A brief introduction to the development of the homotopy perturbation method is given, and the main milestones are elucidated with more than 90 references. This paper further improves the method by constructing a homotopy equation with one or more auxiliary parameters embedding in the linear term with a clear advantage in accelerating and controlling the approximation convergence speed. Moreover, a revision of a recent amplitude-period approximation formula is presented providing an answer to an open problem related to the optimal approximation along with a new universal formula. Duffing equation is used as an example to illustrate the solution process for the homotopy perturbation method, and only one or few iterations are needed in practical applications, making the method much attractive. From the side of amplitude-period formulation, the nonlinear pendulum, the Duffing equation and an oscillator with discontinuity are analyzed providing an asymptotic exact equivalence for bigger parameter values in the case of Duffing's system. This mini review gives a tutorial guideline for practical applications of the homotopy perturbation method, the references are not exhaustive.
The search for non-noble metal catalysts with high activity for the hydrogen evolution reaction (HER) is crucial for efficient hydrogen production at low cost and on a large scale. Herein, we report a novel WO3-x catalyst synthesized on carbon nanofiber mats (CFMs) by electrospinning and followed by a carbonization process in a tubal furnace. The morphology and composition of the catalysts were tailored via a simple method, and the hybrid catalyst mats were used directly as cathodes to investigate their HER performance. Notably, the as-prepared catalysts exhibit substantially enhanced activity for the HER, demonstrating a small overpotential, a high exchange current density, and a large cathodic current density. The remarkable electrocatalytic performances result from the poor crystallinity of WO3-x, the high electrical conductivity of WO3-x, and the use of electrospun CNFs. The present work outlines a straightforward approach for the synthesis of transition metal oxide (TMO)-based carbon nanofiber mats with promising applications for the HER.
Lithium ion batteries (LIBs) have achieved great success in powering portable electronic devices in our modern society, and are to find use in the electrification of transportation and the storage of wind or solar energy in smart grids in the near future. However, there is increasing concern on the safety issues of current LIBs based on organic liquid electrolytes, which are volatile and flammable. This leads to the exploration and development of solid electrolytes to improve the safety of next-generation high-energy LIBs. In this review, we describe two inorganic-organic hybrid solid electrolyte systems for LIBs. Firstly, we present polymer electrolytes with different types of inorganic fillers, discussing how the fillers affect the electrochemical and physical properties of the electrolyte. Secondly, we introduce recent progress in MOF-based solid electrolytes and show how MOFs can contribute to such an inorganic-organic hybrid system. Finally, outlook and future directions for safe and high performance inorganic-organic hybrid solid electrolytes are proposed.
An effective strategy for the rational design of 3D architectures for superior electrocatalysis, through the integration of CNFs, CNTs and oxygen-deficient Mn3Co7–Co2Mn3O8 nanoparticles, has been demonstrated.
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