Difference between revisions of "Team:Toronto/Analysis"
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\begin{eqnarray} | \begin{eqnarray} | ||
\frac{x_2}{dt} = \alpha - \gamma x_2 \\ | \frac{x_2}{dt} = \alpha - \gamma x_2 \\ | ||
| − | \frac{x_2}{dt} + \gamma x_2 | + | \frac{x_2}{dt} + \gamma x_2 = \alpha |
\end{eqnarray} | \end{eqnarray} | ||
<p>Integrating Factor: </p> | <p>Integrating Factor: </p> | ||
\begin{eqnarray} | \begin{eqnarray} | ||
| − | e^ | + | e^{\int \gamma dt} = e^{\gamma t} |
\end{eqnarray} | \end{eqnarray} | ||
<p>Multiplying both sides by our integrating factor: </p> | <p>Multiplying both sides by our integrating factor: </p> | ||
\begin{eqnarray} | \begin{eqnarray} | ||
| − | (\frac{x_2}{dt} + \gamma x_2)e^ | + | (\frac{x_2}{dt} + \gamma x_2)e^{\gamma t} = \alpha e^{\gamma t}\\ |
| − | \int (\frac{x_2}{dt} + \gamma x_2)e^ | + | \int (\frac{x_2}{dt} + \gamma x_2)e^{\gamma t} = \int \alpha e^{\gamma t} \\ |
| − | x_2 = \frac{\alpha}{\gamma} + ce^ | + | x_2 = \frac{\alpha}{\gamma} + ce^{-\gamma t} |
\end{eqnarray} | \end{eqnarray} | ||
</div> | </div> | ||
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<div id="subsection-Plots" class="subsection"> | <div id="subsection-Plots" class="subsection"> | ||
<h2 class="text-yellow">R plots</h2> | <h2 class="text-yellow">R plots</h2> | ||
| − | Visit our <a href="https://github.com/igemuoftATG/drylab-matlab">GitHub repository</a> for our code. | + | <p>Visit our <a href="https://github.com/igemuoftATG/drylab-matlab">GitHub repository</a> for our code.</p> |
<figure> | <figure> | ||
<div class="figures"> | <div class="figures"> | ||
Revision as of 01:44, 16 December 2017
Analysis
Introduction
Using the previously derived expressions from our ODEs, we use the Mathworks Simulink package to derive solutions to our system for a range of parameters.
ODE Solution
Solving:
\begin{eqnarray} \frac{x_2}{dt} = \alpha - \gamma x_2 \\ \frac{x_2}{dt} + \gamma x_2 = \alpha \end{eqnarray}Integrating Factor:
\begin{eqnarray} e^{\int \gamma dt} = e^{\gamma t} \end{eqnarray}Multiplying both sides by our integrating factor:
\begin{eqnarray} (\frac{x_2}{dt} + \gamma x_2)e^{\gamma t} = \alpha e^{\gamma t}\\ \int (\frac{x_2}{dt} + \gamma x_2)e^{\gamma t} = \int \alpha e^{\gamma t} \\ x_2 = \frac{\alpha}{\gamma} + ce^{-\gamma t} \end{eqnarray}R plots
Visit our GitHub repository for our code.
R Analysis
R Analysis:
Call:
lm(formula = log(x) ~ c(time, time, time))
Residuals:
Min 1Q Median 3Q Max
-0.58853 -0.15536 0.01303 0.19867 0.44055
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.87199 0.21773 13.19 1.47e-15 ***
c(time, time, time) 0.15267 0.01142 13.37 9.74e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.2935 on 37 degrees of freedom
Multiple R-squared: 0.8285, Adjusted R-squared: 0.8238
F-statistic: 178.7 on 1 and 37 DF, p-value: 9.741e-16
Intercept represents the equilibrium value of LacILov, our intercept:
