Indian Institute of Technology
Department of Mathematics
Boundary Integral Methods
SAMPLE QUESTIONS FOR PRACTICE
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R1
R5
00
1. (a). Compute the value of −2 (x3 − 2x2 + 5)δ (x − 12 ) dx, (b). 2 (3x − 1) ∂H(x−2)
dx,
∂x
where δ, and H denote the delta and Heaviside functions respectively.
2. Decide whether the following integrals exist as an improper integral or a Cauchy principal
value integral and compute the same.
Z b Z b
x+ξ
ln
cot(x − ξ)dx,
(ii).
(i).
dx, a < ξ < b.
x−ξ
a
a
2
3. Consider the equation ddxf2 = 1, a < x < b. Obtain the corresponding boundary integral
representation for f at any x0 ∈ [a, b]. Note that, this may contain a domain integral term.
df
Hence, find f (x) inside the domain (0, 1), when f (0) = 20, dx
(1) = 100.
If the free space Green’s function in R3 , for Laplace equation behaves like G(x, x0 ) ∼ Ar ,
where r = |x − x0 |, use Gauss-divergence theorem and determine the constant A. Show the
working.
4. Consider a singularity located at x̄0 = (0, 1) in R2 . Write down the corresponding expression for the free-space Green’s function, G(x̄, x̄0 ). Hence, compute ∇G(x̄, x̄0 ). Let Γ be a
closed curve bounded by x = 0, x = 1, y = 0 and y = 1. Then compute
Z
n̄ · ∇G(x̄, x̄0 ) dΓx̄
Γ