BESSELK

Application:

The modified Bessel function of the second kind, Kν​(x), is a solution to the modified Bessel differential equation. Unlike the standard Bessel functions, these functions are not oscillatory; they decay exponentially as x increases. This decay property makes them particularly useful in modeling phenomena that diminish with distance.

One classic application of the Kν​(x) function is in the study of heat conduction in a long, cylindrical rod with a heat source along its axis. Imagine a long, thin wire (a heating element) running down the center of a much larger, insulated cylindrical rod. The temperature at any point in the rod will depend on its radial distance from the central heating element.

Let's say we want to model the steady-state temperature distribution, T(r), at a distance r from the center of the rod. Assuming the rod is infinitely long, the temperature equation simplifies, and the solution involves modified Bessel functions. For the region outside the heating element (r>0), the temperature distribution can be described by a linear combination of the modified Bessel functions of the first and second kind, Iν​(x) and Kν​(x).

Specifically, if the temperature must remain finite as r→∞, the solution must be proportional to K0​(r) (the modified Bessel function of the second kind of order zero). This is because the I0​(r) function grows exponentially, which is not physically realistic for this scenario.

The formula for the temperature distribution in the rod can be expressed as:

$T(r)=A\cdot K_0(ar)$

where A is a constant determined by the heat source, and a is a constant related to the material properties of the rod (thermal conductivity).

The K0​(x) function's value decreases rapidly as x increases, which makes perfect physical sense: the temperature drops off as you move further away from the heat source.

Here is a table of values for the modified Bessel function of the second kind, K0​(x), for a range of argument values (x):