Differential transformation method for studying flow and heat transfer due to stretching sheet embedded in porous medium with variable thickness,variable thermal conductivity,and thermal radiation

This article presents a numerical solution for the flow of a Newtonian fluid over an impermeable stretching sheet embedded in a porous medium with the power law surface velocity and variable thickness in the presence of thermal radiation. The flow is caused by non-linear stretching of a sheet. Thermal conductivity of the fluid is assumed to vary linearly with temperature. The governing partial differential equations （PDEs） are transformed into a system of coupled non-linear ordinary differential equations （ODEs） with appropriate boundary conditions for various physical parameters. The remaining system of ODEs is solved numerically using a differential transformation method （DTM）. The effects of the porous parameter, the wall thickness parameter, the radiation parameter, the thermal conductivity parameter, and the Prandtl number on the flow and temperature profiles are presented. Moreover, the local skin-friction and the Nusselt numbers are presented. Comparison of the obtained numerical results is made with previously published results in some special cases, with good agreement. The results obtained in this paper confirm the idea that DTM is a powerful mathematical tool and can be applied to a large class of linear and non-linear problems in different fields of science and engineering.     

Self-adaptive one-dimensional nonlinear finite element method based on element energy projection method

The element energy projection （EEP） method for computation of super- convergent resulting in a one-dimensional finite element method （FEM） is successfully used to self-adaptive FEM analysis of various linear problems, based on which this paper presents a substantial extension of the whole set of technology to nonlinear problems. The main idea behind the technology transfer from linear analysis to nonlinear analysis is to use Newton＇s method to linearize nonlinear problems into a series of linear problems so that the EEP formulation and the corresponding adaptive strategy can be directly used without the need for specific super-convergence formulation for nonlinear FEM. As a re- sult, a unified and general self-adaptive algorithm for nonlinear FEM analysis is formed. The proposed algorithm is found to be able to produce satisfactory finite element results with accuracy satisfying the user-preset error tolerances by maximum norm anywhere on the mesh. Taking the nonlinear ordinary differential equation （ODE） of second-order as the model problem, this paper describes the related fundamental idea, the imple- mentation strategy, and the computational algorithm. Representative numerical exam- ples are given to show the efficiency, stability, versatility, and reliability of the proposed approach.     

Low dimensional instability for quasilinear problems of weighted exponential nonlinearity

We prove a sharp Liouville type theorem for stable solutions of the equation on the entire Euclidean space , where and f is a continuous and nonnegative function in such that as , where and . Our theorem holds true for and is sharp in the case .     

Existence and concentration result for Kirchhoff equations with critical exponent and Hartree nonlinearity

This paper is concerned with the following Kirchhoff-type equations $$ \left\{ \begin{array}{ll} \displaystyle -\big(\varepsilon^{2}a+\varepsilon b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\mathrm{d}x\big)\Delta u + V(x)u+\mu\phi |u|^{p-2}u=f(x,u), &\quad \mbox{ in }\mathbb{R}^{3},\(-\Delta)^{\frac{\alpha}{2}} \phi=\mu|u|^{p},~u>0, &\quad \mbox{ in }\mathbb{R}^{3},\\end{array} \right. $$ where $f(x,u)=\lambda K(x)|u|^{q-2}u+Q(x)|u|^{4}u$, $a>0,~b,~\mu\geq0$ are constants, $\alpha\in(0,3)$, $p\in[2,3),~q\in[2p,6)$ and $\varepsilon,~\lambda>0$ are parameters. Under some mild conditions on $V(x),~K(x)$ and $Q(x)$, we prove that the above system possesses a ground state solution $u_{\varepsilon}$ with exponential decay at infinity for $\lambda>0$ and $\varepsilon$ small enough. Furthermore, $u_{\varepsilon}$ concentrates around a global minimum point of $V(x)$ as $\varepsilon\rightarrow0$. The methods used here are based on minimax theorems and the concentration-compactness principle of Lions. Our results generalize and improve those in Liu and Guo (Z Angew Math Phys 66: 747-769, 2015), Zhao and Zhao (Nonlinear Anal 70: 2150-2164, 2009) and some other related literature.     

Partial regularity of weak solutions to a PDE system with cubic nonlinearity

In this paper we investigate regularity properties of weak solutions to a PDE system that arises in the study of biological transport networks. The system consists of a possibly singular elliptic equation for the scalar pressure of the underlying biological network coupled to a diffusion equation for the conductance vector of the network. There are several different types of nonlinearities in the system. Of particular mathematical interest is a term that is a polynomial function of solutions and their partial derivatives and this polynomial function has degree three. That is, the system contains a cubic nonlinearity. Only weak solutions to the system have been shown to exist. The regularity theory for the system remains fundamentally incomplete. In particular, it is not known whether or not weak solutions develop singularities. In this paper we obtain a partial regularity theorem, which gives an estimate for the parabolic Hausdorff dimension of the set of possible singular points.     

On the multiplicity of solutions of a simplified tumor growth model with a free boundary

We consider a mathematical model of tumor growth taking into account the spatial structure and include a different phases because of the lack of the nutriments. This model is expressed as a PDE on a spherical domain describing the tumor region. The nonlinearity of this PDE is discontinuous, and our problem can be regarded as a free boundary problem. In fact, the boundary of the tumor and another part in the interior are unknown. We obtain a multiplicity result of solutions together with some properties of the associated free boundaries.     

An efficient explicit full-discrete scheme for strong approximation of stochastic Allen–Cahn equation

In Becker and Jentzen (2019) and Becker et al. (2017), an explicit temporal semi-discretization scheme and a space–time full-discretization scheme were, respectively, introduced and analyzed for the additive noise-driven stochastic Allen–Cahn type equations, with strong convergence rates recovered. The present work aims to propose a different explicit full-discrete scheme to numerically solve the stochastic Allen–Cahn equation with cubic nonlinearity, perturbed by additive space–time white noise. The approximation is easily implementable, performing the spatial discretization by a spectral Galerkin method and the temporal discretization by a kind of nonlinearity-tamed accelerated exponential integrator scheme. Error bounds in a strong sense are analyzed for both the spatial semi-discretization and the spatio-temporal full discretization, with convergence rates in both space and time explicitly identified. It turns out that the obtained convergence rate of the new scheme is, in the temporal direction, twice as high as existing ones in the literature. Numerical results are finally reported to confirm the previous theoretical findings.     

轴向变速运动粘弹性弦线的横向振动分岔

研究轴向运动弦线横向振动的分岔 .弦线轴向速度为常平均速度带有简谐涨落 ,其粘弹性材料由Kelvin模型描述 .建立系统的动力学方程并应用 2阶Galerkin截断进行简化 .计算了弦线中点的Poincar啨截面映射对平均轴向速度、轴向速度涨落幅值和弹性模量的分岔图 .     

均匀的背景引力场是无法感知的

论述了均匀的背景引力场是无法感知的,实际的背景引力场是不均匀的,但在小范围内可视为均匀．