For a small step size , the derivative is close enough to the ratio . In the Euler's method, such an approximation is attempted. To recall, we consider the problem (). Let be the step size and let with . Let be the approximate value of at . We define
The integral on the right hand side is approximated, for sufficiently small value of , by
and neglecting terms that contain powers of that are greater than or equal to .
We illustrate the Euler's method with an example. The example is only for illustration. In (), we do not need numerical computation at each step as we know the exact value of the solution. The purpose of the example is to have a feeling for the behaviour of the error and its estimate. It will be more transparent to look at the percentage of error. It may throw more light on the propagation of error.
The Euler's algorithm now reads as
See the Tables and for the calculation of errors (up to places of decimal).
We now give a sample flow chart for the Euler's method.
A K Lal 2007-09-12