Bessel function oscillates and thus a s are analogous to frequencies in the Fourier series expansion. Equation (5.180) shows that each mode exponentially decays with the distance 5 the characteristic damping length is / = l/a , or, in the dimensional form... 

Naim and Liu [96] used a Bessel-Fourier series stress function and added polynomial terms to provide a nearly exact solution to the stress transfer fixim the matrix to a fragmented fiber through an imperfect interphase. This solution satisfies equilibrium and compatibility every place and satisfies exactly most boundary conditions with the exception of the fiber axial stress. They also proposed the use of an interphase parameter, Ds, and provide a physical interpretation as ... 

These expressions may be evaluated as follows. Since W and S are periodic functions of R, one can expand them into Fourier series over reciprocal lattice vectors G. As the coefficients kq of these expansions decrease very fast, the terms corresponding to G (1,0), (—1,0), (0,1) and (0,-1) are to be kept in the exponent, while the exponential of the rest may be expanded up to the first power. The result may be expressed through the Bessel functions after a straightforwao d algebra. 

---