Partial Differential Equation Mathematical Modelling

[Hector Triana]

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.

 

[pasmith]

Substituting $u(x,t) = e^{\alpha t}v(x,t)$ reduces the problem to the heat equation, as $$\frac{\partial u}{\partial t} = e^{\alpha t}\frac{\partial v}{\partial t} + \alpha u$$ and $$\frac{\partial^2 u}{\partial x^2} = e^{\alpha t} \frac{\partial^2 v}{\partial x^2}$$