A Procedure for Applying the Extended Kantorovich Method to Nonlinear Problems

Abstract

The Kantorovich method as extended by Kerr is an elegant iterative technique applicable to some multivariable problems in applied mechanics. Kerr has shown, in a clamped rectangular plate example, that a one-term solution agreed closely with existing results obtained by more elaborate means. However, the analyzed problems were limited to linear formulations only. The present paper suggests a general purpose nonlinear method, which follows the iterative procedure of the extended Kantorovich technique, to derive approximate solutions of problems that involve simultaneous, multifunctional nonlinear partial differential equations. The solutions are analytic and require much less numerical efforts than, for instance, a corresponding nonlinear finite-difference equation method for comparable accuracy. An example is given for the deflection solutions of a plate under lateral pressure, by using von Karman’s large-deflection plate equations. Numerical calculations showed that a one-term solution of the present method agreed within 0.03 percent with the result of a nonlinear finite-difference method extrapolated to infinite nodes. A highly accurate one-term solution which is continuous and differentiable, sometimes expressible in closed forms, would be valuable in nonlinear analysis as well as in linear problems mixed with local nonlinear, postbuckled regions.