A Partial Differential Equation Mathematical Modelling

Salutations,
I have been trying to approach a modelling case about organism propagation which reproducing with velocity $$\alpha$$ spreading randomly according these equations:
$$\frac{du(x,t)}{dt}=k\frac{d^2u}{dx^2} +\alpha u(x,t)\\\ \\ u(x,0)=\delta(x)\\\ \lim\limits_{x \to \pm\infty} u(x,t)=0$$

This studying case requires to demonstrate that isoprobability contours, it means, in the points (x,t) which P(x,t)=P=constant is verified that
$$\frac{x}{t}=\pm [4\alpha k-2k\frac{\log(t)}{t}-\frac{4k}{t}\log(\sqrt{4\pi k} P)]^\frac{1}{2}$$

Another aspect to demonstrate is that $t \to \infty$, the spreading velocity of these contours, it means, the velocity which these organisms are spreading is aproximated to
$$\frac{x}{t}\pm(4\alpha k)^\frac{1}{2}$$

Finally, how to compare this spreading velocity with purely diffusive process $(\alpha=0)$, it means , x is aproximated to $$\sqrt{kt}$$

This is just for academical curiosity and I would like to understand better this kind of cases with Partial Differential Equations. So, I require any guidance or starting steps or explanations to find the solutions because it's an interesting problem.

Thanks very much for your attention.
 
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Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
 

pasmith

Homework Helper
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Substituting [itex]u(x,t) = e^{\alpha t}v(x,t)[/itex] reduces the problem to the heat equation, as [tex]
\frac{\partial u}{\partial t} = e^{\alpha t}\frac{\partial v}{\partial t} + \alpha u[/tex] and [tex]
\frac{\partial^2 u}{\partial x^2} = e^{\alpha t} \frac{\partial^2 v}{\partial x^2}[/tex]
 

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