The unsmooth boundary has a great influence on the solitary wave form of a nonlinear wave equation. It this work, we for the first time ever propose the fractal regularized long-wave equation which can describe the shallow water waves under the unsmooth boundary (such as the fractal seabed). The fractal variational principle is established and is proved to have a strong minimum condition by the He–Weierstrass theorem. Then, the solitary wave solution is obtained by the fractal variational method which can reduce the order of differential equation and obtain the optimal solution by the stationary condition. Finally, the impact of the unsmooth boundary on the solitary wave is presented. It shows that the fractal order can affect the wave morphology, but cannot affect its peak value. The finding in this paper is important for the coast protection and expected to bring a light to the study of the fractal theoretical basis in the geosciences.
The soliton solutions of the (2 + 1)-dimensional nonlinear electrical transmission line equation are discussed by Md. Abdul Kayum, et al. (Results in Physics, Volume18, 2020, 103269). In this paper, we obtain its periodic solution by the variational method, which is expected to shed a light on the study of the solitary wave theory.
This study proposes a new fractal modified equal width-Burgers equation (MEWBE) with the local fractional derivative (LFD) for the first time. By defining the Mittag-Leffler function (MLF) on the Cantor set (CS), two special functions, namely, the [Formula: see text] and [Formula: see text] functions, are derived for constructing the auxiliary function to seek the non-differentiable (ND) exact solutions. And 16 groups of the ND exact solutions are successfully established. The solutions on the CS are depicted graphically to interpret the nonlinear dynamic behaviors. Furthermore, the comparative results of the fractal MEWBE and the classical MEWBE are also discussed. The obtained results confirm that the proposed method is effective and powerful, and can provide a promising way to find the ND exact solutions of the local fractional PDEs.
In the present work, we present a detailed investigation into the large amplitude vibration of nonlinear engineering structures by means of the Hamiltonian-based frequency formulation (HBFF), extracted from the Hamiltonian theory and residual equation method. Compared to the variational method, the frequency amplitude formulas have an excellent agreement. In addition, the compared curves between the HBFF, variational method and exact solution indicate that the HBFF is more accurate. The ideas in this research are expected to shed a new light to the exploration of the nonlinear oscillation problems arising in physics and engineering.
Under the current research, the local fractional (3+1)-dimensional modified Zakharov–Kuznetsov equation (MZKE) is explored. With the Mittag–Leffler function (MLF) defined on the Cantor sets (CS), four special functions, namely the [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are extracted to construct an auxiliary function. Then the auxiliary function, along with Yang’s non-differentiable (ND) transformation, is manipulated to explore the ND exact solutions (ESs). By means of the proposed method, four different sets of the ND ESs are found in just one step. The nonlinear dynamics of the ND exact solutions on the CS are illustrated graphically. Furthermore, the ND exact solutions for [Formula: see text] and the classic exact solutions for [Formula: see text] are also compared and discussed in detail via the 2-D curves. The attained results reveal that the method is a simple but effective tool to deal with local fractional PDEs arising in physics and maths.
This paper provides an investigation on nonlinear dynamic behaviors of the (3+1)-dimensional B-type Kadomtsev—Petviashvili equation, which is used to model the propagation of weakly dispersive waves in a fluid. With the help of the Cole—Hopf transform, the Hirota bilinear equation is established, then the symbolic computation with the ansatz function schemes is employed to search for the diverse exact solutions. Some new results such as the multi-wave complexiton, multi-wave, and periodic lump solutions are found. Furthermore, the abundant traveling wave solutions such as the dark wave, bright-dark wave, and singular periodic wave solutions are also constructed by applying the sub-equation method. Finally, the nonlinear dynamic behaviors of the solutions are presented through the 3-D plots, 2-D contours, and 2-D curves and their corresponding physical characteristics are also elaborated. To our knowledge, the obtained solutions in this work are all new, which are not reported elsewhere. The methods applied in this study can be used to investigate the other PDEs arising in physics.
This paper proposes a fractal viscoelastic element via He?s fractal derivative, its properties are analyzed in details by the two-scale transform for the first time. The element is used to establish a fractal Maxwell-rheological model, which unifies the fractal creep equation and relaxation equation, and includes the classic elastic model and the classical Maxwell-rheological model as two special cases. This paper sheds a bright light on viscoelasticity, and the model can find wide applications in rock mechanics, plastic mechanics, and non-continuum mechanics.
A new local fractional modified Benjamin–Bona–Mahony equation is proposed within the local fractional derivative in this study for the first time. By defining some elementary functions via the Mittag–Leffler function (MLF) on the Cantor sets (CSs), a set of nonlinear local fractional ordinary differential equations (NLFODEs) is constructed. Then, a fast algorithm namely Yang’s special function method is employed to find the non-differentiable (ND) exact solutions. By this method, we can extract abundant exact solutions in just one step. Finally, the obtained solutions on the CS are outlined in the form of the 3-D plot. The whole calculation process clearly shows that Yang’s special function method is simple and effective, and can be applied to investigate the exact ND solutions of the other local fractional PDEs.
The temperature distribution in a 3-D high-power light emitting diode lamp is affect by multiple factors, the orthogonal experiment method is adopted to elucidate three main factors, an experiment is designed to verify the main finding, which is useful for an optimal design of the light emitting diode lamp.
The objective of the present study is to extract the optical soliton solutions (OSSs) of the perturbed Chen–Lee–Liu equation by exerting three techniques, which are the extended Wang’s direct mapping method, tanh/coth function method and the Subequation method. Various OSSs, including the dark, bright, and singular soliton solutions, as well as other wave solutions, such as the singular periodic and algebraic solitary wave solutions, are successfully found. By assigning the appropriate parameters, the dynamic characteristics of the attained solutions are depicted graphically via the Mathematica software. The outcomes confirm that the methods are highly effective for constructing the OSSs and can also be adopted to investigate the other PDEs in optics.
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