| course |
course_year |
question_number |
tags |
title |
year |
Complex Methods |
IB |
13 |
|
Paper 4, Section II, A |
2018 |
(a) Find the Laplace transform of
$$y(t)=\frac{e^{-a^{2} / 4 t}}{\sqrt{\pi t}}$$
for $a \in \mathbb{R}, a \neq 0$.
[You may use without proof that
$$\left.\int_{0}^{\infty} \exp \left(-c^{2} x^{2}-\frac{c^{2}}{x^{2}}\right) d x=\frac{\sqrt{\pi}}{2|c|} e^{-2 c^{2}} .\right]$$
(b) By using the Laplace transform, show that the solution to
$$\begin{aligned}
\frac{\partial^{2} u}{\partial x^{2}} &=\frac{\partial u}{\partial t} \quad-\infty<x<\infty, \quad t>0 \\
u(x, 0) &=f(x) \\
u(x, t) \quad \text { bounded, }
\end{aligned}$$
can be written as
$$u(x, t)=\int_{-\infty}^{\infty} K(|x-\xi|, t) f(\xi) d \xi$$
for some $K(|x-\xi|, t)$ to be determined.
[You may use without proof that a particular solution to
$$y^{\prime \prime}(x)-s y(x)+f(x)=0$$
is given by
$$\left.y(x)=\frac{e^{-\sqrt{s} x}}{2 \sqrt{s}} \int_{0}^{x} e^{\sqrt{s} \xi} f(\xi) d \xi-\frac{e^{\sqrt{s} x}}{2 \sqrt{s}} \int_{0}^{x} e^{-\sqrt{s} \xi} f(\xi) d \xi .\right]$$