This paper presents a fundamentally new approach to the numerical solution of partial fractional differential equations (PFDE) in higher dimensions by means of hypermatrix equations. By generalizing matrices to their higher dimensional form, i.e., hypermatrices, we show that there is a one-to-one correspondence between PFDE and hypermatrix equations. It is shown that the resulting hypermatrix equation can be solved in an expedient manner, namely by an O (n4) algorithm for an l x m x n discretized integral surface with l ~ m ~ n. Given that previous algorithms were of order O (n9) this represents a massive improvement in computational complexity. The proposed algorithm is demonstrated for a problem in two spatial and one time dimension; however, the algorithm can be extended to higher dimensions as well.