Numerical Solution of Ordinary and Partial Differential Equations by Means of Equivalent Circuits

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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.