A Study of the Diffusion Equation with Increase in the Amount of Substance, and its Application to a Biological Problem

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For the sake of simplicity we consider the two-dimensional diffusion equation 1 $$\frac{{\partial v}}{{dt}} = k\left( {\frac{{{{\partial }^{2}}v}}{{\partial {{x}^{2}}}} + \frac{{{{\partial }^{2}}v}}{{\partial {{y}^{2}}}}} \right),k > 0$$ where r and y are the coordinates of a point in the plane, t is time and v is the density of substance at the point (r, y) at time t. We now assume that diffusion is accompanied by increase in the amount of substance at a rate which depends on the density at the given point and time. We then obtain the equation 2 $$\frac{{\partial v}}{{\partial t}} = k\left( {\frac{{{{\partial }^{2}}v}}{{\partial {{x}^{2}}}} + \frac{{{{\partial }^{2}}v}}{{\partial {{y}^{2}}}}} \right) + F(v)$$