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
1. Linear Interpolation
2. Cubic Interpolation
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
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.
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.

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