Past exam of the mathematics course of the University of Cambridge / 2006 / ia / Paper 2 / 6B / i

Table of contents
- Solution i

Solution

Apply the chain rule to

U (s, y) = u (s, y + s)

. Holding

y

fixed while differentiating in

s

gives

U_{s} = u_{t} + u_{x}

, while

U_{y} = u_{x}

and

U_{yy} = u_{xx}

. Thus the advection-diffusion equation becomes

U_{s} = U_{yy} + F (s, y), F (s, y) = f (s, y + s) .

The moving coordinate follows the unit-speed advection, leaving the heat equation with a transformed source.

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