A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type

A, k, q are known constants. Equation (1) is of the type which arises in problems of heat flow when there is an internal generation of heat within the medium; if the heat is due to a chemical reaction proceeding at each point at a rate depending upon the local temperature, the rate of heat generation is often defined by an equation such as (2). The presence of the non-linear term q(Ow/at) in (1) and the empirical nature, in many cases, of the surface heat transfer function Hi (0), render the use of formal methods unsuitable. Now Hartree [3-5] has suggested two methods (methods I and II below) of evaluating approximate solutions of partial differential equations in two