The partial derivative Fx in eq 16 can be obtained from eq 15 as
t - x
1/ 2 )
F0 x I1(βz1/ 2 ) x z β
1/ 2 - I1(βz
c
I0 (βz )
Fx = A(x) ∫ + 1 +
+
z1/ 2
z x 2
4ηc0 z1/ 2
0
xC(x)
e -α(t -t′) dt′ -
- H e - Hx
(18)
c+
where
β -αx / c+
C(x) =
e
2
and I0 is a zeroth-order modified Bessel function of the first kind. Similarly, the partial
derivative Ft in eq 16 can be obtained from eq 15 as
x
t-
c+
1/ 2 )
I1(βz1/ 2 ) 1 z β
1/ 2 - I1(βz
-α z1/ 2 + z t 2 I0 (βz )
Ft = x ⋅ A(x) C(x) + ∫
e -α(t -t′) dt′
. (19)
1/ 2
z
0
In eq 18 and 19
x
t-
≥0
c+
and
x
φ 0, t - = 1
c+
are assumed.
F(x,t,φ), Fx and Ft can be computed using subroutines given by Press et al. (1992) for 10-
point Gauss-Legendre integration and for polynomial approximation of modified Bessel
functions. We obtain the dynamic wave profile at selected times by fixing φ and t and calcu-
lating the corresponding x using eq 15 and 18 in Newton's method. The half-interval
method can also be used to obtain these profiles if Newton's method fails to converge.
Constant φ trajectory traces on the x-t plane are obtained by specifying φ and x, and finding
the corresponding t using Newton's method with eq 15 and 19.
Carslaw and Jaeger (1959) gave the solution of the linear diffusion wave equation (eq 11)
subject to the initial and boundary conditions stated above, and we write it as an implicit
function:
x - ckt
ck x erfc x + ckt - φ(x, t) = 0.
1
G(x, t, φ) = erfc
+ exp
(4Dt)1/ 2
(20)
D
(4Dt)1/ 2
2
In parallel with the dynamic wave development above, for a given constant φ, the partial
derivative of G(x,t,φ) with respect to x can be obtained from eq 20, and following simplifica-
tion we obtain
-1
-(x - ckt)2 ck
x + ckt
exp k erfc
cx
Gx =
+
.
exp
(21)
D
(4Dt)1/ 2
4Dt 2D
(πDt)1/ 2
The partial derivative Gt can be obtained from eq 20 and simplified as
-(x - ckt)2
x
Gt =
exp
.
(22)
2t(πDt)
1/ 2
4Dt
6
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