Solving pde with gaussian function

[nigels]

I just developed this model to describe an ecological process but have trouble solving the equation. My first question is: is the form even analytically solvable? And if so, what steps / references should I resort to? 'a', 'delta' and 'sigma' are all constants.

*Current reference book: Haberman (2003)

 

[Mute]

I'm not sure if it's analytically solvable, but one simplification you can make is to note that $y(x,t)\delta(1-y(x,t)) = \delta(1-y(x,t))$, as the delta function enforces y = 1, so for anything multiplying the delta function that has y in it you can set y to 1.

The fact that you have y inside the delta function does not give me any confidence that you can find an analytic solution to this. I would consider making some sort of approximation that gets you the same qualitative behavior but isn't quite as complicated. (You may still be forced to solve it numerically, however).

 

[nigels]

Sorry, $$\delta$$ here doesn't refer to the delta function but instead just a constant (decay rate of short-term memory). Would that make things quite different?

 

[Mute]

Ah, yes, that makes things very different. Sorry I missed that $\delta$ is a constant and not the delta function.

In that case, you might be able to find an analytical solution, but it's still hard because the equation is non-linear (and has a mixed partial derivative):

$$\frac{\partial}{\partial x}\frac{\partial y(x,t)}{\partial t} - \delta y(x,t)(1-y(x,t)) = a\exp\left(-\frac{x^2}{2\sigma}\right)$$

If $\delta$ is small you could perhaps do an expansion in powers of delta:

$$y(x,t) = y_0(x,t) + \delta y_1(x,t) + \delta^2 y_2(x,t) + \dots$$

Right now I'm not sure if there's a good way to get a closed form solution for any delta.

 

[sfs01]

You can get rid of the mixed derivative by making the change of variable x = u + v and t = u - v, which is going to leave you with a wave equation plus some extra terms, which would be a more standard form.

Now still not sure how to solve it after that, the nonlinear term makes it quite tricky, although you could try to get a solution for delta = 0 (inhomogeneous wave equation) and use that as a starting point to get to a more general solution. However, the solution for delta = 0 will already include an integral over the Gaussian - i.e. the error function. So even if you can get an analytical solution for the whole thing, it's going to look quite nasty including integrals over erf etc.