Fractal Calculus: Nonhomogeneous linear systems

In this paper, we present a concise overview of fractal calculus and explore the solution of non-homogeneous fractal differential equations. We analyze fractal homogeneous linear systems with initial conditions, introducing the fundamental matrix and special fundamental matrix, and demonstrate their applications in solving systems and analyzing the Jordan form of matrices. We propose the method of undetermined coefficients for solving non-homogeneous fractal linear differential equations and introduce the method of variation of parameters as a supplementary technique. To illustrate these methods, we apply them to the differential equations of resistor–inductor–capacitor (RLC) circuits, successfully solving the corresponding fractal differential equations. Additionally, we provide examples, solve systems with initial conditions, and present the results through plotted graphs.