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Year 3 of the mathematics course of the University of Oxford

Topics in applied mathematics course of the University of Oxford

Past exam of the topics in applied mathematics course of the University of Oxford

2008

Choose a positive reference <biological growth factor> concentration $G_{ref}$, for example its <concentration> ahead of the <travelling wave>. Set
$$
T=T_0t,\quad X=X_0x,\quad G=G_{ref}g,\quad B=B_0b,
$$
where
$$
\boxed{T_0=\frac1{\nu G_{ref}},\qquad X_0=\sqrt{\frac D{\nu G_{ref}}},\qquad B_0=\frac{\nu G_{ref}}\mu.}
$$
The <chain rule> then transforms the equations into
$$
b_t=\frac{DT_0}{X_0^2}b_{xx}+\nu G_{ref}T_0gb,\qquad g_t=-\mu B_0T_0gb.
$$
Each displayed coefficient is one with the chosen scales. Hence the <dimensionless> system is
$$
\boxed{b_t=b_{xx}+gb,\qquad g_t=-gb.}
$$
The free choice of $G_{ref}$ expresses the freedom to normalize the as-yet unspecified <concentration> scale.

Solution

Past exam of the topics in applied mathematics course of the University of Oxford / 2008 / A / 3 / b / Solution

Past exam of the topics in applied mathematics course of the University of Oxford / 2008 / A / 3 / b / Solution

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