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I am looking for an English translation of Euler's derivation of Euler's method for ODEs, namely the update $$ y_{n+1} = y_n + h f(y_n, t_n) $$ What motivated Euler to consider this problem, and how does he develop the idea?

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    $\begingroup$ Is not his motivation evident? Euler wanted to solve differential equations numerically. What he considered is the most naive and immediate discretization of the problem. $\endgroup$ – Alexandre Eremenko May 27 at 2:04
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    $\begingroup$ I was hoping for a little more than that. Was it celestial mechanics? Why did he feel like he needed to go beyond current analytic techniques? $\endgroup$ – user14717 May 27 at 2:05
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    $\begingroup$ Because it is an empirical fact well understood by Euler, that most differential equations cannot be solved "analyticallly". That is in closed form. $\endgroup$ – Alexandre Eremenko May 27 at 2:06
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What motivated Euler was not any problem in particular, but rather the general need to solve differential equations approximately when an analytic solution could not be found. He explains the method in a general form in Section 2, chapter VII of volume I of Institutionum Calculi Integralis (Foundations of Integral Calculus, 1768), his textbook on integral calculus. It closely follows the method of step by step numerical integration described in Section 1, chapter VII. The example he considers is illustrative, in modern notation $y'=\frac{a^2}{x^2-y^2}$.

The chapter is titled De integratione aequationum differentialium per approximationem (On approximate integration of differential equations), English translation of the relevant portion and commentary are available in Jardine's Euler's Method in Euler's Words (note that Euler uses $\partial$ where we would use $d$):

"The pair of variables $x$ and $y$ appear in an integral of the form $\frac{\partial y}{\partial x}=V$, the function $V$ itself a function of $x$ and $y$. We desire the complete integral, which is interpreted that as long as $x$ is assigned a certain value $x = a$, the other variable $y$ takes on a given value $y = b$. Therefore our primary goal is to find the value of $y$ so that when $x$ takes on a value that differs little from $a$, or we assume $x = a + \omega$, then we can find $y$. Since $\omega$ is a very small quantity, then the value of $y$ itself differs minimally from $b$; so while $x$ varies a little from $a$ to $a+\omega$, one may consider the quantity $V$ as a constant.

When we specify $x = a$ and $y = b$ then by virtue of the small change we have $V=A$, for that reason when integrating $y= b+X(x-a)$, a constant being added of course so that when $x = a$ we have $y = b$. Therefore given the initial values $x = a$ and $y = b$, we obtain the approximate next values $x = a + \omega$ and $y = b + A\omega$, so that proceeding further in a similar way over the small intervals, in the end arriving at values as distant as we would like from the earlier values."

Euler was aware of the problems with the accumulation of errors in this method, which he explicitly mentions in the Corollary 2:

"Where smaller intervals are taken, through which the values of $x$ progress iteratively, so much the more accurate values are obtained one at a time. However the errors committed one at a time, even if they may be very small, accumulate because of the multitude."

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    $\begingroup$ Thanks for tracking this down. It looks like corollary 2 was identifying his method as unstable, and corollary 3 as tracking the source of this instability to what we would describe as uncontrolled variation. Pretty good summary of the modern literature! $\endgroup$ – user14717 May 27 at 15:37

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