86 Ordinary Differential Equations
2.4.1.2 Runge-Kutta Methods
Among the more popular single-step numerical approximation methods are
those developed by the German mathematicians Carl David Tolme Runge
(1856-1927), Karl Heun (1859-1929), and Martin Wilhelm Kutta (1867-1944).
Runge was an appli ed mathematician who studied spectral lines of elements
and Diophantine equations. He devised and published his numerical technique
in 1895. In 1900 , Karl Heun published a paper concerning the improvement
of Runge's method and in 1901 Kutta extended Runge's method to systems
of equations. Kutta is also well-known for his contributions to airfoil theory.
Improved Euler's Method In deriving Euler's method for approximat-ing the solution of the IVP (1) y' = f(x , y); y(xo) = yo, we noted that the
IVP (1) is equivalent to the integral equation
y(x) =Yo+ 1x f(t, y(t)) dt
XQand we replaced the integrand f(t, y(t)) over the entire interval [xo, x1] by
its approximate value at the left endpoint, f(xo,Yo). Upon integrating from
x 0 to x 1 , we obtained Euler's formula for approximating the solution to the
IVP (1) at x 1. A more accurate approximation may be obtained, if, instead of
approximating the integrand by its approximate value at the left endpoint of
the interval of integration, we approximate it by the average of its approximate
values at the left endpoint and the right endpoint. Thus, when solving the
general IVP y' = f(x , y); y(xn) = Yn on the interval [xn, Xn+1] which is
equivalent to the integral equation
(13) Y(Xn+I) = Yn + 1:n+i J(t, y(t)) dt
we replace the integrand f(t, y(t)) by the constant
Substituting this expression into (13) and integrating, we obtain the following
expression for the approximation of the solution to the IVP (1) at Xn+i:
This equation involves the unknown Yn+I as an argument of f on the right-
hand side; and, therefore, will generally be difficult or impossible to solve
expli citly for Yn+l · Instead of trying to solve this equation for Yn+l, we simply
replace the Yn+I appearing on the right-hand side by the approximation we
obtain using Euler's method- namely, Yn+l = Yn + f(xn, Yn)(xn+l - xn)-The
following recursive formula which results is known as the improved Euler's