Higher Order Differences in the Analogue Solution of Partial Differential Equations

Most analogue devices for the solution of Partial Differential Equations in two or more variables necessarily treat one or more of the continuous independent variables by finite difference techniques. In this paper we shall consider the application of such techniques only to Partial Differential Equations in two Variables (although the general ideas are not restricted to these). In particular we shall concentrate our attention on those devices, such as electronic and mechanical differential analyzers, [1] which handle the first independent variable continuously this is the variable whose analogue is physical time or some function of time. These devices treat only the second independent variable by finite difference techniques. The use of these techniques introduces certain mathematical inaccuracies into the solution of the partial differential equation. These inaccuracies are independent of any physical errors in the computing process such as might be due to the finite bandwidth of a computing amplifier, or to the difficulties of making a precise physical measurement. In some circumstances the mathematical inaccuracies may be smaller than the physical computing errors and may be ignored safely. Frequently, however, they will be more important than the physical errors and must be taken into account. Now the errors due to the use of a finite difference approximation may always be reduced by the use of a smaller interval. Mathematically this is the simplest procedure and would normally be used. In practical analogue computation this means either the use of proportionately more equipment or of proportionately more time for the solution. Both of these alternatives may be undesirable, or even unrealisable. Furthermore the reduction of the interval may lead to increased instability in the solution process. Another method of reducing the finite difference errors is to use a higher order approximation. This method is the concern of the of avoiding the two undesirable consequences of the expense of slightly greater complication in t~