# Euler Method

## Proof Technique

Consider the first order ODE:

- $(1): \quad y' = \map f {x, y}$ subject to the initial condition $\map y {x_0} = y_0$

where $\map f {x, y}$ is a continuous real function.

Let $\map y x$ be the solution of $(1)$.

For all $n \in \N_{>0}$, we define:

- $x_n = x_{n - 1} + h$

where $h \in \R_{>0}$.

Then for all $n \in \N_{>0}$ such that $x_n$ is in the domain of $y$:

- $y_{n + 1} = y_n + h \map f {x_n, y_n}$

is an approximation to $\map y {x_{n + 1} }$.

## Proof

Let $(1)$ be integrated with respect to $x$ from $x_0$ to $x_1$.

\((2):\quad\) | \(\displaystyle \map y {x_1} - \map y {x_0}\) | \(=\) | \(\displaystyle \int_{x_0}^{x_1} \map f {x, y} \rd x\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle y \left({x_1}\right)\) | \(=\) | \(\displaystyle y_0 + \int_{x_0}^{x_1} f \left({x, y}\right) \, \mathrm d x\) | ||||||||||

Assuming that the integrand in $(2)$ varies little over the interval $\closedint {x_0} {x_1}$: | |||||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \map y {x_1}\) | \(\approx\) | \(\displaystyle y_0 + \map f {x_0, y_0} \paren {x_1 - x_0}\) | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle y_0 + h \, \map f {x_0, y_0}\) |

Because $f$ is continuous, the assumption holds.

By making $h$ small, the difference:

- $y_0 + h \, \map f {x_0, y_0} - \map f {x_1, y_1}$

can be made arbitrarily small.

$y_{n + 1}$ can be defined recursively:

\(\displaystyle \map y {x_{n + 1} } - \map y {x_n}\) | \(=\) | \(\displaystyle \int_{x_n}^{x_{n + 1} } \map f {x, y} \rd x\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \map y {x_{n + 1} }\) | \(=\) | \(\displaystyle \map y {x_n} + \int_{x_n}^{x_{n + 1} } \map f {x, y} \rd x\) | ||||||||||

\(\displaystyle \) | \(\approx\) | \(\displaystyle \map y {x_n} + \map f {x_n, y_n} \paren {x_{n + 1} - x_n}\) |

The errors accumulate; with increasing $n$ the values of $y_{n + 1}$ are based on increasingly inaccurate values of $y_n$.

These can be reduced by making $h$ smaller, so the inaccuracies can be reduced by increasing the computation needed.

$\blacksquare$

## Source of Name

This entry was named for Leonhard Paul Euler.

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): Miscellaneous Problems for Chapter $2$: Appendix $\text{A}$. Numerical Methods