This article is devoted to propose and investigate the fractal battery discharging model, which is one of the well-known models with a memory effect. It is presented as to how non-locality affects the behavior of solutions and how the current state of the system is affected by its past. Firstly, we present a local fractal solution. Then we solve the non-local fractal differential equation and examine the memory effect that includes the Mittag-Leffler function with one parameter. For that aim, the local fractal and non-local fractal Laplace transforms are used to achieve fractional solutions. In addition, the simulation analysis is performed by comparing the underlying fractal derivatives to the classical ones in order to understand the significance of the results. The effects of the fractal parameter and the fractional parameter are discussed in the conclusion section.

In this paper, we develop fractal calculus by defining improper fractal integrals and their convergence and divergence conditions with related tests and by providing examples. Using fractal calculus that provides a new mathematical model, we investigate the effect of fractal time on the evolution of the physical system, for example, electrical circuits. In these physical models, we change the dimension of the fractal time; as a result, the order of the fractal derivative changes; therefore, the corresponding solutions also change. We obtain several analytical solutions that are non-differentiable in the sense of ordinary calculus by means of the local fractal Laplace transformation. In addition, we perform a comparative analysis by solving the governing fractal equations in the electrical circuits and using their smooth solutions, and we also show that when α=1, we get the same results as in the standard version.

The intention of this study is to investigate the fractal version of both one-term and three-term fractal differential equations. The fractal Laplace transform of the local derivative and the non-local fractal Caputo derivative is applied to investigate the given models. The analogues of both the Wright function with its related definitions in fractal calculus and the convolution theorem in fractal calculus are proposed. All results in this paper have been obtained by applying certain tools such as the general Wright and Mittag-Leffler functions of three parameters and the convolution theorem in the sense of the fractal calculus. Moreover, a comparative analysis is conducted by solving the governing equation in the senses of the standard version and fractal calculus. It is obvious that when α = γ = β = 1 {\alpha=\gamma=\beta=1} , we obtain the same results as in the standard version.