On the Extension of the Kantorovich Method

Abstract

In ref. 1, an extension of the Kantorovich variational method of minimising a functional is discussed. The extension is based on an iterative technique wherein functions derived from the solution of sets of simultaneous ordinary differential equations are used again in cyclic order to obtain an improved solution. It is conjectured that if the iterative procedure is continued indefinitely, the functional will be close to its correct value, and so also will the associated function. The method given in ref. 1 is a generalisation of the iterative technique for minimising a double integral given by the writer in ref. 4, although in this work the iterations are carried out in the first approximation of the Kantorovich method and convergence is obtained in the usual manner of Kantorovich. Like ref. 4, ref. 1 uses the classical torsion problem of a rectangular bar as an illustrative example. Each shows that accurate Prst approximation solutions can be obtained and ref. 1 goes on to show that, for this particular problem, the derived functions are unique and are not dependent on the initial choice for part of the function.