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sol_hw1 - CE 335 Solutions to Homework 1 8 Given the...

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CE 335 Solutions to Homework 1 8) Given the differential equation, the Euler's method numerical approximation is y(t + Δ t) = y(t) + Δ t(3(Q/A)sin 2 (t) - Q/A). Here, the number of time steps to take is n = (t 1 - t 0 )/ Δ t = (10 d)/(0.5 d) = 20. We can write a Matlab script to execute Euler's method and display the result from each step: %set values for parameters in problem Q = 500; %m^3/d A = 1200; %m^2 %step size to use Dt = 0.5; %d %number of steps n = 20; %initial condition t = 0; y = 0; %display the initial condition %cf. Sec. 3.2 for usage of fprintf fprintf(' t(d) y(m)\n'); fprintf('%5.1f %10.3f\n', t, y); %do Euler's method and display the result from each step for i = 1:n y = y + Dt*(3*(Q/A)*(sin(t)^2) - Q/A); t = t + Dt; fprintf('%5.1f %10.3f\n', t, y); end The result is t(d) y(m) 0.0 0.000 0.5 -0.208 1.0 -0.273 1.5 -0.039 2.0 0.375 2.5 0.683 3.0 0.699 3.5 0.503
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4.0 0.371 4.5 0.521 5.0 0.910 5.5 1.276 6.0 1.379 6.5 1.220 7.0 1.040 7.5 1.102 8.0 1.443 8.5 1.847 9.0 2.037 9.5 1.935 10.0 1.730 9) Same idea, but displaying y(t) as a plot will be easier if we make t and y vectors that contain the results from all the time steps, instead of scalars. The new script is: %set values for parameters in problem Q = 500; %m^3/d A = 1200; %m^2 alpha = 300; %step size to use
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