MATH 406, HWK 6, Due 7 November 2014

1. Alter the program poisson.m (a copy of poisson.m, V.m, and VN.m

can be found on the course web site) to be able to solve the following

‘crack-like’ mixed boundary value problem for Laplace’s equation on a

semi-circular domain:

∆u = urr +1

rur +

1

r2uθθ = 0

subject to

u(r, 0) = 0∂u

∂θ(r,π) = 0 (1)

u(a, θ) = f(θ) = sin

µ(2m+ 1)

2θ

¶. (2)

Figure 1:

Show that the exact solution for the m th mode is of the form:

u(r, θ) =³ ra

´ (2m+1)

2

sin

µ(2m+ 1)

2θ

¶Assuming that a = 10 for modes m = 0 and 1 compare the exact andnumerical solutions along the lines of benchmarks

x = 0 y = 0.1 : 0.1 : a− 0.1

Also for modesm = 0 and 1 compare the exact solution along −a < x < 0,y = 0 to the values of u computed by the BEM at the element centres.

2. Implement the indirect boundary integral equation method using the sin-

gle layer potential to solve Laplace’s equation:

u(x) =

Z∂Ω

F (x0)V (x0, x) ds (x0) (3)

Apply this method to the following boundary value problem for a circle

1

∆u = 0, for 0 < r < a, and − π < θ < π

u(a, θ) = 1 for 0 < θ < π

u(a, θ) = 0 for − π < θ < 0

for which the exact solution is

u(r, θ) =1

2+1

πtan−1

µ2ar

a2 − r2

¶3. Use a one dimensional Fourier Transform to determine the Green’s func-

tion for

∆G(x0, y0;x, y) = δ(x0 − x)δ(y0 − y), −∞ < x <∞, y > 0

G(x0, 0;x, y) = 0

by the following procedure:

(a) Use the one dimensional Fourier Transform

F(u) = bu(k, y) = ∞Z−∞

eikxu(x, y)dx

to reduce the above boundary value problem for G to an ODE for bG.(b) Now use the stitching process introduced for Green’s functions for

ODEs to determine the solution bG(k, y) to this ODE and boundaryconditions - assuming that bG(k, y) → 0 as y →∞.

(c) Now use the Fourier inversion formulaG(x, y) = 12π

∞R−∞

e−ikx bG(k, y)dkto determine G(x, y).

(d) Use the free space Green’s function in 2D to interpret your result,

i.e. recall G = 12πln |x0 − x| + h (x0) where ∆h = 0 and h (ξ, 0) =

− 12πlnh(ξ − x)2 + y2

i1/2(∗).

(e) Use the indirect BEM defined in problem 2 to approximate the Green’s

Function for a half-space

∆G (x0, x) = δ (x0 − x) x0 = (ξ, η) η > 0

G (0, x) = 0 x0 = (ξ, 0)

Discretize the boundary η = 0, ξ² (−10, 10) and impose the condition(∗) above to find h (ξ, η). Evaluate the numerical and exact Green’sfunctions along a line of benchmarks x = −10 : 1 : 10, y = 1 if thedelta function δ (ξ − 5, η − 2) has a source point at (5, 2) and plotthe result.

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