7 Features of Subsonic Stage of Contact Interaction …
105
ε nm = 1 (m < l n ), ε nm = 0 (m > l n ), l n =
b n
δ ξ
,
I
r
nm =
¨
B nm
G 30,r (ξ m − ξ, τ n − τ )σ (ξ, τ )dτ dξ,
I
s
nm =
¨
B nm
G 30,s (ξ m − ξ, τ n − τ )σ (ξ, τ )dτ dξ.
For integrals calculation, we use rectangular method, taking rectangulars K i j as
elementary integration domains and supposing σ (ξ, τ ) ≈ σ nm at (ξ, τ ) ∈ K i, j ∪
K i, j+1 . Then, the integral I
r
nm obtains the form
I
r
nm ≈ δ τ δ ξ
n−1
i=1
q i
j= p i
a
r
n−i,m− j σ i j ,
p i = max(−l i , k i1 ), l i =
b i
δ ξ
, k i1 = (i − n)
δ τ
δ ξ
+ m − 1,
q i = min(l i , k i2 ), k i2 = m − 1 + (n − i)
δ τ
δ ξ
.
Here, quadrature formulae coefficients
a
r
nm =
1
4
1
−1
1
−1
G 30,r
δ ξ
m −
1
2
−
v
2
, δ τ
n −
1
2
−
u
2
dudv.
Expressions for coefficients a
r
nm are obtained using substitution
2τ = (u + 2i − 1)δ τ ,
2ξ = (v + 2 j − 1)δ ξ .
Integral I
s
nm is calculated similarly despite the singularity of sub-integral function G 30,s
δ ξ
m −
1
2
−
v
2
, δ τ
n −
1
2
−
u
2
. This is caused by the fact that analytical
integral calculation is impossible in this case, and numerical analytic method of
singularity extraction (multiplicative method) did not lead to its smoothing.
Integral I
0
nm is calculated over the area A nm near the point τ = τ n , x 1 = ξ m :
I
0
nm =
¨
A nm
G 30 (ξ m − ξ, τ n − τ )σ (ξ, τ )dτ dξ ≈ a nm σ nm ,
a nm =
nδ τ
(n−1)δ τ
dτ
−τ +(m+n)δ ξ
τ +(m−n)δ ξ
G 30 (mδ ξ − ξ, nδ τ − τ )dξ.