The following are not *stiff* differential equations, however, the
techniques may still be applied.

# Example 1

Given the IVP y^{(1)}(*t*) = 1 - *t* y(*t*) with
y(0) = 1, approximate y(1) with one step.

First, let *t*_{0} = 0, *y*_{0} = 1, and *h* = 1.
Thus, we write down the equation

*y*

_{0}+

*h**f(

*t*

_{1}, υ) = 0

and, after substituting the appropriate values, we get

Solving this equation yields *υ* = 1, and therefore we set *y*_{1} = 1.
The absolute error is 0.33.

# Example 2

Given the same IVP shown in Example 1, approximate y(0.5).

First, let *t*_{0} = 0, *t*_{1} = 0.5, *y*_{0} = 1, and *h* = 0.5.
Thus, we write down the equation

*y*

_{0}+

*h**f(

*t*

_{1}, υ) = 0

and, after substituting the appropriate values, we get

Solving this equation yields *υ* = 1.2, and therefore we set *y*_{1} = 1.2.
The absolute error is 0.14 which is approximately the absolute error in Example 1.

# Example 3

Repeat Examples 1 and 2 but with with the initial value y(0.5) = 2.5 and approximating y(1.5) and y(1.0).

To find y(1.5), let *t*_{0} = 0.0, *t*_{1} = 1.5, *y*_{0} = 2.5, and *h* = 1.
Thus, the equation

Solving this equation yields *υ* = 1.4, and therefore we set *y*_{1} = 1.4.
The actual value is y(1.5) = 1.502483616, and therefore the absolute error is 0.102.

To find y(1.0), let *t*_{0} = 0.0, *t*_{1} = 1.0, *y*_{0} = 2.5, and *h* = 0.5.
Thus, the equation is

Solving this equation yields *υ* = 2, and therefore we set *y*_{1} = 2.
The actual value is y(1) = 2.126611964 and therefore the absolute error is 0.127.

This absolute error is larger than it was when *h* = 1.0, and thus, to show that
the error is O(*h*^{2}), we must use smaller values of *h*. These are shown in Table 1.

Table 1. Errors when approximating y(*t*_{0} + *h*) for decreasing values of *h*.

h | Approximation of y(0.5 + h) | Error |
---|---|---|

1. | 1.4 | 0.102 |

0.5 | 2. | 0.127 |

0.25 | 2.315789474 | 0.0528 |

0.125 | 2.434782609 | 0.0160 |

0.0625 | 2.475471698 | 0.00434 |

0.03125 | 2.489913546 | 0.00112 |

0.015625 | 2.495519495 | 0.000285 |

0.0078125 | 2.497902608 | 0.0000719 |

0.00390625 | 2.498987283 | 0.0000181 |

0.001953125 | 2.499502670 | 0.00000452 |

0.0009765625 | 2.499753595 | 0.00000113 |

The quadratic behaviour becomes obvious with the last step, being smaller by almost exactly 4.

Copyright ©2005 by Douglas Wilhelm Harder. All rights reserved.