How to compute a Taylor series with respect to a coordinate system, when this series is known with respect to alternative coordinates? This paper gives an answer to this question, as well as applications to Partial Differential Equations. An alternative technique is also proposed by applying the change of variables directly on the PDE. Such procedures prove to be efficient for the numerical solution of PDEs, especially to accelerate the convergence or in the case of exterior domains. These new techniques were motivated by PDEs solved in the framework of the so-called Taylor Meshless Method, but the application field could cover any numerical method based on Taylor series.