Mathematical Methods for Physics and Engineering - Matematica.NET

27.6 DIFFERENTIAL EQUATIONSx h y(exact)0.01 0.1 0.5 1.0 1.5 2 30 (1) (1) (1) (1) (1) (1) (1) (1)0.5 0.605 0.590 0.500 0 −0.500 −1 −2 0.6071.0 0.366 0.349 0.250 0 0.250 1 4 0.3681.5 0.221 0.206 0.125 0 −0.125 −1 −8 0.2232.0 0.134 0.122 0.063 0 0.063 1 16 0.1352.5 0.081 0.072 0.032 0 −0.032 −1 −32 0.0823.0 0.049 0.042 0.016 0 0.016 1 64 0.050Table 27.10 The solution y of differential equation (27.61) using the Eulerforward difference method for various values of h. The exact solution is alsoshown.27.6.1 Difference equationsConsider the differential equationdy= −y, y(0) = 1, (27.61)dxand the possibility of solving it numerically by approximating dy/dx by a finitedifference along the lines indicated in section 27.5. We start with the forwarddifference( ) dy≈ y i+1 − y i, (27.62)dxx ihwhere we use the notation of section 27.5 but with f replaced by y. Inthisparticular case, it leads to the recurrence relation( ) dyy i+1 = y i + h = y i − hy i =(1− h)y i . (27.63)dxiThus, since y 0 = y(0) = 1 is given, y 1 = y(0 + h) =y(h) can be calculated, andso on (this is the Euler method). Table 27.10 shows the values of y(x) obtainedif this is done using various values of h and for selected values of x. The exactsolution, y(x) =exp(−x), is also shown.It is clear that to maintain anything like a reasonable accuracy only very smallsteps, h, can be used. Indeed, if h is taken to be too large, not only is the accuracybad but, as can be seen, for h>1 the calculated solution oscillates (when itshould be monotonic), and for h>2 it diverges. Equation (27.63) is of the formy i+1 = λy i , and a necessary condition for non-divergence is |λ| < 1, i.e. 0