Boundary element methods for acoustics

v j , j = 1, ..., N, so that equation (10) is satisfied exactly at the points x i ,
i = 1, ..., N, implies that v satisfies the linear system, in matrix form,
Av = −b, (8)
where b denotes the column vector whose ith entry is G(x 0 , x i ), and the
N × N matrix A has ijth entry
∫
a ij := G(y, x i ) ds(y). (9)
γ j
A crude, but not completely useless, approximation for the integrals in
(6) and (9), is to make the approximation that
G(y, x) ≈ G(x j , x)
for y ∈ γ j , so that
∫
∫
G(y, x) ds(y) ≈ G(x j , x) ds(y) = G(x j , x) A j ,
γ j γ j
where
∫
A j :=
γ j
ds(y)
is the area of the element γ j in 3D, the arc-length of γ j in 2D. Making this
approximation in (6) gives the completely explicit formula for u(x) that
u(x) ≈ G(x 0 , x) +
N∑
v j A j G(x j , x), x ∈ D. (10)
j=1
The same approximation can be made in (9), except when i = j since
G(x j , x j ) is undefined (is infinite). A crude, but simple, explicit, and not
completely useless approximation is to take
a ij ≈ ã ij :=
{
Aj G(x j , x i ), i ≠ j,
0, i = j,
(11)
and so solve
Ãv = −b, (12)
instead of (8), where à is the matrix with ijth entry ã ij.
[Solution to question 1 is y(s) = 1 + 5
12 s2 .]
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