Chaplygin Method

a method of approximate integration of differential equations. Proposed by S. A. Chaplygin in 1919, the method permits an approximate solution to be found for a differential equation to a given degree of accuracy. It involves the construction of sequences of functions {u_n} and {v_n} that approximate with continually increasing accuracy the desired solution y of a given differential equation and that fulfill the following relations:

u_n ≥ u_{n + 1} ≥ y ≥ v_n+1v_n

The construction of the sequences {u_n} and {v_n} is based on Chaplygin’s theorem of differential inequalities and is a generalization of Newton’s method to the case of differential equations. The rate of convergence is the same as in Newton’s method; that is, u_n – y tends to zero like

C/2²ⁿ