A time-stepping dual-reciprocity boundary element method is proposed for solving approximately an inverse problem of determining an unknown control parameter p(t) which is the coefficient u(x,y,t) in a diffusion equation in a two-dimensional region R. The diffusion equation is to be solved subject to suitably prescribed initial-boundary conditions. The presence of the unknown coefficient p(t) requires an extra condition. This takes the form of a non-local condition expressed in terms of an integral over R. The method is applied to solve a specific test problem.