Euler approximation

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The Euler approximation is a numerical method of solving differential equations, mostly useful when the solution to a differential equation cannot be found analytically. Euler approximations are found using a recursive formula that uses slope information, given by the derivative, to approximate a value on a solution close to an initial point. It is named after the mathematician Leonhard Euler.

If the function is concave, the approximation will be an overestimate, and if the function is convex ("concave up-wards"), the approximation will be an underestimate. These approximations are more accurate with smaller step sizes.

1 Formulae
2 Reasoning
3 Uses
4 External links

[edit] Formulae

Given a differential equation in the form

$\frac{dy}{dx} = f(x,y) \Leftrightarrow y' = f(x,y)$

and an initial point $(x 0, y 0)$ of the solution $y (x)$ , an approximate discrete solution is given as a pair of sequences

$x_{n+1} = x_n + h \Leftrightarrow x_n = x_0 + hn$

$y_{n+1} = y_n + h \times f_n\,$

where

$y_n \approx y(x_n)$

$f_n = f(x_n,y_n) \,$

$h \in \mathbb{R}^{+}$ , is the step size

$n \in \mathbb{N}_0$

[edit] Reasoning

Near any given $x n$ , the function $y (x)$ may be approximated by a line tangent to that point. Since the derivative at that point, $y'(x n) = f n$ , is the value of the tangent line slope, we have that

$y(x) \approx y_n + f_n \times (x-x_n)$

using this to determine the value at $x n + 1 = x n + h$ yields the result above.

Further insight comes from evaluating the Taylor series expansion of $y (x)$ at $x n + 1$ , centred in $x n$ , at $x n + 1$ , as above.

$y_{n+1} \approx y_n + f_n \times h + \boldsymbol{O}(h^2)$

The first two terms are the same as above, yet now we have an estimate of the order of magnitude of the approximation error at each step, which is of $h 2$ for each step.