An analytical study of physical models with inherited temporal and spatial memory

Du et al. (Sci. Reb. 3, 3431 (2013)) demonstrated that the fractional derivative order can be physically interpreted as a memory index by fitting the test data of memory phenomena. The aim of this work is to study analytically the joint effect of the memory index on time and space coordinates simultaneously. For this purpose, we introduce a novel bivariate fractional power series expansion that is accompanied by twofold fractional derivatives ordering α, β∈ (0 , 1 ]. Further, some convergence criteria concerning our expansion are presented and an analog of the well-known bivariate Taylor’s formula in the sense of mixed fractional derivatives is obtained. Finally, in order to show the functionality and efficiency of this expansion, we employ the corresponding Taylor’s series method to obtain closed-form solutions of various physical models with inherited time and space memory.