A Set of Principles to Interconnect the Solutions of Physical Systems

A set of principles and a systematic procedure are presented to establish the exact solutions of very large and complicated physical systems, without solving a large number of simultaneous equations and without finding the inverse of large matrices. The procedure consists of tearing the system apart into several smaller component systems. After establishing and solving the equations of the component systems, the component solutions themselves are interconnected to obtain outright, by a set of transformations, the exact solution of the original system. The only work remaining is the elimination or solution of the comparatively few superfluous constraints appearing at the points of interconnection. The component and resultant solutions may be either exact or approximate and may represent either linear or, with certain precautions, nonlinear physical systems. The component solutions may be expressed in numerical form or in terms of matrices having as their elements real or complex numbers, functions of time, or differential or other operators, etc. Boundary value, characteristic value, and time-varying problems of partial differential equations, as well as problems in ordinary differential equations and algebraic equations may be solved in this manner. The method shown may be extended and generalized so that one can tear apart and afterward reconstruct the solution of extremely large or highly intricate physical systems, often without calculating any inverse matrices at all and always without carrying along unduly large matrices. This extension and generalization of the method is analogous to building skyscrapers by erecting first a steel framework and only afterward filling the gaps between the girders as needed. Those versed in the science of tensorial analysis of interrelated physical systems on the one hand and of large electrical networks on the other, should thereby be able to solve, with the aid of already available digital computers, highly complex physical systems possessing tens of thousands and, in special cases, even hundreds of thousands of variables. The accuracy of machine calculations, of coding and even the correctness of the analytical procedure itself may be simultaneously checked by physical tests at various stages of the computation. The saving in computing time is considerable even in smaller problems; by tearing a physical system into n parts, the usual machine calculations are reduced, in matrix inversion for instance, to a fraction of about 2/n2. The present paper develops in detail the solution of a simple boundary value problem of Poisson's equation. A numerical example of interconnecting the solutions of large electric-power transmission systems appears in reference 3. Many simpler numerical examples are worked out in reference 1.