The Conservative Averaging Method: Applications, Theory and New Hyperbolic Approximation

摘要

In this paper we look over history of conservative averaging method in last 100 years. Some well known mathematicians in classical papers investigated problems with non-classical boundary conditions, which have derivatives in boundary conditions equal or higher as derivatives in the main partial differential equation. Forty or fifty years ago it was important to construct mathematical models for intensification of the crude oil output. We have experience in around 30 years in these fields. Most of the papers were published in Russian language. We have made short overlook of some most important papers. In these papers were investigating problems with non-classical boundary conditions, mathematical basis of conservative averaging was constructed; integral spline, which fulfil in new problem formulation the energy or mass conservation was introduced. We introduce new hyperbolic approximation for conservative averaging method. In the paper is given a new representation for classic cubic spline.