The amazing fractional Maclaurin series for solving different types of fractional mathematical problems that arise in physics and engineering

Abstract

Many applications and natural phenomena in the fields of physics and engineering are described by ordinary and partial differential equations. Therefore, obtaining solutions to these equations helps to analyze and understand the dynamics of these systems, as well as to identify the factors affecting them. Extraordinary differential equations represent a generalization of differential equations through the application of fractional calculus. The main objective of this paper is to accommodate the Maclaurin series, one of the available distinctive methods, to solve linear and nonlinear fractional differential equations. We discuss the solution of some types of fractional problems, including linear/nonlinear FDEs, nonlinear FPFEs, fractional integro-differential equations, and nonlinear fractional autonomous system. The suggested examples simulate many physical applications and natural phenomena. We believe that the findings of this work will be of benefit to scholars and graduate students to adopt the proposed scheme to extract closed-form solutions to different types of fractional problems.