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

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- Solution b

Solution

Choose a positive reference biological growth factor concentration

G_{re f}

, for example its concentration ahead of the travelling wave. Set

T = T_{0} t, X = X_{0} x, G = G_{re f} g, B = B_{0} b,

where

T_{0} = \frac{1}{ν G _{re f}}, X_{0} = \frac{D}{ν G _{re f}}, B_{0} = \frac{ν G _{re f}}{μ} .

The chain rule then transforms the equations into

b_{t} = \frac{D T _{0}}{X _{0}^{2}} b_{xx} + ν G_{re f} T_{0} g b, g_{t} = - μ B_{0} T_{0} g b .

Each displayed coefficient is one with the chosen scales. Hence the dimensionless system is

b_{t} = b_{xx} + g b, g_{t} = - g b .

The free choice of

G_{re f}

expresses the freedom to normalize the as-yet unspecified concentration scale.

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