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 \(\alpha \Delta t < 1 \ {and}\ \alpha \Delta t \ne 0\).