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Random Walk and 1-Dimensional Heat Equation and its Fundamental Solution

Suppose that a particle, starting at the origin, has an equal chance of moving to the left or right by a distance ∆x in a time interval of ∆t.

(a) Let n>0 be an integer, and let m be an integer, such that -n≤m≤n and n-m is even. By computing the number of ways that the particle can move a net distance of m∆x in n time intervals ∆t, show that the probability that it is at x = m∆x, after a time t=n∆t is

(1.1)

(b) Use Stirling's formula ≈ , for large n, to deuce that (1.1) is approximately

(c) Note that we get a well-defined density of order , if is proportional to , say = . Then dividing (1.2) by and letting , show that we obtain the fundamental source solution

Hint. Recall that . Observe that we divided by (instead of ), because the spacing between the possible values of is , since n-m must be even.

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A random walk and 1-dimensional heat equation and its fundamental solution are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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