# Solution for an ODE given only at discrete points

The problem I have: For each $n \in \mathbb N$ I have

\begin{align} x_0^n & \in \mathbb R \\ h_n & \in \mathbb R \\ x_k^n & = x_0^n + k \cdot h_n \text{ for } k \in \{0,1,\ldots n\} \\ D_n & = \{ x_k^n : 0 \le k \le n \land k \in \mathbb N\} \\ f_n & : D_n \rightarrow \mathbb R^{+} \\ \Delta f_n & : D_n\setminus\{x_n^n\} \rightarrow \mathbb R: x_k^n \mapsto \frac{f_n\left(x_{k+1}^n\right)-f_n\left(x_k^n\right)}{h_n} \end{align}

with

\begin{align} \Delta f_n\left(x_k^n\right) & = g\left(x_k, f_n\left(x_k^n\right)\right) \text { for a function } g : \mathbb R^2 \rightarrow \mathbb R\\ \sum_{k=0}^n f_n\left(x_k^n\right) \cdot h_n & = 1 \\ \lim_{n\rightarrow\infty} h_n & = 0 \\ \lim_{n\rightarrow\infty}x_0^n & = -\infty \\ \lim_{n\rightarrow\infty}x_n^n & = \infty \end{align}

I need to find a function $f:\mathbb R\rightarrow \mathbb R$ which approximate $f_n$ (if it exists), i.e. the function $f:\mathbb R \rightarrow \mathbb R$ shall satisfy $$\lim_{n\rightarrow\infty} \max\left\{\left|f\left(x_k^n\right)-f_n\left(x_k^n\right)\right|: 0 \le k \le n \land k \in \mathbb N\right\} = 0$$

My question: Is there any work already have discussed the above problem? Are there any useful theorems for my problem? What kind of literature should I read?

I guess reading about numerical solutions of ODEs will be useful for me but I thought asking this question here will give me some additional hints...

My attempt to solve the problem: Because of the two properties \begin{align} \Delta f_n\left(x_k^n\right) & = g\left(x_k, f_n\left(x_k^n\right)\right) \text { for a function } g : \mathbb R^2 \rightarrow \mathbb R\\ \sum_{k=0}^n f_n\left(x_k^n\right) \cdot h_n & = 1 \end{align} I would take the function $f:\mathbb R\rightarrow\mathbb R$ with \begin{align} f^\prime(x) & = g(x,f(x)) \\ \int_{-\infty}^\infty f(x) &= 1 \end{align} Of course it's left to investigate under which circumstances $f$ is the right solution...

• You have two limiting processes here: the domain of the ODE (-m,m) is expanding and the number of sampling point n is increasing. I don't expect convergence when the two are interleaved in an ad hoc way, since forward Euler can have exponential global error growth. – user7530 Feb 9 '15 at 16:39
• Also, what is known about g and its derivatives? – user7530 Feb 9 '15 at 16:42
• @user7530 You can assume about $g$ whatever you need (such as $g\in C^\infty$)... – Stephan Kulla Feb 9 '15 at 17:17