The Initial Value Problem y' = f(x, y); y(c) = d 732.4 Numerical Solution
The oldest and simplest algorithm for generating a numerical approxima-
tion to a solution of a differential equation was developed by Leonhard Euler
in 1768. Given a specific point (xo, Yo) on the solution of the differential equa-
tion y' = f(x, y), Euler wrote the equation for the tangent line to the solutionthrough (xo, Yo)- namely, y =Yo+ f(xo, Yo)(x - xo). To obtain an approxi-
mation to the solution through (xo, Yo) at x 1 , Euler took a small step along
the tangent line and arrived at the approximation Y1 =Yo+ f(xo, Yo)(x 1 - x 0 )
to the solution at x1, y(x1). Continuing to generate points successively in
this manner and by connecting the points ( xo, Yo), ( x1, Y1), ( Xz, Y2), ... in
succession, Euler produced a polygonal path which approximated the solu-
tion. This first numerical algorithm for solving the initial value problem
y' = f(x, y); y(xo) = Yo is called Euler's method or, due to its particular
geometric construction, the tangent line method.
Euler's method is a one step method. In one step methods only one solution
value, (xo, y 0 ), is required to produce the next approximate solution value. On
the other hand, multistep methods require two or more previous solution val-
ues to produce the next approximate solution value. In 1883, more than a
century after Euler developed the first one step method, the English mathe-
maticians Francis Bashforth (1819-1912) and John Couch Adams (1819-1892)
published an article on the theory of capillary action which included multistep
methods that were both explicit methods and implicit methods. In 1895, the
German mathematician Carl D avid Tolme Runge (1856-1927) wrote an article
in which he developed two one step methods. The second order method was
based on the midpoint rule while the third order method was based on the
trapezoidal rule. In an article which appeared in 1900, Karl Heun (1859-1929)
improved Runge's results by increasing the order of the method to four. And
in 1901, Martin Wilhelm Kutta (1867-1944) completed the derivation for the
fourth-order methods by finding the complete set of eight equations the co-
efficients must satisfy. He also specified the values for the coefficients of the
classic fourth-order Runge-Kutta method and those of a fifth-order method.
Prior to 1900, most calcul ations were performed by hand with paper and
pencil. Euler's method and Runge-Kutta methods are one step methods.
Euler's method is of order one and requires only one f function evaluation
per step. The classic Runge-Kutta method is fourth order and requires four
f function evaluations per step. Adams-Bashforth, Adams-Moulton, and
predictor-corrector methods require only two f function evaluations per step;
however, since these methods are multistep methods, they require starting
values obtained by some other method. By the 1930s significant numerical
integration techniques had been developed; however, their effective implemen-
tation was severely limited by the need to perform the computations by hand
or with the aid of primitive mechanical calculators.