Power series solution for fractal differential equations

Abstract

In this paper, we provide an overview of fractal calculus as applied to fractal sets. We introduce the concept of power series defined on fractal sets and explore series solutions for second alpha-order fractal linear differential equations, particularly focusing on solutions near ordinary points. We also investigate the radius of convergence for these fractal series, elucidating the region where these series representations are valid. Furthermore, we employ the fractal Frobenius method to derive series solutions for second alpha-order linear fractal differential equations near regular singular points. By utilizing this method, we obtain analytical expressions that are applicable in the vicinity of such singularities, enhancing our understanding of fractal differential equations. To illustrate our findings, we graphically represent these solutions, providing visual demonstrations of the outcomes. These graphs serve to complement our analytical results and offer insights into the behavior of solutions within the context of fractal calculus and differential equations on fractal sets.