4.9. This problem explores how to derive the diffusion equation for the general random walk in the plane, as given in (4.68), (4.69). Let $u(x, y,t)$ be the probability that the particle is located at the spatial location $(x, y)$ at time $t$.
(a) Suppose that at time step $t + Delta t$ the particle is located at $(x, y)$. Explain why at time $t$ the particle was located somewhere on the circle of radius $h$ that is centered at $(x, y)$.
(b) As an approximation to the circle in part (a), distribute $N$ points uniformly around this circle. Specifically, take the points $(x+hcos(jDelta heta), y+hsin(jDelta heta))$, where $Delta heta = 2pi/N$ and $j = 1,2,..., N$. Explain why the probability of the particle moving from one of these $N$ points to $(x,y)$ is approximately $1/N$. From this explain why
$$u(x, y,t + Delta t) approx frac{1}{N} sum_{j=1}^{N} u(x + hcos(jDelta heta), y + hsin(jDelta heta), t).$$
(c) Use the result from part (b) to show that for the general random walk
$$u(x, y,t + Delta t) = frac{1}{2pi} int_{0}^{2pi} u(x + h cos heta, y + h sin heta, t)d heta.$$
(d) Derive the diffusion equation from the result in part (c) by letting $Delta t$ and $h$ approach zero.