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As is well known, one of the main challenges in dealing with the time‐fractional diffusion‐wave equation is that the solutions have weak regularity at the initial values. This necessitates the use of nonuniform grids for error analysis. However, the theoretical analysis of nonuniform grids is relatively complex and lacks efficient and stable numerical methods. The main research problem addressed in this article is the time‐fractional diffusion‐wave equation on nonuniform grids. We use the order reduction method and discrete complementary convolution kernel to discretize the Caputo derivative of the equation and obtain the L1‐type finite element method on nonuniform grids. The properties related to discrete complementary convolutional kernels are often used for algorithm convergence analysis, which simplifies the process of finite element theory analysis and is efficient and innovative. This article demonstrates the solvability, stability, and convergence of the L1‐type finite element schemes on nonuniform grids. Finally, the consistency between the proposed finite element scheme and the convergence order results obtained from theoretical analysis was verified through experiments. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following nonlinear Schrödinger equation: 0.1−Δu+V(|x|)u=Q(|x|)up,u>0inℝN,u∈H1ℝN,$$ -\Delta u+V\left(|x|\right)u=Q\left(|x|\right){u}^p,\kern0.30em u>0\kern0.20em \mathrm{in}\kern0.20em {\mathrm{\mathbb{R}}}^N,u\in {H}^1\left({\mathrm{\mathbb{R}}}^N\right), $$where N≥3,p∈1,N+2N−2$$ N\ge 3,p\in \left(1,\frac{N+2}{N-2}\right) $$, and V,Q$$ V,Q $$ have the algebraical decay that V(|x|)=V0+a|x|m+O1|x|m+κ,Q(|x|)=Q0+b|x|n+O1|x|n+θ,$$ V\left(|x|\right)={V}_0+\frac{a}{{\left|x\right|}^m}+O\left(\frac{1}{{\left|x\right|}^{m+\kappa }}\right),\kern0.30em Q\left(|x|\right)={Q}_0+\frac{b}{{\left|x\right|}^n}+O\left(\frac{1}{{\left|x\right|}^{n+\theta }}\right), $$as |x|→∞$$ \mid x\mid \to \infty $$, where V0,Q0,κ,θ,a>0,b∈ℝ$$ {V}_0,{Q}_0,\kappa, \theta, a>0,b\in \mathrm{\mathbb{R}} $$, and m,n>1$$ m,n>1 $$. By introducing the Miranda theorem, via the Lyapunov–Schmidt finite‐dimensional reduction method, we construct infinitely many multi‐bumps solutions of (0.1), whose maximum points of bumps lie on the top and bottom circles of a cylinder provided m0$$ m0 $$, or b≤0$$ b\le 0 $$. This result complements and extends the ones in [Duan and Musso, JDE, 2022] for a slow decaying rate of the potential function at infinity from m>max2,4p−1$$ m>\max \left\{2,\frac{4}{p-1}\right\} $$ to m>1$$ m>1 $$. [ABSTRACT FROM AUTHOR]

We consider the equation ε2s(−Δ)su+V(x)u−f(u)=0inℝN,where s ∈ (0, 1), p∈1,N+2sN−2s, N > 2s, f(u)=|u|p−1u, V∈L∞(ℝN) is such that infℝNV>0 and ε > 0 is small. We study the existence and nonexistence of solutions concentrating at a local minimum point of V as ε → 0 without using any symmetry assumption on V. First, we prove that certain type of positive solutions exhibiting peaks does not exist. Then, we study the existence of sign‐changing solutions under a suitable configuration of positive and negative peaks. To guarantee the existence, we cannot neglect the interaction between peaks. In particular, by using a minimization argument, we found solutions exhibiting peaks at the vertices of a 2ℓ‐regular polygon, such that two adjacent peaks have alternate sign. [ABSTRACT FROM AUTHOR]

We present a new family of fast and robust methods for the calculation of the vapor–liquid equilibrium at isobaric‐isothermal (PT‐flash), isochoric‐isothermal (VT‐flash), isenthalpic‐isobaric (HP‐flash), and isoenergetic‐isochoric (UV‐flash) conditions. The framework is provided by formulating phase‐equilibrium conditions for multi‐component mixtures in an effectively reduced space based on the molar specific value of the recently introduced volume function derived from the Helmholtz free energy. The proposed algorithmic implementation can fully exploit the optimum quadratic convergence of a Newton method with the analytical Jacobian matrix. This article provides all required exact analytic expressions for the general cubic equation of state. Computational results demonstrate the effectivity and efficiency of the new methods. Compared to conventional methods, the proposed reduced‐space iteration leads to a considerable speed‐up as well as to improved robustness and better convergence behavior near the spinodal and coexistence curves of multi‐component mixtures, where the preconditioning by the reduction method is most effective. [ABSTRACT FROM AUTHOR]

We consider the problem PεΔ2u=|u|8N−4u+εf(x)inΩ,u=Δu=0on∂Ω,where Ω is a bounded smooth domain in RN, N≥5, that exhibits certain symmetries and contains the origin, f∈L∞(Ω), f≥0, f≢0, and ε>0 is a small parameter. By using the Lyapunov–Schmidt reduction method and topological degree theory, for each sufficiently large k∈N, we construct sign‐changing solutions to (Pε) exhibiting k negative spikes at the vertices of a regular polygon and a single positive spike at the origin. [ABSTRACT FROM AUTHOR]

An accurate, safe, environmentally friendly, fast and sensitive electrochemical biosensors were developed to detect xanthine in serum. The metal‐organic framework ZIF‐8 was synthesised and elemental gold was supported on the surface of ZIF‐8 by reduction method to synthesise Ag‐ZIF‐8. The mesoporous carbon material and the synthesised Ag‐ZIF‐8 were, respectively, applied to a glassy carbon electrode to construct biosensors. The constructed biosensor has a good linear relation in the range of 1–280 μmol l−1 of xanthine and the detection limit is 0.167 μmol l−1. The relative standard deviation value in serum samples was <5%, and the recoveries were 96–106%, indicating the good selectivity, stability and reproducibility of this electrochemical biosensor. [ABSTRACT FROM AUTHOR]

In this paper, we study the Kirchhoff equations with perturbation in R3. Applying the finite reduction method, we prove that the equation has one‐bump positive solutions under some suitable conditions that are given in Section 1. [ABSTRACT FROM AUTHOR]

We consider the problem Δ2u = V(x)up + ϵ in RN with u,Δu→0 as |x|→ + ∞, where p=N+4N−4, N ≥ 5, V is a positive continuous potential. Our aim is to construct high‐energy solutions for this equation by applying the finite‐dimensional reduction method and the penalization method. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following nonlinear fractional Schrödinger equation [ABSTRACT FROM AUTHOR]

This paper addresses the robust consensus control design for input-delayed multi-agent systems subject to parametric uncertainties. To deal with the input delay, the Artstein model reduction method is employed by a state transformation. The input-dependent integral term that remains in the transformed system, owing to the model uncertainties, is judiciously analysed. By carefully exploring certain features of the Laplacian matrix, sufficient conditions for the global consensus under directed communication topology are identified using Lyapunov-Krasovskii functionals in the time domain. The proposed control only relies on relative state information of the subsystems via network connections. The effectiveness and robustness of the proposed control design are demonstrated through a numerical simulation example. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The present study suggests a new scheme to design a decentralized PI controller for two-input two-output (TITO) processes. The decentralized controller is obtained from the direct synthesis method. The decoupler is derived from a modified decoupler and minimizes the interaction between the loops. Then a new reduced method based on equivalent transfer function (ETF) with relative gain array concept and relative normalized relative gain array concept reduces the decoupled process transfer functions to first-order plus time delay (FOPTD) forms. Further, two PI controllers are obtained in the proposed delay-compensated configuration in terms of the only parameter in every individual loop for the FOPTD transfer functions. Three illustrative examples demonstrate that the performance is verified on the original interactive system and show the effectiveness of the proposed design method. The results confirm the superior performance of the proposed compensation configuration in nominal and robust cases. The proposed method also offers several important advantages compared to other previous methods. [ABSTRACT FROM AUTHOR]

Stability analysis is generally faster than phase-split calculations because the Rachford--Rice equation is not part of stability testing. However, there are occasions in which one may encounter computational inefficiency and indeed divergence in the single-phase region far away from the critical point. The computational inefficiency is found in various formulations of the stability analysis calculations. In this work, the cause of the problem is explained. A simple algorithm is also presented for stability analysis calculations in the reduction method. As in the past, only the Newton method is used in the solution of nonlinear equations except in some isolated iterations when one single successive substitution (SSI) iteration may be required to avoid nonphysical conditions. Results show that the stability analysis for the proposed algorithm is fast and robust. Results also show that the simple algorithm in the reduction method is superior to the combined SSI--Newton and quasi-Newton methods with conventional variables. Results are also presented showing that the stability analysis in two-phase is faster than phase-split calculations even when the initial guess is from the stability testing. [ABSTRACT FROM AUTHOR]

Background: In order to improve patient compliance in allergen avoidance, a new occupational activity was created: Medical Indoor Environment Counselor (MIEC). The aim of this study was to assess the impact of an MIEC on compliance with advice for mite allergen reduction in patients sensitized and exposed to mite allergens, and on mite allergen levels. Methods: The study included 378 patients from four centers (Marseilles, Montpellier, Paris, Strasbourg) in a randomized prospective study. Patients in group A received advice from doctors only, and those in group B from doctors and MIEC. Mite allergen levels were measured on mattresses, mattress bases, and floors. Compliance and mite allergen levels were assessed 5 months later. Results: Compliance for changing the mattress bases, removing or treating the carpets, washing duvets, pillows and stuffed animals were significantly higher in group B than in group A. A significant decrease in mite allergen was obtained for group B in mattress bases (83.8–22.9 µg/g) and in carpets (15.8–6.3 µg/g), but not in group A. Conclusion: Our results suggest that the home visit by the MIEC 1) increased the compliance to mite reduction methods advised; 2) induced a significant difference in mite reduction levels on mattress bases and on carpets; 3) avoided nonestablished avoidance advice. [ABSTRACT FROM AUTHOR]