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Mathematical methods I course of the University of Oxford

Past exam of the mathematical methods I course of the University of Oxford

2014

All nonzero <eigenmodes> decay, including the $te^{Kt}$ term in the repeated-root case. Total concentration is conserved and initially equals $n$. Combining $A_\infty+B_\infty+C_\infty=n$ with the equal-flux ratios gives
$$
\boxed{A_\infty=\frac{n\delta\eta}{q},\qquad B_\infty=\frac{n\epsilon\delta}{q},\qquad C_\infty=\frac{n\epsilon\eta}{q},\qquad
q=\epsilon\delta+\delta\eta+\eta\epsilon.}
$$
These are the unique <steady state> concentrations with this total. The formula remains valid when the transient <eigenvalues> are complex or repeated.

\Image[/past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2014-cyclic-conversion-dynamics.png]
{title=Three-state conversion conserves total concentration; equal rates give damped oscillations and repeated roots give a Jordan transient}

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Past exam of the mathematical methods I course of the University of Oxford / 2014 / 11 / v / Solution

Past exam of the mathematical methods I course of the University of Oxford / 2014 / 11 / v / Solution

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