Simulação Computacional Matemática

COMPUTER SIMULATION

Final Project

EXERCISE (1) PUB PROBLEM:

Note: I kept the both resolutions, the first resolution (1) where was solved analytically, and second resolution (2) where was solved numerically.

Resolution (1):

In other to have an ODE the y is eliminated:

x^'=-λ*x*(500-x)

Isolating the -λ we have:

-λ=x'/(x*(500-x))

Then let’s consider that it will takes a infinitive time for x(t) to go down to zero and introduce a time scale by assuming a half-time condition of t satisfying x(t)=250.

Therefore the problem can be solved analytically:

-λ∫_0^t▒dt=∫_499^250▒dx/(x*(500-x))

-λ*t=-ln⁡(499)/500

Then assuming a time of 1 hour between each new people who knows, and knowing that the initial condition x_0=499 satisfies at 0 hour, therefore the condition adopted x=250 satisfies 249h.

-λ*249=-0.0124

λ=0.0000497

Resolution (2):

In other to have an ODE the y is eliminated:

x^'=-λ*x*(500-x)

Knowing the time scale is not important then λ=1 was settled. Parameterizing the function and seeking for the “time-reversed” equation, was found:

x^'=-x*(1-x)

Applying on the Scilab Software through the formula sequence:

Then plotting the graph:

Where can be identified through the blue line (graph line) the relation of people who knows increasing through the time. The graph line starts at 1 where is the number of person who started with the “rumor” and the graph line curve evidence how λ behaves.

EXERCISE (3) PROCESS CONTROL:

(3-1) Uncontrolled filling

Through Euler-Method the following equation was found:

h(t+Δt)=h(t)+(Q_1/A-k*√(h(t)))*Δt

Where:

k=√(2*g)

Applying the given informations Q_in=4 m³/s and A=1 m² and also assuming that Δt=0.001 s to the above equation in other to calculate the level to h(t→∞) was found the following graph:

From the graph we can interpret that the curve to infinite starts to be linear around the 3 seconds and the level of the water stands at 0.8151 meters.

Therefore the water level for uncontrolled filling to infinite with the given informations and conditions will