Mechanical and Aerospace Engineering

MAE 297 Seminar

A numerically controlled system of analytics for the exact integration of  the Navier-Stokes Equations.

Mike Mikalajunas

Center for Innovations in Mathematical Equations.

Certain traditional methods of Calculus for solving differential equations and systems of differential equations in Scientific and Engineering analysis depend in one form or another on the use of some general initially assumed analytical representation of the intended solution. Unfortunately this often leads to defining one or several integrals that cannot always be resolved exactly. In order to avoid this complication, all initially assumed mathematical equations whether defined in explicit or in implicit form are instead initially assumed as a system of differentials involving multivariate polynomials all defined in a very unique algebraic configuration. The computations involved when substituting such an initially assumed system of differentials with unknown coefficients to solve for into any type of DEs and systems of DEs would lead to solving for a system of nonlinear simultaneous equations all of which are known to always consist of an infinite number of numerical solution sets. Such a unique differential representation of all mathematical equations will be presented in the context of establishing by means of direct computation an exact mathematical condition by which the Navier-Stokes Equations may always become integrable in terms of complete analytical solutions. Based on the unique computational structure of this general differential algorithm, all initial conditions would be represented in a very special data configuration that would become part of the large system of nonlinear simultaneous equations to solve for. There are two main conditions that would guarantee the complete analytical integrability of Navier-Stokes Equations which are (1) existence of exact numerical solution sets of the system of nonlinear simultaneous equations and as a result of this, (2) each of the corresponding initially assumed differentials satisfy the complete test for exactness and thus become much more easily integrable by virtue of each of their unique exactness property.

Under such circumstances as the systems of nonlinear simultaneous equations in question are expected to consist of an infinite number of exact numerical solution sets, this in turn would always give rise to an infinite number of exact differentials to integrate as well. Some of these exact differentials would be much more easily integrable than others thereby significantly increasing our chances for arriving at complete general analytical solutions satisfying certain cases of the Navier-Stokes Equations. In fact because of the universality of the proposed differential method of integration, one of the simple cases of the Navier-Stokes Equations under incompressible flow may represent the best possible test case that can be used for validating the new mathematical theory by investigating the exact numerical solution sets obtained for the corresponding system of nonlinear simultaneous equations to solve for and determine which of the “infinite” number of corresponding “exact” systems of differentials are the simplest to integrate. Solutions to other unsolved cases of the Navier-Stokes Equations may potentially be uncovered provided that a universal software be written for completely automating the entire process of solving for DEs and systems of DEs by the use of differentials defined in a very unique configuration.

Bio: I have a degree in Mechanical Engineering with specialization in fluid option and have taken part in a number of research projects with various faculty members as an undergraduate student. This would include a flight simulation project in conjunction with Canadian Aviation Electronics (CAE) and also some research development work related to mechanical vibrations. I have also provided some support for the development of the software program AUTO (software for continuation and bifurcation problems in ordinary differential equations) with a faculty member in the computer science department. I later developed a great deal of interest in computer software programming mainly for the PC towards the health care industry with more than 20 years of experience acting as a consultant for a private software development company ever since it was founded back in the mid 80’s. Because of my extensive software involvement with this company, I have been assigned to maintain and support our existing statistical software at Boehringer Ingelheim pharmaceutical division and at Novartis Animal Health division which is now owned by Elanco, a division of Eli Lilly. Under a special software license agreement I deal mainly with each of their international divisions by running our sales forecasting and assumption reporting software on a monthly basis for meeting their total manufacturing process requirements.

Date(s) - 03/16/2017
4:00 pm - 5:00 pm

1062 Bainer Classroom


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