The implicit btcs scheme is unconditionally stable

Is incorrect. The implicit BTCS scheme is unconditionally stable. (This was covered in the lecture
on von Neumann stability analysis where we compared FTCS and BTCS.)
(ii) Is incorrect. An infinite number of Fourier modes are needed to encode a square wave initial
condition (i.e. a square wave). If the solution had only 1 mode it would necessarily mean the initial
condition was a single sinusoidal mode – and a square wave is clearly not. Remember that a
Fourier series is used to represent a function, and a function with sharp edges (like a square wave)
requires many (in fact , infinite) sinusoids. Take a look at the data sheet to see the Fourier series for
a square wave.
(iii) Is incorrect. The Laplace transform is generally useful when the BCs are time varying or there is a
forcing term.
(iv) Is incorrect. Splines are designed to ensure continuity of the first and second derivatives at all the
nodes.
Therefore, the solution is B.
3. Which of the following options is the correct inverse Laplace transform of 𝑈𝑈
(
𝑠𝑠
) =
𝑠𝑠−1
( 𝑠𝑠−2 )
2
+1
?
=1
4. Which of the following graphs best represents the Fourier transform of the function:
𝑓𝑓
(
𝑥𝑥
) = 2cos(10
𝜋𝜋𝑥𝑥
) + 4cos(20
𝜋𝜋𝑥𝑥
)
?
AMME2000/AMME2960 2018
Quiz Solutions, Page 2 of 6