An Exact Solution of the Fractional Transient Electromagnetic Field Inside an Atmospheric Duct

The objective of this paper is to find the exact solution of the fractional electromagnetic (EM) equation within an atmospheric duct (dielectric medium) of the transient EM field caused by a vertical magnetic dipole and discuss the results in both fractional D-dimensional space and integer space. The novelty of this work is the exact solution of the fractional EM equation in terms of Bessel and Mittage-Leffler functions based on Caputo fractional derivative order \(\alpha\) $\alpha$ and the Laplacian operator in D -dimensional fractional space. Using the separation of variables method, the resulting magnetic field can be controlled by \(D=\alpha _1+\alpha _2+\alpha _3\) $D={\alpha }_1+{\alpha }_2+{\alpha }_3$ and alpha as spatial and time fractional parameters, respectively. The transient magnetic field behaviour inside the duct is plotted through some of figures depending on fractional parameters D and \(\alpha\) $\alpha$ . The classical integer results in usual space are recovered from fractional solution at \(D=\alpha =1\) $D=\alpha =1$ . The main results highlighted are that spatial frequencies grow at a faster rate during EM wave propagation; thus, the amplitude of the propagated wave in integer space is greater than that in fractional space. Increasing the fractional parameters D and \(\alpha\) $\alpha$ simultaneously increases the amplitude of the EM wave and can be used to control the power and energy of the EM wave.