LECTURE 25
Ordinary Differential Equations
Introduction
The Euler Method
Given y'= f(x,y)
Approximate y' at point (xi , yi) by the
slope of the line joining it and (xi+1 , yi+1);
y'(xi , yi) (yi+1 - yi) / (xi+1 - xi) = (yi+1 - yi) / h
Then conclude
f(xi , yi) (yi+1 - yi) / h , and yi+1 yi + h f(xi , yi)
Euler algorithm
Starting from the point (0 , yo), we estimate the solution to y' = f(x,y) at x-values seperated by the step size h, until we reach the desired value xf of x. It is assumed that xf = n h for some positive integer n, which is the number of steps taken in attaining the final solution.
The Runge-Kutta method
Advantages (relative to more complicated techniques):
The derivation of the formula is too complex,
but basic idea is to start from (xi , yi) and use Euler method
to obtain different estimates of yi + ½ and yi+1 which are
combined so as to minimize error:
y*i + ½ = yi + h/2 f(xi , yi)
y**i + ½ = yi + h/2 f(xi + ½ , y*i + ½)
y*i + 1 = yi + h f(xi + ½ , y**i + ½)
yi + 1 = yi + h/6 [ f(xi , yi) + 2f(xi + ½ , y*i + ½)
+2f(xi + ½ , y**i + ½) + f(xi + 1
, y*i + 1) ]
To use the Runge-Kutta method, start with y(0) = y0, and apply the above equations (in sequence!) to get y1 = y(h).
Repeat this process for x2,
, xn where
xn = nh.
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