Application of Hankel Transform

The Hankel transform may be used to solve numerous boundary-value problems in a relatively straightforward way, using various properties of Bessel functions. We present here the solution to one of the boundary-value problems (Davies, 2002) of interest. 

The axisymmetric steady-state temperature distribution in an infinite cylinder with one of its ends maintained at constant temperature is described by the solution of 

In addition to the stated boundary conditions, it is assumed that 

The second integral is the Hankel transform of the Bessel operator of zero order. Using the property of the Hankel transform, Equation (2.248) is transformed with the conditions stated by Eqnation (2.249) to 

The inverse transform can be easily obtained from the table for Hankel transforms in Appendix 2.C. 

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