Differential Equations Ordinary and Partial Differential Equations Math Forum

 September 8th, 2014, 10:18 PM #1 Senior Member   Joined: Jan 2014 Posts: 196 Thanks: 3 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!
 September 14th, 2014, 01:32 PM #2 Senior Member   Joined: Jan 2014 Posts: 196 Thanks: 3 Any thoughts? Thanks for any help!
 September 15th, 2014, 01:46 AM #3 Senior Member   Joined: Apr 2014 From: Glasgow Posts: 2,049 Thanks: 680 Math Focus: Physics, mathematical modelling, numerical and computational solutions 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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