Question

3. Finish the following problem we discussed in class today: Utt - и(х, 0) — 0, и (х, 0) — е-1e1 5 and then plot u(r, 5) for (a) Choose t do it 10 < x < 10. Use a program to (b) Try to figure out what happens as t -» o0, that is find lim u(r, t) t->oo

Answer 1

a) Using Matlab:

Command Window >>symS X C 1;f-0;g-exp (-abs (x)); Sfunctions related to question >>syms t >> u 0.5* (subs (f,x, (x-c*t)) subs (f,x,x+c*t)) +1/ (2*c) int (g,x, x-c*t,x+c*t); %D'Alembert's solution >>u5-subs (u,t,5); Su (x, 5) >> a linspace (-5,5,100) >> plot (a, subs (u5,a)); %plot solution f>>

Using these commands, we get our plot as:

1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 -5 4 -3 -2 -1 4 2 3

b) We can guess that as (constant function 1) because the graph eventually flattens

too, u(x, t)1

$\blacksquare$

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