Next: 1.2 Example: Up: Main Previous: 1.Introduction1.1 Euler's Method
Consider the problem (1.1). Let be the step size. Let , with . Let be the approximate value of y at . We define
(1.1.1) |
where . The method of determination of by (1.1.1) is called the Euler's method.
Remark : Euler's method is an one-step method. The Euler's method has many motivation. Firstly the derivative y' at can be approximated by
if h is sufficiently small. With this approximation in (1.1) we have
we can also look at (1.1.1) from the following point of view. Integration of (1.1) yields
The integral on the right hand side now is approximated by
for sufficiently small .
Moreover if y is differentiable sufficient number of times, we use Taylor's expansion
neglecting terms containing with powers of , we arrive at (1.1.1).
We illustrate the Euler's Algorithm with an example.