Generalized Bessel equation

Very often, Bessel s equation can be obtained by an elementary change of variables (either dependent, independent, or both variables). For the general case, we can write 

Representing Z + /j ) as one of the Bessel functions, then a general solution can be written 

The types of Bessel functions that arise depend on the character of (d) /  

If /s is real and p is zero or integer k, then denotes and Z denotes Y/.  

Pin-promoters of the type shown in Fig. 3.2a and 3.2b are used in heat exchangers to enhance heat transfer by promoting local wall turbulence and by extending heat transfer area. Find an e q)ression to compute the temperature profile, assuming temperature varies mainly in the x direction. The plate temperature T, fluid temperature T , and heat transfer coefficient h are constant. 

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The homogeneous solution of (3 269) is a Bessel function. To see this, we note that the general Bessel equation is... 

The general solution of Eq. (82), which is equivalent to the Bessel equation, is given by the formula... 

Here ji(qa) is the spherical Bessel function of order l,g(a) is the radial distribution function at contact, and f = /fSmn/Anpo2g a) is the Enskog mean free time between collisions. The transport coefficients in the above expressions are given only by their Enskog values that is, only collisional contributions are retained. Since it is only in dense fluids that the Enskog values represents the important contributions to transport coefficient, the above expressions are reasonable only for dense hard-sphere fluids. Earlier Alley, Alder, and Yip [32] have done molecular dynamics simulations to determine the wavenumber-dependent transport coefficients that should be used in hard-sphere generalized hydrodynamic equations. They have shown that for intermediate values of q, the wavenumber-dependent transport coefficients are well-approximated by their collisional contributions. This implies that Eqs. (20)-(23) are even more realistic as q and z are increased. 

This is a modified Bessel equation of order I — m. Except for the physically improbable case where 1 — m is exactly an integer or zero, the general solution to Eq. (24) is... 

This is a modified Bessel equation of zero order with the general solution... 

Let us now return to the solution of our problem for Rr 1. Although the arguments leading to (4-25) were complex, the resulting equation itself is simple compared with the original Bessel equation. Our objective here is an asymptotic approximation of the solution for the boundary-layer region. In general, we may expect an asymptotic expansion of the form... 

Solutions of the first equation are harmonic functions sin mz and cos mz. The second equation is the modified Bessel equation, solutions of which are functions Io mr) and A o(mr). Inasmuch as the field is an even function with respect to coordinate 2, it cannot contain sin mz. For this reason, the general solution presents a combination of functions such as Ko mr) cos mz and Io mr) cos mz. 

The general solution for this Bessel equation is given by... 

Equation (9A-1) is a form of Bessel s equation and can be solved by comparison of its terms with those in the solution of the generalized Bessel s equation. 

This is an inhomogeneous Bessel equation and its general solution is [1, p.496]... 

Bessel s Differential Equation. We showed previously 2fi above) that, if n is an integer, J (ie) is a solution of Bessel s equation (2G. )). We shall now examine the solutions of that equation when the parameter n is not necessarily an integer. To emphasise that this parameter is, in general, non-integral, wc shall replace it by the symbol v, so that wc now consider the solutions of the second order linear differential equation... 

In these formulas the symbol Za(co) stands for the Bessel function, sn (to), dn ( ), cn ( ) are the Jacobi elliptic functions having the module /(xvxv) is the general solution of the ordinary differential equation... 

The most general solution to the wave equation of a spherically confined particle is the Fourier transform of this Bessel function, i.e. the box function defined by ro- Such a wave function, which terminates at the ionization radius, has a uniform amplitude throughout the sphere, defined before (3.36)... 

The differential equation is the same for -n as for n, so D. is also a solution, and is generally different from D . Thus, a general solution of Bessel s equation with two arbitrary constants is... 

Equation (109) can readily be derived from Eq. (110). The functions zn and kn are modified spherical Bessel functions of order w, and P (x) are the generalized Legendre polynomials [91,97]. 

The interesting case of potentials with tails falling off like 1 /i was recently treated by Mortiz et al. [59]. They reported many aspects of the near-threshold properties for these potentials. However, they did not give values of the critical parameters Xc and a. There is no general solution of the Schrodinger equation for power-law potentials, but for tails going to zero faster than 1/r2 and E(XC) 0, the solutions are Bessel functions [60] that can be normalized for... 

A general solution of this equation, which is bounded at ij = 1, can be obtained easily by separation of variables in terms of modified Bessel functions of the first and second kind... 

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