Lesson 25 - Learning
Goals
25.1 Learn why Ordinary Differential Equations is useful
25.2 Learn several different Ordinary Differential Equations Techniques
25.3 Learn how to compare different Ordinary Differential Equations algorithms
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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