Abstract:
The importance of theoretical models in Science and Engineering far outweighs that of experimental based models. The result of our lack of transparency towards the use of a more unified approach to analytical integration for solving some of the most difficult problems related to the Physical and Biological Sciences has forced us to become to dependent on the use of experimental based models. In reality this has never been a matter of choice for all of us but rather as a direct consequence in our failure to fully understand exactly why the vast majority of differential equations behave the way they do by not admitting highly predictable patterns of analytical solutions for resolving them.
In this talk I will begin by extending the traditional concept of a “differential” in Calculus by introducing an entirely new algorithm capable of representing all mathematical equations consisting of only algebraic and elementary functions in complete specialized differential form. Such a universal algorithm would involve the use of multivariate polynomials and the differential of multivariate polynomials all defined in a very unique algebraic configuration.
At first glance this may not sound like a major breakthrough in the Physical Sciences but progressively throughout this entire presentation, it will become very apparent that such a specialized differential representation of all mathematical equations would lead to some form of a unified theory of integration. It is only from the general numerical application of such a universal theory in mathematics can we expect to arrive at some form of a unified theory of Physics. This would be constructed from the development of very advanced physical models that would be built exclusively on general rather than on the local analytical solutions of many well known fundamental differential equations of the Physical and Biological Sciences.
We will be presenting a very large amount of empirical results that were gathered from the numerical application of our unified theory of integration on a number of very specific mathematical models. This would include a general first order ODE followed by a second order PDE where a detailed empirical analysis of the data collected on each of these differential equations would lead to their complete integration in terms of generalized analytical solutions involving only the algebraic and elementary functions.
We will also be presenting a series of Physical models which have been chosen very carefully just for demonstrating the applicability of our unified theory of integration into the Physical Sciences. These will include the equations for describing general linear elasticity and a very specific case of the Navier-Stokes equations for an incompressible fluid with heat transfer and variable viscosity. For each of these physical models we will be developing a universal numerical process that would be based entirely on the general application of our specialized differential form representation of all mathematical equations for the exact integration of the corresponding set of PDEs in terms of only generalized exact analytical solutions that can satisfy a wide range of boundary conditions.
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