A method for obtaining approximate solutions for highly dynamic problems

As a rule, a theoretical analysis of the behavior of highly dynamic systems is very difficult due to the strong non-linearity of the governing equatio ns. Basic results are usually achieved by the application of the inverse sc attering transform methods, methods of perturbation theory and numerical ap proaches, however, all the above-mentioned methods and approaches have well -known limits in their application. In this paper, a dynamic system describ ed by linear hyperbolic partial differential equations with a non-linearity localized in a space-time domain is considered. The application of the the ory of laws of conservation together with the Huygens' principle allows the generation of a family of integral inequalities by using the solution of t he corresponding linear problem with the same initial data. In turn, these integral inequalities make it possible to formally reduce the initial probl em for locally non-linear hyperbolic equations to an extremal problem at li mitations (restrictions) defined by these integral inequalities. Thus, uppe r and lower bound estimates of the solution of the locally non-linear probl em can be obtained from the solution of this extremal problem to which stan dard techniques can be applied. The method under development has many advan tages when compared with known approaches. These advantages together with i ts limitations are discussed in this paper. Examples of this new method as applied to some locally non-linear problems of dynamic elasticity are also considered. (C) 2001 Elsevier Science Ltd. All rights reserved.