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Abstract-This work deals with an exact solution of cylindrical wave equation for electromagnetic field in fractional dimensional space. The obtained fractional solution is a generalization of the cylindrical wave equation from integer dimensional space to a fractional dimensional space. The resulting theoretical framework can be used to study the phenomenon of electromagnetic wave propagation in any fractal media because fractal media can be described as an ordinary media in a fractional dimensional space. The classical results are recovered from fractional solution when integer dimensional space is considered.

Internet of Things enabled smart grid (SG) is one of the most advanced technologies, which plays a key role in maintaining a balance between demand and supply by implementing demand response (DR) program. In SG, the main focus of the researchers is on home energy management (HEM) system, which is called demand side management. Appliance scheduling is an integral part of HEM system as it manages energy demand according to supply, by automatically controlling the appliances and shifting the load from peak to off peak hours. In this paper, the comparative performance of HEM controller embedded with heuristic algorithms, such as harmony search algorithm, enhanced differential evolution, and harmony search differential evolution, is evaluated. The integration of renewable energy source (RES) in SG makes the performance of HEM system more efficient. The electricity consumption in peak hours usually creates peaks and increases the cost but integration of RES makes the electricity consumer able to use the appliances in the peak hours. We formulate our problem using multiple knapsack theory that the maximum capacity of the consumer of electricity must be in the range, which is bearable for consumer with respect to electricity bill. Feasible regions are computed to validate the formulated problem. Finally, simulation of the proposed techniques is conducted in MATLAB to validate the performance of proposed scheduling algorithms in terms of cost, peak-to-average ratio, and waiting time minimization.

Abstract-This work presents the analytical solution of vector wave equation in fractional space. General plane wave solution to the wave equation for fields in source-free and lossless media is obtained in fractional space. The obtained solution is a generalization of wave equation from integer dimensional space to a non-integer dimensional space. The classical results are recovered when integer-dimensional space is considered.