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.

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Numerical methods are developed to solve certain types of linear and nonlinear partial differential equations to any desired degree of accuracy with the aid of equivalent electrical networks. The methods of solution of ordinary differential equations, both linear and nonlinear, follow as special cases. Three types of problems are considered: 1. Initial-value problems. If the field quantities are known along a surface, the networks may be solved by a straight-forward step-by-step calculation. The networks may also be looked upon as supplying a ``schedule'' of operations that can be put on a digital calculating machine. For time-varying problems new types of networks are developed in which time appears as an extra spatial variable. Examples of new networks for the elastic field and for the general nonlinear wave equation are given. Sample calculations and theoretical checks of a transient heat flow problem and of an ordinary differential equation are also included. 2. Boundary-value problems. Four methods of solution are given, the first three being cut and try processes. (a) The method of weighted averages; (b) The method of unbalanced currents and voltages; (c) The ``relaxation'' method; (d) The ``diffusion'' method, that changes the boundary-value problem into an initial-value problem by adding to the original partial differential equation a time variable of the form A∂φ/∂t, allowing the unbalanced currents to ``diffuse'' in time. These numerical methods may also be used to improve the accuracy of the results found on the Network Analyzer. Examples of calculations are given for the electromagnetic and the elastic fields. 3. Characteristic-value problems. Their methods of solution are similar to those of boundary-value problems. An additional method of unbalanced admittances is also indicated. It is shown that by calculating the power in the network, the characteristic value of the assumed function is found. An improved characteristic value of the linear harmonic oscillator, solved initially on a Network Analyzer, is calculated as an example. In general the electrical networks may be used to check the consistency and correctness of solutions arrived at by other methods, approximate or exact. The unbalanced currents at the junctions (easily calculated) give a quantitative measure of the deviation from the correct solution.

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