In this study, we present a fractal generalized fourth-order Boussinesq equation which can describe the shallow water waves with the non-smooth boundary (such as the fractal boundary). Aided by the semi-inverse method, we establish its variational principle, which is proved to have a strong minimum condition via the He–Weierstrass theorem. Then, two powerful approaches namely the variational method (VM) and energy balance theory (EBT) are utilized to search for the periodic wave solutions. As expected, the results obtained by the two methods are almost the same. Furthermore, the impact of the fractal orders on the periodic wave structure is illustrated via the 3D plot and 2D curve. The results of this paper are expected to provide a reference for the study of periodic wave theory in fractal space.
The well-known Emden-Fowler equation is widely used to model many problems arising in thermal science, physics, and astrophysics. Although there are some analytical solutions available, the high requirement for mathematical knowledge has hindered researchers from direct applications. This paper suggests a straightforward method with a simple solution process and highly accurate results. Two examples are given to verify the accuracy and reliability of the proposed method.
The local fractional derivative (LFD) has gained much interest recently in the field of electrical circuits. This paper proposes a non-differentiable (ND) model of high-pass filter described by the LFD, where the ND transfer function is obtained with the help of the local fractional Laplace transform, and its parameters and properties are studied. The obtained results reveal the sufficiency of the LFD for analyzing circuit systems in fractal space.
Local fractional calculus has gained wide attention in the field of circuit design. In this paper, we propose the zero-input response(ZIR) of fractal RC circuit modeled by local fractional derivative(LFD) for the first time. With help of the law of switch and the Kirchhoff Voltage Laws, the transient local fractional ordinary differential equation is established, and the corresponding exact solution behavior defined on Cantor sets is presented. What we found especially interesting was that the fractal RC becomes the ordinary one in the particular case κ = 1. The results obtained in this paper reveal that the local fractional calculus is a powerful tool to analyze the fractal circuit systems.
Through silicon via technology is a promising and preferred way to realize the reliable interconnection for 3-D integrated circuit (3-D IC), which can transfer heat from multiple dies to the heat sink in vertical direction. In this paper, a new gen?eral model of the through-silicon via (TSV) is proposed to investigate the thermal performance of the 3-D IC. The heat transfer characteristics of conical-annular TSV are studied for the first time. The impacts of different sidewall inclination angles and insulating layer thicknesses of TSV on the heat dissipation of 3-D IC were compared and analyzed in detail. As expected, our proposed model is in good agreement with the results of the existing models, which shows that the proposed model considering the lateral heat transfer and TSV structures can predict the distribution of temperature more efficiently and accurately. Furthermore, it is found that conical-annular TSV has more excellent heat dissipation performance.
This paper derives a new fractional Fokas system with the aid of the local fractional derivative. A novel technology named the extended rational fractal sine–cosine method is presented for the first time ever to develop the abundant traveling wave solutions. Two families (six sets) of the traveling wave solutions on Cantor set are successfully constructed. The dynamic behaviors of the solutions on Cantor sets are provided via numerical simulations. The obtained results in this work strongly prove that the proposed approach is simple but effective, which is expected to shed a new light on the study of the traveling wave theory of the local fractional equations.
This work plumbs the nonlinear dynamics of the ([Formula: see text])-dimensional generalized Kadomtsev–Petviashvili–Boussinesq equation (gKPBe), which is used to describe some interesting physical phenomena in the fields of fluids. The resonance conditions of the soliton molecules on the ([Formula: see text]), ([Formula: see text]) and ([Formula: see text]) planes are investigated and the soliton molecules are obtained on the basis of the N-soliton solutions that are extracted by virtue of the Hirota form. Furthermore, some novel hybrid interactions including the interaction between the soliton and soliton molecule, the interaction between the different soliton molecules are also explored. Finally, the sub-equation approach is exerted to explore the various wave solutions, which include the kinky wave, bright-dark wave and the singular periodic wave solutions. Correspondingly, the graphical descriptions of the attained solutions are drawn to present a better understanding of the physical attributes. The derived solutions can enlarge the exact solutions of the ([Formula: see text])-dimensional gKPBe and lead us to understand the nonlinear dynamic behaviors better.
This paper focuses on the coupled Konno-Oono equation that arises in the magnetic field. Fifteen sets of the analytical solutions in the form of bright solitary, dark solitary, bright-dark solitary, kinky bright-dark solitary, rough bright solitary and periodic wave solutions are obtained by applying three effective technologies, which are simplified extended tanh-function method, variational direct method and He's frequency formulation. The behaviors of the solutions are plotted through the 3-D contours. The results show that the proposed methods are effective and powerful, which are expected to helpful for the study of the traveling wave theory in physics.
Under the non-smooth condition, many theories obtained by the assumption on the smooth condition become invalid, so a generalized Burgers–Huxley equation (GBHE) with fractal derivative is introduced in this work. The fractal variational formulation for the problem is established by using the semi-inverse method, which provides conservation laws in an energy form and possible solution structures of the equation. The two-scale transform method and variational iteration method (VIM) are used to solve the fractal GBHE. The obtained results show a great agreement with the existed results.
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