A full dual mean square error criterion for the equivalent linearization

Abstract

Among approximate methods, the method of equivalent linearization proposed by N. Krylov and N. Bogoliubov and extended by Caughey has remained an effective tool for both deterministic and stochastic problems. The idea of the method is based on the replacement of a nonlinear oscillator by a linear one under the same excitation. The standard way of implementing this method is that the coefficients of linearization are to be found from a criterion of equivalence. When the difference between the nonlinear function and equivalent linear one is significant the replacement leads to unaccepted errors. In order to reduce the errors one may apply the dual approach. One of significant advantages of the dual conception is its consideration of two different aspects of a problem in question allowing the investigation to be more appropriate. In this paper a special case of the weighted full dual mean square error criterion is introduced and investigated in detail. Numerical results are carried out to show that this special full dual mean square error criterion can give more accurate approximate solutions for both deterministic and random nonlinear systems.