Since their introduction over fifty years ago,ADI methods have been employed effectively for the time stepping in the numerical solution of a variety of time-dependent multidimensional problems.Their primary attraction is that they reduce such a problem to independent systems of one-dimensional problems.On the other hand,orthogonal spline collocation(OSC)has evolved as a valuable technique for the spatial discretization of several types of differential equations.Their popularity is due in part to their conceptual simplicity,wide applicability and ease of implementation.A well-known advantage of OSC methods over finite element Galerkin methods is that the calculation of the coefficients in the equations determining the approximate solution is very fast,since no integrals need to be evaluated or approximated.Another attractive feature of OSC methods is their superconvergence properties.In this talk,we describe an ADI method for the solution of a class of two-component nonlinear reaction-diffusion problems on evolving domains.In our new method,OSC is used for the spatial discretization,while the time-stepping is done with an algebraically linear ADI method based on an extrapolated Crank-Nicolson OSC method.The ADI OSC method is efficient,requiring at each time level only O(N)operations where N is the number of unknowns.The results of numerical experiments involving well-known examples of reaction-diffusion systems from the literature demonstrate its efficacy.