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September 8th, 2014, 10:18 PM   #1
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advection-diffusion-decay equation

Show that the advection-diffusion-decay equation
$\displaystyle u_t=Du_{xx} - cu_x - \lambda u$

can be transformed into the diffusion equation by a transformation of the form

$\displaystyle u(x,t) = w(x,t)e^{\alpha x-\beta t}$

Solution: take $\displaystyle \alpha = c/(2D) , \beta = \lambda + c^2/(4D)$

Will I somehow apply the quadratic formula and work forward from there?

By diffusion equation, the author mean $\displaystyle u_t - Du_{xx}=0$?

Thank you for any help!
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September 14th, 2014, 01:32 PM   #2
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Any thoughts?

Thanks for any help!
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September 15th, 2014, 01:46 AM   #3
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To do this you just need to calculate $\displaystyle u_t$, $\displaystyle u_x$ and $\displaystyle u_{xx}$ in terms of $\displaystyle w$ by substituting the transform. Substituting these into your original ODE will give you an equation that will demonstrate what you're after if the conditions given for $\displaystyle \alpha$ and $\displaystyle \beta$ are true.
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