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.
Given the following four (x,y) points (-5,-1), (0,0), (5,1), (8,4)
find the y-value at x=3 using
- Linear Interpolation
- Cubic Interpolation
Given the equation
\[\frac{dy}{dt} = y(y+t)\]
write down
- forward Euler difference formula
- backward Euler difference formula
- centered difference formula
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 \(y=\cos(x-ct)\).
- Expand both derivatives as centred differences.
- Show that the algebraic solution is an exact solution of the
difference formula if we choose \(\Delta x = c \Delta t\).
Given
\[\frac{dy}{dt} = -\alpha y,\ y = 1 \ {at}\ t=0\]
- Show that the forward Euler method gets a smaller answer than the
backward Euler method for all \(t > 0\), provided that
\(0 < \alpha^2 \Delta t^2 < 1\).
- Solve the equation analytically.
- Show that the forward Euler always under-estimates the answer
provided that
\(\alpha \Delta t < 1 \ {and}\ \alpha \Delta t \ne 0\).