Modified Bessels Equation

As we saw in Example 3.3, another form of Bessel s equation arises when a 

This can be obtained directly from Eq. 3.143 by replacing x with ix (since = -1). The solution when p is not integer or zero yields 

However, because of the complex arguments, we introduce the modified Bessel function, which contains real arguments, so if p is not integer (or zero), write 

The modified Bessel functions can be computed from the general result for any p 

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This is the modified Bessel equation of order v = n + Vi. The solutions of Eq. (C.3) are modified Bessel functions of the first kind, which is defined through the Bessel function Jfx) as... 

This is a standard modified Bessel equation of zero order whose solution is(l5) ... 

When limiting our attention to purely spherical pre-stress we find analytical forms for the solutions of Bessel or Modified Bessel equations in dependence on the coupling coefficient Ksf. The obtained density profiles may show an oscillating behavior we prove the conjecture that oscillating profiles are unstable as well as the non-oscillating ones which correspond to sufficiently high absolute values of Ksf. 

Assuming the parabolic profile and that L is short enough that the density varies linearly around z=h, Milner [55] showed that this equation can be transformed into a modified Bessel equation and solved analytically for v=l/2. In Fig. 4, his results for the velocity profile for a brush with a parabolic and step profile are... 

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... 

Sessile capillary menisci in the vicinity of a point with zero slope are described by the modified Bessel equation... 

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. 

This is a nonhomogeneous zero-order modified Bessel equation, subject to the boundary conditions ... 

For r and c simple functions of x, the equation becomes a modified Bessel equation and may be solved analytically. For example, in the case of the groove geometry, as considered by de Levie,... 

Equation 3.138 is a modified Bessel equation of zero order. By comparison with Equation 3.79, we get... 

An alternative to the iteration method is to take the unknown terms in/(7) to the left-hand side of (8.80), and thus the remaining terms on the right-hand side of the equation are known from the previous time step, as expressed by Eq. 8.75. In this ease, the problem will be reduced to solving a sequence of modified Bessel equations (see Ingber and Phan-Thien 1992 for more details). 

For r > To, (/> satisfies the modified Bessel equation for which the solutions are the zeroth-order modified Bessel functions of the first and second kinds, loiqr) and Ko(qr). Since as r— oo, the solution must tend to the... 

This is a Modified Bessel Equation (Wylie, 1960) which is a special linear second-order differential equation with nonconstant coefficients. The solution is given as... 

The volume integral specified in (3) is over a cylindrical region of unit height and infinite radial extent. Equation (6.34) may be reduced to the modified Bessel equation of order zero by the application of the Laplace transform on the age variable t. If as before 0 is the transform of Q, then the transform of (6.34) is... 

The analytical solution of this equation Is Known (9) (10) In terms of modified Bessel functions of the first kind. AccorxUngly, the dlstrltutlon of the active chains In the particles with volume V, fn(v)/f(v), and the average nunher of active chains In the same — 00... 

In this equation, Jv is the Bessel function of the first kind, and Kv is the modified Bessel function of the second kind, U = aikfnf—fi1)112, W = a(f21—konf)if2,... 

The Modified Bessel Functions. By an argument similar to that employed in 1 we can readily show that Laplace s equation in cylindrical coordinates d2yi, 1 dy> 1 d tp d2yi... 

Actually, by starting with Equations (C.8) and (C.9) as the definitions, all the properties of the spherieal modified Bessel functions can be obtained, without tracing back to the formal definition. Equations (C.6) and (C.7). 

Bamford and Tompa (93) considered the effects of branching on MWD in batch polymerizations, using Laplace Transforms to obtain analytical solutions in terms of modified Bessel functions of the first kind for a reaction scheme restricted to termination by disproportionation and mono-radicals. They also used another procedure which was to set up equations for the moments of the distribution that could be solved numerically the MWD was approximated as a sum of a number of Laguerre functions, the coefficients of which could be obtained from the moments. In some cases as many as 10 moments had to be computed in order to obtain a satisfactory representation of the MWD. The assumption that the distribution function decreases exponentially for large DP is built into this method this would not be true of the Beasley distribution (7.3), for instance. 

Activity coefficients on the molal scale were calculated from Equation 39 by means of a straightforward program containing library sub-routines for evaluation of integrals and modified Bessel functions. 

Actually, the distinction between analytically and numerically obtained model solutions is rarely clear. Ana-lytical solutions to governing differential equations are often expressed in terms of special functions such as exponentials, which must be approximated numerically. Here we will see that die solutions to die Sangren and Sheppard model are conveniently expressed in terms of a class of special functions called modified Bessel functions. 

Taking P0(a,b) as given by Equation 6.44 and with Q0(a,b) as the following cross-product of modified Bessel functions ... 

In this equation, I and K are nth order modified Bessel functions of the first and second kind (Olver, 1972), respectively. These functions behave somewhat like increasing and decreasing exponentials, respectively. The eigenvalues S are obtained easily in this case by the consideration that physically local incident radiation must be periodical when the angle 6 completes a full turn around the tube... 

For the frequently used annular fins of constant thickness f, y(r) = <5f/2 has to be put into (2.70) for the profile function. The fin efficiency r/f is dependent on two dimensionless groups mh according to (2.78) and the radius ratio (r0+h)/r0 = 1+ h/r0, cf. Fig. 2.12. This yields a complicated expression containing modified Bessel functions. F. Brandt [2.13] found the rather accurate approximation equation... 

Iq is the zero-order modified Bessel function of the first kind [24]. Equation 14.56 shows that a plot of x = C/Cq versus t depends only on the two dimensionless parameters Nrea and Req-... 

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