ABSTRACT
J 21 (βm) (R01.1)
Eigenvalues are found from
J0(βm) = 0 (R01.2)
The derivative of G with respect to n′ and evaluated at r ′ = b is
−∂GR01 ∂n′
∣∣∣∣ r ′=b
= 1 πb3
J1(βm) (R01.3)
Also the cross derivative at r = r ′ = b is
−∂ 2GR01 ∂r ∂r ′
∣∣∣∣ r=r ′=b
= 1 2πb
1 {4π[α(t − τ)]3}1/2
[ 1 + α(t − τ)
] (R01.4)
For small values of α(t − τ) / b2 and r and r ′ not near b,
GR01(r , t |r ′, τ) ≈ GR00(r , t |r ′, τ)
T&F Cat # K10695, Appendix R, Page 541, 12-6-2010
r near GR01(r , t |r ′, τ)
≈ 1 4π[παrr ′(t − τ)]1/2
{ exp
[ − (r − r
′)2 4α(t − τ)
] − exp
[ − (2b − r − r
′)2 4α(t − τ)
]}
(( 1 r ′
− 1 r
) erfc
{ r − r ′
[4α(t − τ)]1/2 }
− (
1 r ′
+ 1 r
− 2 b
) erfc
{ 2b − r − r ′
[4α(t − τ)]1/2 })
(R01.5)
for 0 < r ′ < r . For 0 < r < r ′, exchange r and r ′. For small α(t − τ) / b2 values and r not near zero, the derivative is
−∂GR01 ∂n′
∣∣∣∣ r ′=b
≈ 1 4πb2
√ rb
r − b [πα(t − τ)]1/2 e
α(t − τ) + 1 8 b
r
] (R01.6)
The average GF for a circular region of radius a is given by Equation R00.19 for u ≡ α(t − τ) / a2 less than (b+ − 1)2 / 12 (with b+ ≡ b/a) and for larger u values by
GR01(t , τ) ≡ 4 a2
GR01(r , t |r ′, τ)rr ′ dr dr ′
= 4 πa2
]2 (R01.7)
R02 SOLID CYLINDER, ∂G/∂r = 0 AT r = b
GR02(r , t |r ′, τ) = 1 πb2
[ 1 +
e−β2mα(t−τ) / b2 J0(βmr /b)J0(βmr ′ / b)
] (R02.1)
Eigenvalues are found from
J1(βm) = 0 (R02.2) Some values are given in Table R02.
