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MATH3121 Assignment
创建日期：2024-06-13 17:54:12
所属分类：数学mathematics
标签： MATH3121
Assignment

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MATH3121 Assignment 

1. The Helmholtz equation takes the form,
uxx + uyy + k
2u = f(x, y),
for −∞ < x <∞, −∞ < y <∞.
i) Assuming that the functions u(x, y) and f(x, y) have Fourier transforms
show that the solution to this equation can formally be written:
u(x, y) = − 1
4π2
∫ ∞
−∞
∫ ∞
−∞
∫ ∞
−∞
∫ ∞
−∞
e−i(λ(x−ζ)+µ(y−η))
f(ζ, η)
λ2 + µ2 − k2 dλ dµ dζ dη.
ii) Find a solution to the Helmholtz equation for the special case where
f(x, y) = 0. Is this solution consistent with the general result given in
part i)? Justify your answer.
2. Using Laplace transforms, or otherwise, find a function f that satisfies
the following equation:∫ t
0
exp (−(t− z)) f ′(z)dz = 1.
for t > 0, subject to the initial condition f(0) = 2. Substitute your answer
back into the equation to verify that it works. Explain any issues that you
encounter.
3. Find the inverse Laplace transform of,
F (s) =
1
s+ exp(−sτ) ,
where τ is a positive real parameter.
4. Consider the following PDE,
∂u
∂t
− κ∂
2u
∂x2
= S0δ(x)δ(t),
subject to the initial condition,
u(x, 0) = δ(x),
with κ > 0, and S0 > 0. Find the solution of this PDE by taking both
a Fourier and a Laplace transformation. You may use that fact that the
Laplace transformation of the Dirac delta function is one, i.e.
L{δ(t)} = 1.
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