Simulating the joint impact of temporal and spatial memory indices via a novel analytical scheme

Abstract

The prime concern of this study is to simulate the joint effect for the presence of two fractional derivative parameters (memory indices) by providing a novel analytical solution scheme for the fractional initial value problems. Our goal has been fulfilled by extending the residual power series method into the two-dimensional time and space, with time and space endowed with fractional derivative orders $$\\alpha $$\n and $$\\gamma $$\n , respectively (simply denoted by fractional $$(\\alpha ,\\gamma )$$\n -space), by virtue of a new $$(\\alpha ,\\gamma )$$\n -fractional power series representation (\n $$(\\alpha ,\\gamma )$$\n -FPS). The necessary theoretical framework for the convergence and the error bound is also provided to enrich our analytical study. Among other main findings, it is deserved to mention that the fractional derivative parameters act like the homotopy parameters, in a topological sense, to generate a rapidly convergent series solution for the classical integer version of the problem under consideration, which promotes the idea that these parameters describe a remnant memory. The efficiency of the proposed approach is assessed by projecting the obtained solutions of several well-known (non)linear problems into lower-dimensional fractal space and/or into integer space and then comparing them with the corresponding results of the literature. Overall, the method shows a wide versatility and adequacy in dealing with such hybrid problems.