Bessels Equation

Sometimes an equation out of this classification can be altered to fit by change of variable. The equations with separable variables are solved with a table of integrals or by numerical means. Higher order linear equations with constant coefficients are solvable with the aid of Laplace Transforms. Some complex equations may be solvable by series expansions or in terms of higher functions, for instance the Bessel equation encountered in problem P7.02.07, or the equations of problem P2.02.17. In most cases a numerical solution Is possible. 

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. 

This is a Bessel equation (Jahnke Emde, p 146, Eq 6). Their parameters are... 

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... 

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

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... 

Likewise, the same treatment can be applied to the calculation of the shapes of pendent drops and bubbles. The behavior of pendent profiles about the apical point of zero slope, /(vq) = 0 at xq = 0, is given by the Bessel equation ... 

The r equation can be massaged into the form of a Bessel equation while the 9 equation is elementary. As a result of these insights, the solution may be written as Jn s/2mE jh. It now remains to impose our boundary conditions. We... 

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

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... 

For a potential which is constant with the value Vpp in the entire space, the radial Schrodinger equation (1.17) reduces to the 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. 

The first equation is called the Bessel equation and its solutions are Bessel functions of the first and second kind Jo(Ar) and Yo Xr) ... 

Substituting the expression (17.60) into Eq. (17.56) and separating variables, one obtains a Bessel equation for 0(r),... 

This is a Bessel equation, and its solution consists of the modified Bessel functions Iq and Ko of the first kind l kr) = —i) Jn ikr). Since Kq is unbounded... 

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