1. In class we derived a nite-dierence scheme for the heat equation ut = kuxx of the
j = unj
+1) where =
x2 > 0:
approximates u(xj ; tn) at the grid locations xj = jx and tn = nt. (Recall
that tn here does not mean raise t to the nth power.) We know that the true u(x; t)
satises the maximum principle and we can investigate whether a discrete version of the
maximum principle holds for unj
. Specically, we can ask whether
hold everywhere on the grid. This means that the new value un+1
j is bounded by the three
old values that went into computing it. Use rst equation to show that second equation
holds provided that is less than a certain number to be identied. (This result provides
an important numerical stability condition that bounds the admissible time step t in
practical applications of rst equation).
2. In class we showed that a random walker on a lattice that moves with equal prob-
ability to the left or to the right at each time step gives rise to the following evolution
law for the occupation probabilities Pn
j is the probability that the walker is at lattice point j at time step n. We also
showed that if Pn
j is slowly varying on the grid then it can be approximated by a continuous
function p(x; t) via Pn
j p(xj ; tn) with xj = jx and tn = nt as before. Using Taylor
expansion to rst order in time and second order in space, the PDE governing p(x; t) for
small x and t was found to be the heat equation
Now, nd the equivalent of third equation and the fourth one in the more general case
where the random walker moves with probability q 2 (0; 1) to the right and probability
(1 q) to the left.
3. Repeat the previous question in the case of a walker that moves left or right with
equal probability 0.25 or stays put with probability 0.5.
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