Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2025 / Paper 1 / 2 / a

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Solution

For separation of variables, take

n = T (t) X (x) Y (y) Z (z)

and divide the reaction-diffusion equation by this product wherever it is nonzero:

\frac{T ^{'}}{μ T} - \frac{γ}{μ} = \frac{X ^{''}}{X} + \frac{Y ^{''}}{Y} + \frac{Z ^{''}}{Z} .

Each spatial term depends on a different variable and must be constant. Write

X^{''} = - k_{x}^{2} X

and similarly for the other two factors. The Dirichlet boundary conditions give

X (0) = X (L) = 0

. A zero separation constant permits only a linear function and hence the zero solution; a positive value of

X^{''} / X

likewise permits no nonzero function vanishing at both endpoints. The nontrivial choices are therefore

X_{p} (x) = sin \frac{p π x}{L}, Y_{q} (y) = sin \frac{q π y}{L}, Z_{r} (z) = sin \frac{r π z}{L}, p, q, r = 1, 2, \dots .

The temporal ordinary differential equation is

T_{p, q, r}^{'} = [γ - \frac{μ π ^{2}}{L ^{2}} (p^{2} + q^{2} + r^{2})] T_{p, q, r},

so, absorbing its constant multiplier into

a_{p, q, r}

,

T_{p, q, r} (t) = exp {[γ - \frac{μ π ^{2}}{L ^{2}} (p^{2} + q^{2} + r^{2})] t} .

The superposition principle and completeness of the Fourier sine series give the stated threefold expansion with these factors. For an initial density

n_{0}

, orthogonality also determines the constants:

a_{p, q, r} = \frac{8}{L ^{3}} \int_{0}^{L} \int_{0}^{L} \int_{0}^{L} n_{0} (x, y, z) sin \frac{p π x}{L} sin \frac{q π y}{L} sin \frac{r π z}{L} d x d y d z .

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