Solution Of Problem Research Articles

Concentration of solutions for non-autonomous double-phase problems with lack of compactness

The present paper is devoted to the study of the following double-phase equation -div(|∇u|p-2∇u+με(x)|∇u|q-2∇u)+Vε(x)(|u|p-2u+με(x)|u|q-2u)=f(u)inRN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} -\ ext {div}(|\ abla u|^{p-2}\ abla u+\\mu _{\\varepsilon }(x)|\ abla u|^{q-2}\ abla u)+V_{\\varepsilon }(x)(|u|^{p-2}u+\\mu _{\\varepsilon }(x)|u|^{q-2}u)=f(u)\\quad \ ext{ in }\\quad \\mathbb {R}^{N}, \\end{aligned}$$\\end{document}where N≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\ge 2$$\\end{document}, 1<p<q<N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<p<q<N$$\\end{document}, q<p∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q<p^{*}$$\\end{document} with p∗=NpN-p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p^{*}=\\frac{Np}{N-p}$$\\end{document}, μ:RN→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu :\\mathbb {R}^{N}\\rightarrow \\mathbb {R}$$\\end{document} is a continuous non-negative function, με(x)=μ(εx)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu _{\\varepsilon }(x)=\\mu (\\varepsilon x)$$\\end{document}, V:RN→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V:\\mathbb {R}^{N}\\rightarrow \\mathbb {R}$$\\end{document} is a positive potential satisfying a local minimum condition, Vε(x)=V(εx)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V_{{{\\,\\mathrm{\\varepsilon }\\,}}}(x)=V({{\\,\\mathrm{\\varepsilon }\\,}}x)$$\\end{document}, and the nonlinearity f:R→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:\\mathbb {R}\\rightarrow \\mathbb {R}$$\\end{document} is a continuous function with subcritical growth. Under natural assumptions on μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}, V and f, by using penalization methods and Lusternik–Schnirelmann theory we first establish the multiplicity of solutions, and then, we obtain concentration properties of solutions.

Scientific basis for intelligent of production processes in cattle breeding

Increasing the productivity of livestock and veterinary specialists is an integral part of intensifying the production of animal husbandry products. The solution of multifaceted problems is ensured by the organization of their work. For example, the intelligent of production processes in livestock breeding lies in the fact that the use of innovative technologies, the rate of which on average in the industry reaches 46 %. So far, involves the use of electronic telecommunication devices, which had currently better meet the requirements for the maintenance and operation of breeding animals than traditional standards using analog-digital forms of equipment and document fl ow. The purpose of the research was to study the eff ectiveness of using innovative equipment in terms of time spent on implementing animal science accounting and obtaining animal husbandry products when breeding animals using telecommunication devices for recording technical and technological processes through the integration of data collection terminals using RFID (radio frequency identifi cation) and scanners that allow reading electronic ear (UHF) tags, boluses, electronic chips compared to traditional equipment that is still used in livestock enterprises. It has been established that the use of innovative technologies in animal husbandry meets the requirements of breeding pedigree animals better than using previous standards, i.e. using analog-electronic forms of equipment. Thus, innovative technologies make production processes shorter in time by 16.3 %, and more economical. Analysis of the results of the growth rate of off spring of individual sires using traditional standard technologies showed the loss of working time 10.3 times greater than when using innovations. With traditional breeding technologies, losses in the average daily gain in live weight of young and adult animals by 17.5 and 30.0 %, respectively, were noted. Losses in average daily gain in live weight were noted by 1.1–1.2 times greater when using traditional technologies of registration, identifi cation and control than when using innovations. Therefore, it is necessary to organize professional communication with colleagues in the information fi eld to participate in breeding pedigree livestock in cattle breeding.

Emergency Doctor in the Reanimation Room and Solution of Medical Problems

Emergency Doctor in the Reanimation Room and Solution of Medical Problems

Entropy and Renormalized Solutions for a Nonlinear Elliptic Problem in Musielak–Orlicz Spaces

Entropy and Renormalized Solutions for a Nonlinear Elliptic Problem in Musielak–Orlicz Spaces

Exponential Convergence and Computational Efficiency of BURA-SD Method for Fractional Diffusion Equations in Polygons

In this paper, we develop a new Best Uniform Rational Approximation-Semi-Discrete (BURA-SD) method taking into account the singularities of the solution of fractional diffusion problems in polygonal domains. The complementary capabilities of the exponential convergence rate of BURA-SD and the hp FEM are explored with the aim of maximizing the overall performance. A challenge here is the emerging singularly perturbed diffusion–reaction equations. The main contributions of this paper include asymptotically accurate error estimates, ending with sufficient conditions to balance errors of different origins, thereby guaranteeing the high computational efficiency of the method.

The UPC CoA’s first substantive order—central issues clarified, but on a high level

Abstract In its decision, the UPC Court of Appeal confirms the application of key principles of patent claim interpretation for unitary patents. With regard to the inventive step, a tentative path is chosen that does not clearly follow the problem–solution approach of the European Patent Office and instead, comprehensively takes the state of the art into account at the outset. The standard for the necessary probability of infringement and validity is set at a preponderance of probability (sufficient degree of certainty, ie more likely than not). Furthermore, the temporal and factual necessity of the injunction is also important in determining whether a preliminary injunction should be ordered. Ultimately, the granting of injunctive relief hinges on an overall weighing of the interests.

Deployment Approach in WSNS Using Metaheuristic Algorithm: A Survey

Deploying wireless sensor networks (WSN) is one of the areas with the most significant research since the methods employed can have a big impact on the system's overall performance and the amount of energy the sensors in it consume.Thus, an effective deployment problem solution should not only "enhance its performance" but also save the energy needed to extend the lifetime of WSN. Finding an optimal solution for most deployment problems with limited computational resources is challenging, particularly for NP-hard optimization problems. By searching for a nearly optimal solution with constrained computational resources in a reasonable amount of time, metaheuristics present an alternative to exhaustive search and deterministic algorithms in solving these optimization problems. This paper classifies the deployment techniques of sensor networks and the metaheuristic algorithms.

Resource scheduling optimization for industrial operating system using deep reinforcement learning and WOA algorithm

Industrial operating systems (IOS) are essential for supporting smart manufacturing, particularly in managing and utilizing heterogeneous production resources through resource instantiation scheduling (RIS) technique. However, RIS faces the challenge of efficiently selecting optimal resource service compositions from numerous options with varying quality of service. To boost the solving of the RIS problem and improve the quality of the solution, this paper proposes a novel hybrid algorithm, named DWOA, based on the whale optimization algorithm (WOA) and deep reinforcement learning (DRL). It first incorporates the DRL algorithm to learn experience from the historical data regarding exploration and exploitation in the WOA search process and train an optimal behavior decision model. Subsequently, utilizing the trained model, the DWOA can effectively guide the search agent in achieving a better balance between global exploration and local exploitation, thereby enhancing its convergence speed and solution quality. The effectiveness and efficiency of the DWOA approach are evaluated by the CEC2017 benchmark functions and RIS problems with various scales, compared with 11 state-of-the-art methods. The experimental results indicate that our method converges faster and produces better solutions for RIS problems.

Doubly perturbed uncertain differential equations

In this paper, we study a class of doubly perturbed uncertain differential equations driven by canonical process that is the counterpart of Wiener process in the framework of uncertain theory. We give some sufficient conditions for the existence and uniqueness of solutions of the present problem under some weak conditions, where a convex combination of the Nagumo and Osgood conditions is considered. Subsequently, we consider the stability properties with regard to the initial value and coefficients. Finally, an example is provided to illustrate the effectiveness of our main results.

Durability of anticorrosive-coated steel–carbon fiber reinforced polymer structural adhesive joint at high temperature and humidity

Limited information on the long-term reliability of structural adhesive joints hinders their practical application. Moreover, performance improvement in transportation equipment and the solution of concrete problems, such as high strengthening, vibration reduction, and weight saving, are expected by increasing the availability of materials in the shipbuilding industry using highly functional materials that are not welded. This study evaluated the durability of second-generation acrylic (SGA) structural adhesives. The strength of SGA was higher than that of epoxy-based adhesives in the case of thick adhesive layers, and few fluctuations were observed in the strength owing to variations in the adhesive-layer thickness. The static strength following environmental degradation was evaluated using a tensile lap shear structural adhesive joint coated with an anticorrosive coating, assuming application to the exposed part. The acceptable design adhesive strength in shipbuilding was evaluated using the strength retention rate ηd (mean strength after deterioration/initial mean strength) and coefficient of variation CV (standard deviation/mean strength) after deterioration. ηd for each test was greater than 72 %. The CV value for each test was <0.16; this is equivalent to a coefficient of dispersion of deterioration Dy ≥ 0.41 on percent defects allowed = 1/10,000 (upper limit of the defect rate assumed at the design stage). This result indicates that the adhesive joint is available even for the exposure section, if the adhesive joint is protected similar to adherend steel with anticorrosive paint, which is common in the field of shipbuilding.

Analytical solution of fractional oscillation equation with two Caputo fractional derivatives

Analytical solution of initial value problem for the fractional oscillation equation with two Caputo fractional derivatives , where the coefficients and orders satisfy and , is investigated by using the Laplace transform and complex inverse integral method on the principal Riemann surface. It is proved by using the argument principle that the characteristic equation has a pair of conjugated simple complex roots with a negative real part on the principal Riemann surface under the assumption that and are not both integers. Then three fundamental solutions, the unit impulse response, the unit initial displacement response, and the unit initial rate response, are derived analytically. Each of these solutions is expressed into a superposition of a classical damped oscillation decaying exponentially and a real Laplace integration decaying in a negative power law. Finally, the asymptotic behaviors of these analytical solutions for sufficiently large are determined as monotonous decays in a power of negative exponent.

Well-posedness of the Optimal Control Problem Related to Degenerate Chemo-attraction Models

This paper delves into the mathematical analysis of optimal control for a nonlinear degenerate chemotaxis model with volume-filling effects. The control is applied in a bilinear form specifically within the chemical equation. We establish the well-posedness (existence and uniqueness) of the weak solution for the direct problem using the Faedo Galerkin method (for existence), and the duality method (for uniqueness). Additionally, we demonstrate the existence of minimizers and establish first-order necessary conditions for the adjoint problem. The main novelty of this work concerns the degeneracy of the diffusive term and the presence of control over the concentration in our nonlinear degenerate chemotaxis model. Furthermore, the state, consisting of cell density and chemical concentration, remains in a weak setting, which is uncommon in the literature for solving optimal control problems involving chemotaxis models.

Long-time behaviour of a porous medium model with degenerate hysteresis.

Hysteresis in the pressure-saturation relation in unsaturated porous media, owing to surface tension on the liquid-gas interface, exhibits strong degeneracy in the resulting mass balance equation. As an extension of previous existence and uniqueness results, we prove that under physically admissible initial conditions and without mass exchange with the exterior, the unique global solution of the fluid diffusion problem exists and asymptotically converges as time tends to infinity to a possibly non-homogeneous mass distribution and an a priori unknown constant pressure.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

A direct and analytical method for inverse problems under uncertainty in energy system design: combining inverse simulation and Polynomial Chaos theory

This article introduces and formalizes a novel stochastic method that combines inverse simulation with the theory of generalized Polynomial Chaos (gPC) to solve and study inverse problems under uncertainty in energy system design applications. The method is particularly relevant to design tasks where only a deterministic forward model of a physical system is available, in which a target design quantity is an input to the model that cannot be obtained directly, but can be quantified reversely via the outputs of the model. In this scenario, the proposed method offers an analytical and direct approach to invert such system models. The method puts emphasis on user-friendliness, as it enables its users to conduct the inverse simulation under uncertainty directly in the gPC domain by redefining basic algebra operations for computations. Moreover, the method incorporates an optimization-based approach to integrate supplementary constraints on stochastic quantities. This feature enables the solution of inverse problems bounding the statistical moments of stochastic system variables. The authors exemplify the application of the proposed method with proof-of-concept tests in energy system design, specifically performing uncertainty quantification and sensitivity analysis for a Multi-Energy System (MES). The findings demonstrate the high accuracy of the method as well as clear advantages over conventional sampling-based methods when dealing with a small number of stochastic variables in a system or model. However, the case studies also highlight the current limitations of the proposed method such as slow execution speed due to the optimization-based approach and the challenges associated with, for example, the curse of dimensionality in gPC.

Inverse a time-dependent potential problem of a generalized time-fractional super-diffusion equation with a nonlinear source from a nonlocal integral observation

This paper focus on the problem of reconstructing the time-dependent potential in a class of generalized (including three special cases: the classical/multi-term/distributed order) time-fractional super-diffusion equations with nonlinear sources from a nonlocal integral observation. For such nonlinear equation, we investigate it for both the direct and inverse time-dependent potential problems. For the direct problem, given the time-dependent potential function p(t), we obtain the well-posedness of the corresponding initial–boundary value problem. For the inverse potential problem, by utilizing additional nonlocal measurement data and using the Arzelà–Ascoli theorem and Grönwall’s inequality, we prove the existence and uniqueness of the solution for such nonlinear problem. Meanwhile, we also demonstrate the ill-posedness of the inverse problem. Moreover, to validate the theoretical results, we numerically reconstruct the potential term from Bayesian perspective. Several numerical examples are presented to show the efficiency of the proposed method.

Robust Estimation of Structural Orientation Parameters and 2D/3D Local Anisotropic Tikhonov Regularization

Understanding the orientation of geological structures is crucial for analyzing the complexity of the Earths' subsurface. For instance, information about geological structure orientation can be incorporated into local anisotropic regularization methods as a valuable tool to stabilize the solution of inverse problems and produce geologically plausible solutions. We introduce a new variational method that employs the alternating direction method of multipliers within an alternating minimization scheme to jointly estimate orientation and model parameters in both 2D and 3D inverse problems. Specifically, the proposed approach adaptively integrates recovered information about structural orientation, enhancing the effectiveness of anisotropic Tikhonov#xD;regularization in recovering geophysical parameters. The paper also discusses the automatic tuning of algorithmic parameters to maximize the new method's performance. The proposed algorithm is tested across diverse 2D and 3D examples, including structure-oriented denoising and trace interpolation. The results show that the algorithm is robust in solving the considered large and challenging problems, alongside efficiently estimating the associated tilt field in 2D cases and the dip, strike, and tilt fields in 3D cases. Synthetic and field examples show that the proposed anisotropic regularization method produces a model with enhanced resolution and provides a more accurate representation of the true structures.

Variational formulation and monolithic solution of computational homogenization methods

AbstractIn this contribution, we derive a consistent variational formulation for computational homogenization methods and show that traditional FE and IGA approaches are special discretization and solution techniques of this most general framework. This allows us to enhance dramatically the numerical analysis as well as the solution of the arising algebraic system. In particular, we expand the dimension of the continuous system, discretize the higher dimensional problem consistently and apply afterwards a discrete null‐space matrix to remove the additional dimensions. A benchmark problem, for which we can obtain an analytical solution, demonstrates the superiority of the chosen approach aiming to reduce the immense computational costs of traditional FE and IGA formulations to a fraction of the original requirements. Finally, we demonstrate a further reduction of the computational costs for the solution of general nonlinear problems.

The unified transformation approach to higher-order Gerdjikov-Ivanov model and Riemann-Hilbert problem

In this paper, a higher-order Gerdjikov-Ivanov (HOGI) model with cubic dispersion and quintic nonlinearity terms will be researched, which is a 3-order flow model of the well known GI system. The initial-boundary value problems of the HOGI model on the half-line will be given by the unified transformation approach. The solutions of the HOGI model will be expressed by forms of the solutions of a matrix Riemann-Hilbert problem (RHP). The relevant jump matrices of the RHP will be explicitly found by the two scattering matrices s(ϵ) and S(ϵ). Finally, the so-called global relation will be given.

Plane SV-waves scattering by seawater convex surface topography and underlying lined tunnel in a saturated half-space

Analytical solutions of boundary-valued problems for aboveground topography and underground buried structures, especially for sites with overlying seawater, have always been challenging. Based on the region decomposition technique (RDT) and Hankel integral transform method (HITM), SV-waves scattering by the convex circular topography with seawater and underlying tunnel in a saturated half-space is solved. The primary problem is tackling additional scattered waves introduced by the water–soil interface and convex topography, while ensuring boundary conditions are all met. To this end, scattering wave potential functions in cylindrical coordinates are first transformed into the Hankel integral form in Cartesian coordinates. This greatly facilitates the straightforward application of free boundary conditions at the soil–seawater flat interface, rather than making a large circular arc approximate assumption, which, in turn, avoids the resultant accumulating errors. Secondly, scattering problem to be solved is mathematically and analytically formulated as solving a 5 × 5 linear equation system comprised of wave functions inside the convex topography and lining tunnel while satisfying all physical continuous boundary conditions. This is the main innovation and resulting consequence from the theoretical derivation process. Finally, validation of numerical examples between this paper, existing studies and theoretical analysis further demonstrates the reliability and general adaptability of solutions. Parametric studies of the incident SV waves are conducted to investigate the spatially varying seismic response. The effects of SV-waves incident angles and frequencies, tunnel burial depth, tunnel lining thickness, saturated soil porosity, and seawater depth are investigated and analyzed in detail. This study provides a feasible scheme of investigating the dynamic response of complex submarine sites, which is of great significance for both theoretical value and engineering purposes.

An exact method for trilevel hub location problem with interdiction

In this paper, we study the problem of designing a hub network that is robust against deliberate attacks (interdictions). The problem is modeled as a three-level, two-player Stackelberg game, in which the network designer (defender) acts first to locate hubs to route a set of flows through the network. The attacker (interdictor) acts next to interdict a subset of the located hubs in the designer’s network, followed again by the defender who routes the flows through the remaining hubs in the network. We model the defender’s problem as a trilevel optimization problem, wherein the attacker’s response is modeled as a bilevel hub interdiction problem. We study such a trilevel problem on three variants of hub location problems studied in the literature namely: p-hub median problem, p-hub center, and p-hub maximal covering problems. We present a cutting plane based exact method to solve the problem. The cutting plane method uses supervalid inequalities, which is obtained from the solution of the lower level interdiction problem. To solve the lower level hub interdiction problem efficiently, we propose a penalty-based reformulation of the problem. Using the reformulation, we present a branch-and-cut based exact approach to solve the problem efficiently. We conduct experiments to show the computational advantages of the above algorithm. To the best of our knowledge, the cutting plane approach proposed in this paper is among the first exact method to solve trilevel location–interdiction problems. Our computational results show interesting implications of incorporating interdiction risks in the hub location problem.