problemset3

Figure 5 also shows the pdf for the case for when a

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Unformatted text preview: the random variable as ∞� � In � X∞ = 2n/2 n=1 � Where the In take values (1, 0), (−1, 0), (0, 1), and (0, −1) with a quarter probability each. This can be rewritten as ∞� ∞� √ � I2m−1 � I2m �∞ = 2 X + 2m 2m m=1 m=1 √ � � = 2Y1 + Y0 � � where Y1 and Y0 are decaying walks with parameter a = 1/2. From part (b), we know that these are uniformly distributed on the region |x| + |y | < 1, which allows us to easily construct an analytic solution to the resultant PDF as a convolution two simple PDFs. Similarly, we have � X∞ = � X∞ = 3 � i=0 7 � � 2i/4 Yi for a = 2−1/4 � 2i/8 Yi for a = 2−1/8 i=0 � where the Yi are all decaying walks with parameter a = 1/2; again, this easily allows us to construct the resultant PDF as a convolution of a finite number of simple PDFs. √ Figure 5 also shows the PDF for the case for when a is equal to the golden mean, ( 5 − 1)/2. For this value, 1 − a = a2 , and this special relationship results in sections of the PDF appearing self-similar. 3 Shock structure We are interested in finding a traveling wave solution c(x, t) = f (x − v t) of Burgers’ equation ct + ccx = Dcxx which satisfies the boundary conditions c(−∞, t) = c− and c(∞, t) = c+ . Writing z = x − v t, we find that f (z ) satisfies (−v + f )f � = Df �� which can be integrated to obtain f2 = Df � + A 2 for some constant A. Applying the boundary conditions, and assuming f � (z ) → 0 as z → ±∞, we find that c2 −vc− + − = A 2 c2 −vc+ + + = A 2 −vf + M. Z. Bazant – 18.366 Random Walks and Diffusion – Problem Set 3 Solutions 0.1 a = 0.1 0.4 a = 0.2 8 a = 0.3 0.2 0.3 0.05 0 0.2 0.1 0 0.1 0 -0.1 -0.1 -0.05 -0.2 -0.3 -0.2 -0.1 -0.4 -0.1 -0.05 0 0.05 -0.2 0.1 a = 1/3 0.6 -0.1 0 0.1 0.2 -0.4 -0.3 -0.2 -0.1 1 a = 0.4 0 0.1 0.2 0.3 0.4 a = 0.5 0.4 0.4 0.5 0.2 0.2 0 0 0 -0.2 -0.2 -0.5 -0.4 -0.4 -0.6 -1 -0.4 1.5 -0.2 0 0.2 -0.6 -0.4 -0.2 0.4 a = 0.6 1.5 1 0.2 0.4 0.6 √ a = ( 5 + 1)/2 -1 2 1 0.5 0 -0.5 0 0.5...
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This note was uploaded on 01/23/2014 for the course MATH 18.366 taught by Professor Martinbazant during the Fall '06 term at MIT.

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