where denotes , subject to the general initial-value boundary condition

(6) |

It is important to appreciate that the numerical solution to a differential
equation is only an *approximation* to the actual solution. The actual
solution, , to Eq. (5) is (presumably)
a *continuous* function of a continuous
variable, . However, when we solve this equation numerically, the best that we can
do is to evaluate approximations to the
function at a series of *discrete* grid-points, the (say), where
and
. For the moment, we shall restrict our
discussion to *equally spaced* grid-points, where

(7) |

The simplest possible integration scheme was invented by the celebrated
18th century Swiss mathematician Leonhard Euler, and is, therefore, called
*Euler's method*. Incidentally, it is interesting to note that virtually
all of the standard methods used in numerical analysis were invented
*before* the advent of electronic computers. In olden days, people
actually performed numerical calculations *by hand*--and a very long and tedious
process it must have been! Suppose that we have evaluated
an approximation, , to the solution, , of Eq. (5) at the grid-point
. The approximate gradient of at this point is, therefore, given by

(8) |

(9) |

The above formula is the essence of Euler's method. It enables us to calculate all of the , given the initial value, , at the first grid-point, . Euler's method is illustrated in Fig. 4.