Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2008 / Paper 2 / 14 / b

Table of contents
- i b
  - Solution i
- ii b
  - Solution ii

i

Solution

Differentiate the equation for

f

with respect to

x

, commuting mixed partial derivatives for a sufficiently smooth solution. With

y = f_{x}

this gives

f_{x t} + f_{x} f_{xx} = k f_{xxx}, y_{t} + y y_{x} = k y_{xx} .

This is the viscous Burgers equation when

k > 0

.

ii

Solution

For

k \neq = 0

, the chain rule gives

g_{t} = - f_{t} g / (2 k)

and

g_{xx} = (- f_{xx} / (2 k) + f_{x}^{2} / (4 k^{2})) g

. Consequently

g_{t} - k g_{xx} = - \frac{g}{2 k} (f_{t} + \frac{1}{2} f_{x}^{2} - k f_{xx}) = 0.

Thus

g_{t} = k g_{xx} .

This is the potential form of the Cole-Hopf transformation, with

y = - 2 k g_{x} / g

. For

k > 0

it is a forward diffusion equation; at

k = 0

the exponential transformation is undefined, while negative

k

gives a backward heat equation.

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