Mathematical Methods for Physics and Engineering - Matematica.NET

PDES: SEPARATION OF VARIABLES AND OTHER METHODSzu =0 u =0ayxu = T 0Figure 21.5 A uniform metal cylinder whose curved surface is kept at 0 ◦ Cand whose base is held at a temperature T 0 .By imposing the remaining boundary condition u(ρ, φ, 0) = T 0 , the coefficients A n can befound in a similar way to Fourier coefficients but this time by exploiting the orthogonalityof the Bessel functions, as discussed in chapter 16. From this boundary condition werequire∞∑u(ρ, φ, 0) = A n J 0 (k n ρ)=T 0 .n=1If we multiply this expression by ρJ 0 (k r ρ) and integrate from ρ =0toρ = a, and use theorthogonality of the Bessel functions J 0 (k n ρ), then the coefficients are given by (18.91) asA n = 2T 0a 2 J1 2(k J 0 (k n ρ)ρdρ. (21.36)na) 0The integral on the RHS can be evaluated using the recurrence relation (18.92) ofchapter 16,ddz [zJ 1(z)] = zJ 0 (z),whichonsettingz = k n ρ yields1 dk n dρ [k nρJ 1 (k n ρ)] = k n ρJ 0 (k n ρ).Therefore the integral in (21.36) is given by∫ a[ 1J 0 (k n ρ)ρdρ= ρJ 1 (k n ρ) = 1 aJ 1 (k n a),k n k n0∫ a730] a0