Applied Differential Equations

1. In class we derived a nite-dierence scheme for the heat equation ut = kuxx of the form un+1 j = unj + (unj 􀀀1 􀀀 2unj + unj +1) where  = k
t
x2 > 0: Here unj approximates u(xj ; tn) at the grid locations xj = j
x and tn = n
t. (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 un􀀀1 j  max(unj 􀀀1; unj ; unj +1) 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 : Pn+1 j = 1 2 (Pn j􀀀1 􀀀 Pn j+1): Here 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 = j
x and tn = n
t 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 pt =
x2 2
t = pxx: 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. 1