# Assignment Set for Laboratory 1¶

Here is a pdf copy assignment1.pdf

ATSC 409: Hand-in answers to questions 1, 2 and 3.

EOSC 511/ATSC 506: Hand-in answers to questions 1, 3 and 4.

1. Given the following four (x,y) points (-5,-1), (0,0), (5,1), (8,4) find the y-value at x=3 using

1. Linear Interpolation
2. Cubic Interpolation
2. Given the equation

$\frac{dy}{dt} = y(y+t)$

write down

1. forward Euler difference formula
2. backward Euler difference formula
3. centered difference formula
3. The equation

$\frac{dy}{dt} + c \frac{dy}{dx} = 0,\ y = \cos(x)\ {at}\ t=0,\ \frac{dy}{dt} = c \sin(x)\ {at}\ t=0$

has a solution $$y=\cos(x-ct)$$.

1. Expand both derivatives as centred differences.
2. Show that the algebraic solution is an exact solution of the difference formula if we choose $$\Delta x = c \Delta t$$.
4. Given

$\frac{dy}{dt} = -\alpha y,\ y = 1 \ {at}\ t=0$
1. Show that the forward Euler method gets a smaller answer than the backward Euler method for all $$t > 0$$, provided that $$0 < \alpha^2 \Delta t^2 < 1$$.
2. Solve the equation analytically.
3. Show that the forward Euler always under-estimates the answer provided that $$\alpha \Delta t < 1 \ {and}\ \alpha \Delta t \ne 0$$.