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December 1st, 2018, 07:16 PM  #1 
Newbie Joined: Dec 2018 From: Colombia Posts: 6 Thanks: 0  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 $$\frac{x}{t}=\pm [4\alpha k2k\frac{\log(t)}{t}\frac{4k}{t}\log(\sqrt{4\pi k} P)]^\frac{1}{2}$$ 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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differential, equation, mathematical, modelling, partial 
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