This contribution considers approximating smooth multivariate functions via a local modification of the higher degree multivariate fuzzy transform ( \( F^m \) -transform), when the fuzzy partition is constructed via B-splines. The modification seeks to extend the improved approximation properties on the whole domain of the initial function, rather than just a smaller subset where the Ruspini condition is satisfied. The aim of this paper is to provide a practical and computationally efficient implementation of such modification, enabling a wider scope of applications. In particular, we discuss how to apply the proposed approximation scheme for solving partial differential equations and provide some numerical examples of the proposed method. The proposal utilizes the collocation method, with the numerical solution sought as the \(F^m\) -transform of a discrete function.