Trapezoidal methods of approximating solutions of differential equations

Preston C. Hammer, Jack W. Hollingsworth
1955 Mathematics of Computation  
Introduction. Seemingly by historical accident the points of view usually adopted for stepwise methods of numerical solution of differential equations largely emphasize the discrete values found rather than the functions pieced together by the method over various intervals of advance. The authors, collaborating as a professor and student team, have found that the straightforward selection of functions to approximate solution functions offers many advantages conceptually and makes it possible to
more » ... see much-used methods in a new light. We present here a few of the simpler results. The purpose of this paper is to show that, by considering the method called the trapezoidal method (cf. Milne [1] p. 24) as a parabolic or quadratic function method, not only does one obtain a satisfying geometrical picture of an approximation curve in the usual case, but that two new trapezoidal or parabolic methods are suggested. Of the two new methods, one is based on a Gauss integration formula. Thus the new approach makes it possible to use the Gauss integration formulas. We believe that the two methods are new in their application although they are old in their use of polynomials. We have found in certain applications that both the new formulas have points of preference in some cases over the simple
doi:10.1090/s0025-5718-1955-0072547-2 fatcat:6k23pae72nfw7i6ycnk57mbkhi