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Take an equation


Its solution is

$$w(x)=\frac{\sqrt{3}}{2} \tan \left( \frac{\sqrt{3}}2 C+\frac{\sqrt{3}}2 x\right)+\frac12$$

I wonder why a similar difference equation

$$\Delta w+w-w^2-1=0$$

cannot be solved?

share|cite|improve this question
I think I would first ask why is there a nice formula for $\int\sqrt x\,dx$ but not for $\sum\sqrt n$. – Gerry Myerson Aug 10 '12 at 0:20
@Gerry Myerson there is such formula.… – Anixx Aug 10 '12 at 0:34
Why would there be any implication between the easy solvability of one and of the other/ – Mariano Suárez-Alvarez Aug 10 '12 at 1:23
Anixx, you and I have different opinions as to what constitutes a "nice" formula. – Gerry Myerson Aug 10 '12 at 3:12


Let's start by showing a solution of the difference equation : $$\Delta w+w-w^2-1=0$$ at least if this means $\ (w_{n+1}-w_n)+w_n=w_n^2+1$ because : $$w_{n+1}=w_n^2+1$$ admits the solution (for the specific case $w_0=1$) : $$w_n=\lfloor c^{2^n}\rfloor,\\\text{with}\quad c=\exp\left|\sum_{j=0}^\infty 2^{-j-1}\ln(1+w_j^{-2})\right|,\\c\approx 1.5028368010497564997529364237321694087388717439635793$$ This solution and others are specific cases of the quadratic map.
I'll add that other integer initial values of $w_0$ may be handled with this method (for $w_0=2$ for example we only need to replace $c$ by $c^2$).

Some may object that there is some kind of cheating here since $c$ is 'built' with the first terms but that's life...

The original 1973 Aho & Sloane's paper "Some doubly exponential sequences" is available here

I will now try to consider the real question with a little more care...

Let's first note that the differential as well as the difference problem are both nonlinear... Non linearity means not only that general solutions can't be obtained just by superposition of a few solutions but often that closed form solutions will exist only for specific initial conditions (if they exist at all...).

I will not develop further the very interesting subject of dynamical systems and chaos that may appear for some systems (or parameters) but observe that $w_{n+1}=w_n^2+c$ with $c$ a complex number generates a fractal : the famous Mandelbrot set (see too the quadratic map link). As pointed by Gerry your differential equation was separable (and solved) but just adding $+i$ at the end adds some interesting effects...

At this point I must admit that all these digressions still didn't answer to your question since you noted the O.D.E.'s solution and asked why the difference equation could not be solved. In fact the difference equation can be solved : start with $w_0$ and use the iterations to get the next values. But it is clear that you wanted a closed form for the nth term. The difference equation is often used as an approximation of the differential system but the dynamic is only similar at a small scale : the $\tan$ goes to $\infty$ after a fixed time, the difference equation is squared at every iteration and will need an infinite time to go to $\infty$.

Another nice illustration of these differences may be seen at Wolfram : the ODE gives circles while the difference equation gives spirals (the shorter the steps the nearer we will be to a circle but it will never be a circle...).



Let's interpret the picture :

  • in the differential case : you get very simple regular orbits in the phase plane ($x,\ x'$) (it is a one dimensional problem with the position at the bottom and the speed on the vertical side). Physically this may represent one dimensional oscillations with various amplitudes (in the middle is a fixed point with no oscillation at all). So we have the simplest system with energy conservation (oscillations will never stop nor change).
  • in the difference case : you get orbits that look locally a little like the differential case but with a very different topology : orbits are not closed on themselves, oscillations will amplify without end as time passes, energy is not conserved but increases (decreases) instead (except for the central fixed point)! In fact this is a frequent problem with more elaborate numerical simulations (using difference methods or their more sophisticated counterparts like Runge-Kutta, predictor-corrector and so on...) : the model becomes hotter and hotter (simulated galaxies lose their stability and explode...). Going from continuous to discrete systems must always be handled with care!

Note that sometimes the discrete system will be much simpler than the continuous one, in fact they should simply be considered as different (globally even if locally as similar as wished) so that even the existence of an analytic solution in one case will often be unrelated with any closed form of the other!

I hope all this helped more,

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Good. The differential equation, despite being nonlinear, isn't that hard - it's separable. – Gerry Myerson Aug 9 '12 at 23:53
I wonder why the same method cannot be used to solve the difference equation. Does it mean that the class of solvable difference equations is narrower than the class of solvable differential equations or just that the appropriate analogous method had not been developed yet because the field of difference calculus in general is less developed due to lesser interest or historical reasons? – Anixx Aug 10 '12 at 3:24
The equation with interesting effects you have pointed is a second-order equation btw, it differs not only by $i$ but also by the order of derivative. – Anixx Aug 10 '12 at 3:36
@Anixx: I updated my answer. Nonlinear difference equations began to be studied much in the fifties (with the first computers and names like Von Neumann, Ulam). In the eighties there was a renew of interest (iteration theory, chaos and so on with Feigenbaum and others) and this allowed to rediscover and update the work of great predecessors like Poincaré. ODE and PDE have probably a longer life... – Raymond Manzoni Aug 10 '12 at 8:13
@Raymond Manzoni: "[...] the model becomes hotter and hotter (simulated galaxies lose their stability and explode...)". It's interesting how this difference between continuous and discrete cases might allow us (hypothetically at least) to estimate the extent of maximum possible discontinuity of our world: since global structures (like galaxies) that we have honour to observe are stable for a given amount of time the world must be at least as granular (or at most that smooth) to permit this behaviour. – Wormer Feb 20 '15 at 23:00

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