W-shape bright and several other solutions to the (3+1)-dimensional nonlinear evolution equations

By employing the Modified Sardar Sub-Equation Method (MSEM), several solitons such as W-shape bright, dark solitons, trigonometric function solutions and singular function solutions have been obtained in two famous nonlinear evolution equations which are used to describe waves in quantum electron–positron–ion magnetoplasmas and weakly nonlinear ion-acoustic waves in a plasma. These models are the (3+1)-dimensional nonlinear extended quantum Zakharov–Kuznetsov (NLEQZK) equation and the (3+1)-dimensional nonlinear modified Zakharov–Kuznetsov (NLmZK) equation, respectively. Comparing the obtained results with Refs. 32–34 and Refs. 43–46, additional soliton-like solutions have been retrieved and will be useful in future to explain the interaction between lower nonlinear ion-acoustic waves and the parameters of the MSEM and the obtained figures will have more physical explanation.

An optimized design method of three-point support for precision horizontal machining center with T-shape bed

The support point layout of the machine tool has an important influence on the working performance of the machine tool, when the material, manufacturing process and internal structure of machine bed are determined. In order to ensure that the precision machine tool has good leveling performance, stability and anti-interference, this paper presents an optimized design method of three-point support for T-shape bed of precision horizontal machining center. This article first establishes the statics model of the T-shape bed and analyzes grillage beam model used to characterize the main static deformation trend of the bed based on the singular function method. After verifying the rationality of the model through simulation, the optimized three-point support position can be obtained. Then this paper measured the deformation of the upper surface of a simple bed due to gravity. The deviation between the experimental results and the simulation results is less than 20%, which verifies the reliability of the simulation and theoretical results. Based on the ISIGHT multi-disciplinary optimization platform, this paper completes the multi-objective optimization of the support point layout of the bed, and the optimization results prove the accuracy of the theoretical model. This paper takes the bed of M800H precision horizontal machining center as an example to illustrate the application process of the proposed method. Finally, this paper compares the optimization effect of the static characteristics of the bed and the whole machine. The maximum deformation of the bed has reduced by 27.1%. In the whole machine status, the deformation of the spindle end has reduced by 50.8%, and the maximum deformation of the workpiece end have reduced by 50.0%.

An Optimal Finite Element Method with Uzawa Iteration for Stokes Equations including Corner Singularities

The Uzawa method is an iterative approach to find approximated solutions to the Stokes equations. This method solves velocity variables involving augmented Lagrangian operator and then updates pressure variable by Richardson update. In this paper, we construct a new version of the Uzawa method to find optimal numerical solutions of the Stokes equations including corner singularities. The proposed method is based on the dual singular function method which was developed for elliptic boundary value problems. We estimate the solvability of the proposed formulation and special orthogonality form for two singular functions. Numerical convergence tests are presented to verify our assertion.

Some New Harmonically Convex Function Type Generalized Fractional Integral Inequalities

In this article, we established a new version of generalized fractional Hadamard and Fejér–Hadamard type integral inequalities. A fractional integral operator (FIO) with a non-singular function (multi-index Bessel function) as its kernel and monotone increasing functions is utilized to obtain the new version of such fractional inequalities. Our derived results are a generalized form of several proven inequalities already existing in the literature. The proven inequalities are useful for studying the stability and control of corresponding fractional dynamic equations.

New shape of the chirped bright, dark optical solitons and complex solutions for (2+1) Ginzburg-Landau equation

Investigation of the Ginzburg-Landau equation (GLE) was done to secure new chirped bright, dark periodic and singular function solutions. For this, we used the traveling wave hypothesis and the chirp component. From there it was pointed out the constraint relation to the dierent arbitrary parameters of the GLE. Thereafter, we employed the improved sub-ODE method to handle the nonlinear ordinary differential equation (NODE). It was highlighted the virtue of the used analytical method via new chirped solitary waves. In our knowledge, these results are new, and will be helpful to explain physical phenomenons.

A Note on the Conjugacy Between Two Critical Circle Maps

We study a conjugacy between two critical circle homeomorphisms with irrational rotation number. Let fi, i = 1, 2 be a C3 circle homeomorphisms with critical point x(i) cr of the order 2mi + 1. We prove that if 2m1 + 1 ̸= 2m2 + 1, then conjugating between f1 and f2 is a singular function. Keywords: circle homeomorphism, critical point, conjugating map, rotation number, singular function

A log–exp elliptic equation in the plane

<p style='text-indent:20px;'>In this paper we show the existence of a nonnegative solution for a singular problem with logarithmic and exponential nonlinearity, namely <inline-formula><tex-math id="M1">\begin{document}$ -\Delta u = \log(u)\chi_{\{u&gt;0\}} + \lambda f(u) $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M2">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M3">\begin{document}$ u = 0 $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M4">\begin{document}$ \partial\Omega $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a smooth bounded domain in <inline-formula><tex-math id="M6">\begin{document}$ \mathbb{R}^{2} $\end{document}</tex-math></inline-formula>. We replace the singular function <inline-formula><tex-math id="M7">\begin{document}$ \log(u) $\end{document}</tex-math></inline-formula> by a function <inline-formula><tex-math id="M8">\begin{document}$ g_\epsilon(u) $\end{document}</tex-math></inline-formula> which pointwisely converges to -<inline-formula><tex-math id="M9">\begin{document}$ \log(u) $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M10">\begin{document}$ \epsilon \rightarrow 0 $\end{document}</tex-math></inline-formula>. When the parameter <inline-formula><tex-math id="M11">\begin{document}$ \lambda&gt;0 $\end{document}</tex-math></inline-formula> is small enough, the corresponding energy functional to the perturbed equation <inline-formula><tex-math id="M12">\begin{document}$ -\Delta u + g_\epsilon(u) = \lambda f(u) $\end{document}</tex-math></inline-formula> has a critical point <inline-formula><tex-math id="M13">\begin{document}$ u_\epsilon $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M14">\begin{document}$ H_0^1(\Omega) $\end{document}</tex-math></inline-formula>, which converges to a nontrivial nonnegative solution of the original problem as <inline-formula><tex-math id="M15">\begin{document}$ \epsilon \rightarrow 0 $\end{document}</tex-math></inline-formula>.</p>

Calculation of displacements and internal forces of anchored retaining piles

Pile-anchor retaining structures are widely used in excavation engineering. The evaluation of lateral displacements, the internal forces of piles are extremely important for the performance of the structure. Most of the existing methods are empirical, semiempirical or FEM methods, while analytic calculation methods for this evaluation are rare. This paper presents an analytic method to calculate the displacements and internal forces of anchored retaining piles based on the existing design code. In the calculation method, the singular function is applied to evaluate the effect of segmented loading on the deflection of a beam with a nonuniform cross section. The load concentration function, expressed by the singular function, can describe the segmented load and be integrated without a complicated procedure for determining the integral constants. The method is applied to a structure in Wenzhou, China, and the calculation results are compared to the field measurement results. This method is only valid for pre-failure predictions.