Purchase Solution

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

Not what you're looking for?

Ask Custom Question

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.

Attachments
Purchase this Solution

Solution Summary

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.

Purchase this Solution


Free BrainMass Quizzes
Measures of Central Tendency

Tests knowledge of the three main measures of central tendency, including some simple calculation questions.

Measures of Central Tendency

This quiz evaluates the students understanding of the measures of central tendency seen in statistics. This quiz is specifically designed to incorporate the measures of central tendency as they relate to psychological research.

Know Your Statistical Concepts

Each question is a choice-summary multiple choice question that presents you with a statistical concept and then 4 numbered statements. You must decide which (if any) of the numbered statements is/are true as they relate to the statistical concept.

Terms and Definitions for Statistics

This quiz covers basic terms and definitions of statistics.