This article investigates the extended homoclinic (heteroclinic) breather wave solutions and interaction periodic and dark soliton solutions to the nonlinear vibration and dispersive wave systems. The solutions include periodic, breather, and soliton solutions. The bilinear form is considered in terms of Hirota derivatives. Accordingly, we utilize the Cole–Hopf algorithm to obtain the exact solutions of the (
2
+
1
2+1
)-dimensional modified dispersive water-wave system. The analytical treatment of extended homoclinic breather wave solutions and interaction periodic and dark soliton solutions are studied and plotted in four forms of density plots. A nonlinear vibration system will be studied. Employing appropriate mathematical assumptions, the novel kinds of the extended homoclinic breather wave solutions and interaction periodic and dark soliton solutions are derived and constructed in view of the combination of kink, periodic, and soliton for an extended homoclinic breather and also a combination of two kinks, periodic and dark soliton in terms of exponential, trigonometric, hyperbolic functions for interaction periodic and dark soliton of the governing equation. To achieve this, the illustrative example of the (
2
+
1
2+1
)-D modified dispersive water-wave system is furnished to demonstrate the feasibility and reliability of the procedure applied in this research. The trajectory solutions of the traveling waves are offered explicitly and graphically. The effect of the free parameters on the behavior of designed figures of a few obtained solutions for two nonlinear rational exact cases was also considered. By comparing the suggested scheme with the other existing methods, the results state that the execution of this technique is succinct, extensive, and straightforward.
In this paper, a multi objective optimization approach is studied for optimal energy scheduling of the smart autonomous electrical grid with participation of active consumers (ACs) and hydrogen storage system (HSS). The objective functions consist of: 1) minimizing the costs and 2) maximizing reliability. The ACs participation are modelled through demand reduction approach based on offer price to peak demand management. The proposed optimization is solved by epsilon-constraint method and LINMAP decision making strategy. The 21-node test system is employed to analyse the efficiency of the proposed approach at two case studies. The obtained results shown the high effectiveness of smart autonomous electrical grid with participation of ACs and HSS to supply the demand.
A simple, rapid, and low‐cost method for preparation of Co3O4 nanoparticles (Co3O4‐1 and Co3O4‐2) using solid‐state calcination of mixture of Co(NO3)2∙6H2O and salicylic acid (1:1 weight ratio) at two different temperatures of 500 and 600°C is reported. The morphologies and structure of the as‐prepared Co3O4 nanoparticles were characterized using X‐ray diffraction (XRD) and transmission electron microscope (TEM). In addition, we tested them as an anode material for lithium‐ion batteries. The cycling stability and electrochemical performance of the as‐prepared Co3O4 nanoparticles were excellent. A capacity of ≈1,100 mAh/g was obtained after 50 cycles at 100 mA/g and exhibited stable discharge behavior up to 50 cycles with high rate capability.
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