73 lines
2.6 KiB
Markdown
73 lines
2.6 KiB
Markdown
---
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title: Euler's Method
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---
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# Euler's Method
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The Euler's method is a first-order numerical procedure for solving ordinary differential equations (ODE) with a given initial value.
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## The General Initial Value Problem
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## Methodology
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Euler's method uses the simple formula,
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to construct the tangent at the point `x` and obtain the value of `y(x+h)`, whose slope is,
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<img src="https://raw.githubusercontent.com/pranabendra/articles/master/Euler-method/images/Euler.png" width="600">
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In Euler's method, you can approximate the curve of the solution by the tangent in each interval (that is, by a sequence of short line segments), at steps of `h`.
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<i>In general</i>, if you use small step size, the accuracy of approximation increases.
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## General Formula
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## Functional value at any point `b`, given by `y(b)`
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where,
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* <b>n</b> = number of steps
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* <b>h</b> = interval width (size of each step)
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### Pseudocode
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## Example
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Find `y(1)`, given
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Solving analytically, the solution is <i><b>y = e<sup>x</sup></b></i> and `y(1)`= `2.71828`. (Note: This analytic solution is just for comparing the accuracy.)
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Using Euler's method, considering `h` = `0.2`, `0.1`, `0.01`, you can see the results in the diagram below.
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When `h` = `0.2`, `y(1)` = `2.48832` (error = 8.46 %)
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When `h` = `0.1`, `y(1)` = `2.59374` (error = 4.58 %)
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When `h` = `0.01`, `y(1)` = `2.70481` (error = 0.50 %)
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You can notice, how accuracy improves when steps are small.
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## More Information:
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1. <a href='http://calculuslab.deltacollege.edu/ODE/7-C-1/7-C-1-h-c.html' target='_blank' rel='nofollow'>Numerical Methods for Solving Differential Equations</a>
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2. <a href='https://en.wikipedia.org/wiki/Euler_method' target='_blank' rel='nofollow'>Euler's method</a>
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