This dissertation describes and tests an Object-Oriented framework, written in Fortran 90, for the Sequential Function Approximation (SFA) algorithm. The SFA algorithm is a meshless method which places its basis functions in the domain sequentially, using optimization techniques. The framework described herein allows the user to define the domain, boundary conditions, and governing equations of 1-D and 2-D problems with minimal user coding, and to solve them using the SFA method. This work advances the state of knowledge in the fields of meshless methods in general and of the SFA method in particular. Unsteady transport problems are solved for the first time with the SFA method: diffusive, convective-diffusive, and purely convective problems are solved using a semi-discrete approach and stabilized with the Streamline-Upwind Petrov-Galerkin (SUPG) technique. Additionally, some light is shed on the role of consistency. SFA is placed within the broader context of meshless methods, and made consistent by transforming it into a sequentially solved Partition of Unity (POU) method. Consistency is experimentally found to improve the convergence behavior of all model problems solved. The improvement is most notable in problems with convection phenomena, although some improvement is seen even in purely diffusive problems. Other hypotheses regarding the SFA method are investigated as well.