Analytic simulation of the synergy of spatial-temporal memory indices with proportional time delay

In the present work, three space-time trace parameters are appended to physical systems to analytically outline their mutual impact and to characterize the dynamic behaviors of these systems; namely the proportional time delay τ∈(0,1) and the Caputo spatial-temporal fractional derivatives α,β∈(0,1). The adopted analytical approach depends on a novel adaptation of the differential transform method in a higher dimensional fractional space in which the initial value problems (IVPs), under consideration, are transformed into a 2-dimensional recurrence relation. Some central differential transformation theorems in 2-dimensional fractional space are provided to illustrate the influence of the aforementioned parameters. The method has been successfully applied to furnish the solution, in the form of a Cauchy product of absolutely convergent series, for a 2-dimensional extension of advection-dispersion, gas dynamics, convection-diffusion, wave, telegraph, and Klein–Gorden equations. The study concluded that the obtained solutions operate as a homotopic mapping between two states, and the Caputo fractional derivatives can be interpreted as memory indices.