Electrical circuits involving fractal time.

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.

Fractional solutions of Bessel equation with N-method.

This paper deals with the design fractional solution of Bessel equation. We obtain explicit solutions of the equation with the help of fractional calculus techniques. Using the N-fractional calculus operator N(ν) method, we derive the fractional solutions of the equation.