Second $\alpha$-Order Fractal Differential Equations

Abstract

In this research paper, we provide a concise overview of fractal calculus applied to fractal sets. We introduce and solve a second $\alpha$-order fractal differential equation with constant coefficients across different scenarios. We propose a uniqueness theorem for second $\alpha$-order fractal linear differential equations. We define the solution space as a vector space with non-integer orders. We establish precise conditions for second $\alpha$-order fractal linear differential equations and derive the corresponding fractal adjoint differential equation.