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Euler's Method |
Euler's Method
The Euler's method is a first-order numerical procedure for solving ordinary differential equations (ODE) with a given initial value.
The General Initial Value Problem
Methodology
Euler's method uses the simple formula,
to construct the tangent at the point x
and obtain the value of y(x+h)
, whose slope is,
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
.
In general, if you use small step size, the accuracy of approximation increases.
General Formula
Functional value at any point b
, given by y(b)
where,
- n = number of steps
- h = interval width (size of each step)
Pseudocode
Example
Find y(1)
, given
Solving analytically, the solution is y = ex and y(1)
= 2.71828
. (Note: This analytic solution is just for comparing the accuracy.)
Using Euler's method, considering h
= 0.2
, 0.1
, 0.01
, you can see the results in the diagram below.
When h
= 0.2
, y(1)
= 2.48832
(error = 8.46 %)
When h
= 0.1
, y(1)
= 2.59374
(error = 4.58 %)
When h
= 0.01
, y(1)
= 2.70481
(error = 0.50 %)
You can notice, how accuracy improves when steps are small.