A method for the numerical or mechanical solution of certain types of partial differential equations

Abstract

The development of mechanical means of evaluating solutions of ordinary differential equations, in the form of the differential analyser of Dr. Bush (Bush 1931; Hartree 1935), has made it feasible to undertake the investigation of many problems of scientific and technical interest leading to differential equations which have no convenient formal solution, and which are too elaborate, or for which the range of solutions required is too extensive, for calculation of the solutions by numerical methods to be practicable. The practical success of this machine, and the wide range of equations to which it can be applied, have led to the hope that it may be found possible to apply it to partial differential equations, which are usually regarded as less amenable to numerical methods than ordinary equations. The present paper gives one way of applying it to such equations in two independent variables with certain types of boundary conditions. As will appear, the possibility of applying this method depends more on the form of the boundary conditions than on the exact form of the equations. The method is particularly suited to the differential analyser, though it is also practicable for numerical work.