Uniqueness and existence of solutions for a singular system with nonlocal operator via perturbation method

In this work, we investigate the existence and the uniqueness of solutions for the nonlocal elliptic system involving a singular nonlinearity as follows: $$ \left\{\begin{array}{ll} (-\Delta_p)^su = a(x)|u|^{q-2}u +\frac{1-\alpha}{2-\alpha-\beta} c(x)|u|^{-\alpha}|v|^{1-\beta}, \quad \text{in }\Omega,\ (-\Delta_p)^s v= b(x)|v|^{q-2}v +\frac{1-\beta}{2-\alpha-\beta} c(x)|u|^{1-\alpha}|v|^{-\beta}, \quad \text{in }\Omega,\ u=v = 0 ,\;\;\mbox{ in }\,\mathbb{R}^N\setminus\Omega, \end{array} \right. $$ where $\Omega $ is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary, $0<\alpha <1,$ $0<\beta <1,$ $2-\alpha -\beta 相似文献   

Mathematical Modeling to Simulate the Movement of Contaminants in Groundwater

The objectives of this paper are twofold. Firstly, we formulate a system of partial differential equations that models the contamination of groundwater due to migration of dissolved contaminants through unsaturated to saturated zone. A closed form solution using the singular perturbation techniques for the flow and solute transport equations in the unsaturated zone is obtained. Indeed, the solution can be used as a tool to verify the accuracy of numerical models of water flow and solute transport. The second part of this paper, deals with how the water level in a water reserve drops due to pumping water out of a well that is some distance away.     

Formation of a regular fullerene nanochain lattice

A vicinal Au(11 12 12) surface, naturally patterned into a rectangular superlattice, has been used as a template to prepare C60 nanostructures with long-range order and uniform size. At a coverage of 0.1 monolayer and at room temperature, a two-dimensional long-range ordered superlattice of molecular nanochains is achieved, which perfectly replicates the periodicity of the template surface. The fullerene nanochains are found to be located exclusively on the face-centered cubic stacking domains at the lower step edges. Our experiments demonstrate that highly periodic molecular nanochains can be fabricated through a site-selective anchoring method.     

The extended tanh method for solving systems of nonlinear wave equations

The extended tanh method with a computerized symbolic computation is used for constructing the traveling wave solutions of coupled nonlinear equations arising in physics. The obtained solutions include solitons, kinks and plane periodic solutions. The applied method will be used to solve the generalized coupled Hirota Satsuma KdV equation.     

Nonlinear study of a rotor–AMB system under simultaneous primary-internal resonance

A rotor–active magnetic bearing (AMB) system subjected to a periodically time-varying stiffness with quadratic and cubic non-linearities under multi-parametric excitations is studied and solved. The method of multiple scales is applied to analyze the response of two modes of a rotor–AMB system with multi-parametric excitations and time-varying stiffness near the simultaneous primary and internal resonance. The stability of the steady state solution for that resonance is determined and studied using Rung–Kutta method of fourth order. It is shown that the system exhibits many typical non-linear behaviors including multiple-valued solutions, jump phenomenon, hardening and softening non-linearities and chaos in the second mode of the system. The effects of the different parameters on the steady state solutions are investigated and discussed also. A comparison to published work is reported.     

A monolithic finite element discretization for coupled Darcy-Stokes flow

We consider a Stokes system and a Darcy system in a porous media, mutually coupled together by the the well-known Beaver-Joseph-Saffman interface conditions. Standard approaches use different finite elements in the two parts in order to account for the different stability properties of the two equation systems. In contrast to this, we use standard equal-order finite elements in both parts. Stability is obtained by adding local projection terms (LPS) to the Galerkin formulation. We formulate the scheme and give an a priori estimate. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bayesian estimation of the product of two proportions

Suppose that for i = 1,2, a Bernoulli random variable with success probability θ_i is observable from population i. The problem is to estimate θ = θ₁θ₂ using a Bayesian approach with squared error estimation loss in θ. For estimating θ, the best nonrandom sampling scheme, the two-stage sampling scheme, and the optimal sampling scheme are discussed. It is shown that the two-stage sampling scheme is typically asymptotically optimal, and can improve the Bayes risk (over the best nonrandom allocation) up to fifty percent     

Asymptotically optimal empirical bayes procedures for selecting good

Let ∏₁,…,∏_k denote k independent populations, where a random observation from population ∏ _i has a uniform distribution over the interval (0,θ _i ) and θ _i is a realization of a random variable having an unknown prior distribution G _i . Population ∏ _i is said to be a good population if θ _i ≥θ₀, where θ₀ is a given, positive number. This paper provides a sequence of empirical Bayes procedures for selecting the good populationsamong ∏₁,…,∏ _k . It is shown that these procedures are asymptotically optimal and that the order of associated convergence rates is O(n^-r/4) for some r, 0<r<2, where n is the number of accumulated past observations

at hand     

Thermal stability and UV–Vis-NIR spectroscopy of a new erbium-doped fluorotellurite glass

A new transparent bulk glass from the system 76TeO₂?·?10ZnO?·?9.0PbO?·?1.0PbF₂?·?3.0Na₂O doped with Er³⁺ (TZPPN doped with Er³⁺) has been prepared using the conventional melt-quenching method. Results of differential thermal analysis (DTA) measurements indicate good thermal stability of this glass. The refractive indices at different wavelengths, the optical energy gap, the Sellmeier gap energy and the dispersion energy have been estimated. The Judd–Ofelt parameters, Ω _t (t?=?2,?4,?6) of Er³⁺ were evaluated from optical absorption spectra. Electric dipole, magnetic dipole type transition probabilities, spectroscopic quality factors, branching ratio and radiative lifetimes of several excited states of Er³⁺ have been predicted using intensity Judd–Ofelt parameters. The spectroscopic properties indicate that TZPPN glass doped with Er³⁺ is a promising candidate for laser applications and may be suitable for upconversion fibre optical devices.