Numerical approximation of one dimensional fractional transmission problem

Abstract

Fractional partial differential equations (FPDEs) have attracted significant attention in recent years, owing to their diverse applications across numerous scientific and engineering domains. Often, fractional-order models prove to be more suitable than their integer-order counterparts, as fractional derivatives and integrals facilitate the description of memory properties inherent in various materials and processes. In this context, investigation has been undertaken on a fractional-in-time transmission problem spanning two disjoint intervals. An a priori estimate has been established for its weak solution within a suitable Sobolev-like function space. The study delves into the well-posedness of an interface problem associated with this equation, demonstrating its stability within corresponding Sobolev-like function spaces. Furthermore, a finite difference scheme has been developed to approximate this problem, accompanied by a thorough analysis of its properties. An estimation of the convergence rate has been derived, aligning with the smoothness characteristics of the input data. A proposed difference scheme has been put forth and validated through several numerical examples.