# Convergence of difference equation to differential equation

Starting with the difference equation:

$$f(x(t+dt),t+dt)= (1-a)f(x(t),t) + a f(x(t+dt),t)$$

where $x(0)$ is given and positive, $a\in(0,1)$, $f(0,t)=0$, and $f$ is increasing in both arguments. This clearly defines a positive, decreasing sequence.

I would like to know conditions such that as I decrease the step size (i.e. $dt\to 0$), that this sequence converges to the solution of the corresponding differential equation:

$$x'(t) = -\frac{f_t(x,t)}{(1-a)f_x(x,t)}$$

The domain can be taken to be $[0,x_0]×[0,1]$. $f$ is nonnegative, bounded, and strictly increasing in both arguments, differentiable, etc.

Ideally would also like to handle the case where $x_0 = \infty$.

Thanks for any suggestions on how to approach this.

Any references or pointers to resources (for a relative beginner!) would be appreciated.

• this problem has been studied (and still is), there are conditions to be met by $f(\cdot)$ and the underlying domain that in some cases lead to these reductions, i dont have references handy at this moment but i might post later – Nikos M. Sep 12 '15 at 0:40
• Yes, I assume it has been well studied (at least in some general sense). Look forward to some guidance... – P. Michaels Sep 13 '15 at 0:25
• from a quick search look math.stackexchange.com/questions/194137/…, web.stanford.edu/class/cme306/Discussion/Discussion1.pdf, – Nikos M. Sep 13 '15 at 0:35
• I know that in general we need "consistency" and "stability" for the result to hold. I am hoping there are some standard conditions on $f$ that might guarantee this but am having trouble extrapolating from the literature cited. (Most of the literature starts with a diff eq and then constructs a desirable recurrence relation, I am going in the other direction.) – P. Michaels Sep 14 '15 at 6:33