Name Phung Duc Thuan Student ID P46047158
Content Derive the boundary Integral Equation
HOMEWORK Problem
Let U be a function in the region D with the boundary B. U satisfies the following equation:
ΔU x, y = f x, y Or
ΔU r⃗ = f r⃗ U is an unknown function need to be determined.
Green’s function G associated with Laplace’s operator has the form as follow: ΔG r⃗, r⃗⃗⃗⃗ = r⃗ − r⃗⃗⃗⃗
In the polar coordinates, we have:
ΔG r⃗, r⃗⃗⃗⃗ =r∂r∂ r∂G∂r = 0 where r = |r⃗ − r⃗⃗⃗⃗| ⇒r∂G∂r = C
⇒ ∂G∂r = r
⇒ G = C lnr + D Choose C=1 and D = 0. The Green’s function becomes:
G(r⃗, r⃗⃗⃗⃗) = ln r Green’s Second Identity
∫ (U∂V∂n − V∂U∂n) dS = ∫ UΔV − VΔU dV
Where
B is a piecewise smooth contour enclosing domain D �⃗⃗ is a normal unit vector
Let U be an unknown function satisfying the equation below: ΔU r⃗ = f r⃗
G be Green’s function associated with Laplace’s operator: ΔG r⃗, r⃗⃗⃗⃗ = r⃗ − r⃗⃗⃗⃗
Assume that U exists on the domain D with the boundary B and r⃗⃗⃗⃗ ∈ D
Green’s function is infinite at r⃗ = r⃗⃗⃗⃗ so we enclose r⃗⃗⃗⃗ by a very small circle D�. In the region D-D Green’s second identity becomes
∫ (G∂U∂n − U∂G∂n) dS = ∫ GΔU − UΔG dV
We have three terms which need to be evaluate: lim
∫ (�∂U∂n − U∂G∂n) dS + 2πU = ∫ Gf dV
2�� = ∫ (−�∂U∂n + U∂G∂n) dS + ∫ Gf dV
