In a random walk suppose that at t = (n - 1)At the particle is located...

 

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In a random walk suppose that at t = (n - 1)At the particle is located at x = max. The assumption is that at t = nAt the particle will have moved to x = (m+1)Ax with probability 3/4 or to x = (m-3)Ax with probability 1/4. (a) Draw a grid that shows the achievable positions a particle can reach at n = 0, 1, 2, 3 when starting at (m, n) = (0, 0). Letting w(m, n) be the probability the particle is at x = mAx after n time steps, determine w(m, n) for the positions shown in the grid. (b) Based on your result in part (a) what are the values of A, B so w(m, n) Aw(m 1,n-1)+ Bw(m +3, n 1). = (c) Setting u(x,t) = w(m, n), rewrite your result from part (b) in terms of u(x, t). Assuming Ax and At are small, derive a partial differential equation for u. In a random walk suppose that at t = (n - 1)At the particle is located at x = max. The assumption is that at t = nAt the particle will have moved to x = (m+1)Ax with probability 3/4 or to x = (m-3)Ax with probability 1/4. (a) Draw a grid that shows the achievable positions a particle can reach at n = 0, 1, 2, 3 when starting at (m, n) = (0, 0). Letting w(m, n) be the probability the particle is at x = mAx after n time steps, determine w(m, n) for the positions shown in the grid. (b) Based on your result in part (a) what are the values of A, B so w(m, n) Aw(m 1,n-1)+ Bw(m +3, n 1). = (c) Setting u(x,t) = w(m, n), rewrite your result from part (b) in terms of u(x, t). Assuming Ax and At are small, derive a partial differential equation for u.