Fractional Spectral and Fractional Finite Element Methods: A Comprehensive Review and Future Prospects

Abstract

In this article, we will discuss the applications of the Spectral element method (SEM) and Finite element Method (FEM) for fractional calculusThe so-called fractional Spectral element method (f-SEM) and fractional Finite element method (f-FEM) are crucial in various branches of science and play a significant role. In this review, we discuss the advantages and adaptability of FEM and SEM, which provide the simulations of fractional derivatives and integrals and are, therefore, appropriate for a broad range of applications in engineering, biology, and physics. We emphasize that they can be used to simulate a wide range of real-world phenomena because they can handle fractional differential equations that are both linear and nonlinear. Although many researchers have already discussed applications of FEM in a variety of fractional differential equations (FDEs) and delivered very significant results, in this review article, we aspire to enclose fundamental to advanced articles in this field which will guide the researchers through recent achievements and advancements for the further studies.