One-dimensional Boundary Research Articles

Замкнена форма розв’язків деяких крайових задач дробово-диференційної консолідаційної динаміки геопористих середовищ

Створення сучасних геотехнологій, що мають на меті функціонування в складних гірничо-геологічних умовах, вимагає подальшого розвитку та уточнення існуючих математичних моделей і методів моделювання особ-ливостей динаміки геоміграційних процесів. При цьому помітний прогрес у галузі математичного моделювання геоміграційних процесів за складних умов їхнього перебігу, пов’язаний із використанням формалізму інтегро-диференціювання дробового порядку. Зазвичай математичне моделювання динаміки геоміграційних процесів виконується за умов насиченості маси-вів геопористих середовищ чистою водою, однак у цей час, внаслідок зрос-тання техногенного впливу на природу, особливу актуальність становлять дослідження в галузі моделювання динаміки зазначених процесів за умов насичення геопористого середовища сольовими розчинами. Це значною мірою обумовлено рядом проблем екології, зокрема задачами захисту ґрун- тів та ґрунтових вод від забруднення токсичним вмістом поверхневих на-копичувачів промислових та побутових стоків. У даній роботі одержано точні розв’язки деяких одновимірних крайових задач дробово-диферен-ційної консолідаційної динаміки насичених сольовими розчинами глинистих геопористих середовищ за умов одночасного врахування як просторової, так і часової нелокальностей геоміграційного процесу. Зокрема представлено дробово-диференційну математичну модель динаміки нелокального у часі та просторі фільтраційно-консолідаційного процесу, яка містить похідні Капуто за часовою змінною та Рімана–Ліувілля — за геометричною змінною. У рамках зазначеної моделі наведено постановку та одержано точний розв’язок прямої задачі консолідаційної динаміки геопористого масиву скінченної потужності, насиченого сольовим розчином. Також розглянуто задачу консолідаційної динаміки масиву скінченної потужності в оберне-ній постановці щодо визначення невідомих функцій джерел, залежних ли-ше від геометричної змінної, за відповідними додатковими (кінцевими) умо-вами. Наведено умови існування регулярних розв’язків розглянутих задач.

Observations on holographic aspects of four-dimensional asymptotically flat mathcal{N} = 2 black holes

In this note, we explore holographic attributes of four-dimensional near-extremal Reissner-Nordstrom black hole solutions in ungauged mathcal{N} = 2 supergravity theories at the two-derivative level by recasting them as a specific first-order deformation in solution space, associated with an infinitesimal Harrison transformation, of black holes in an AdS2 space-time. Specifically, we use this link to exhibit how bulk properties, such as mass and entropy, of four-dimensional near-extremal black holes are holographically encoded in the one-dimensional boundary theory dual to gravity in an infinitesimally deformed AdS2 space-time. We do so for the case of four-dimensional near-extremal black holes that arise as deformations in solution space of BPS black holes by changing the non-extremality parameter. For these near-extremal black holes, we further show that the nAdS2 attractor mechanism can be recast as a specific deformation of the BPS flow equations in four dimensions. Additionally, we also discuss time-dependent perturbations of the four-dimensional near-extremal Reissner-Nordstrom solutions from a two-dimensional point of view.

A bottleneck detection approach for separating overlapped cells

To separate the overlapped cells, a bottleneck detection approach is proposed in this paper. The cell image is segmented by slope difference distribution (SDD) threshold selection. For each segmented binary clump, its one-dimensional boundary is computed as the distance distribution between its centroid and each point on the two-dimensional boundary. The bottleneck points of the one-dimensional boundary is detected by SDD and then transformed back into two-dimensional bottleneck points. Two largest concave parts of the binary clump are used to select the valid bottleneck points. Two bottleneck points from different concave parts with the minimum Euclidean distance is connected to separate the binary clump with minimum-cut. The binary clumps are separated iteratively until the number of computed concave parts is smaller than two. We use four types of open-accessible cells to verify the effectiveness of the proposed approach and experimental results showed that the proposed approach is significantly more robust than state of the art methods.

Global existence and uniqueness of solutions for one-dimensional reaction-interface systems

In this paper, we provide a mathematical framework in studying the wave propagation with the annihilation phenomenon in excitable media. We deal with the existence and uniqueness of solutions to a one-dimensional free boundary problem (called a reaction–interface system) arising from the singular limit of a FitzHugh–Nagumo type reaction–diffusion system. Because of the presence of the annihilation, interfaces may intersect each other. We introduce the notion of weak solutions to study the continuation of solutions beyond the annihilation time. Under suitable conditions, we show that the free boundary problem is well-posed.

Long-time asymptotics of solutions to the Keller–Rubinow model for Liesegang rings in the fast reaction limit

We consider the Keller--Rubinow model for Liesegang rings in one spatial dimension in the fast reaction limit as introduced by Hilhorst, van der Hout, Mimura, and Ohnishi in 2007. Numerical evidence suggests that solutions to this model converge, independent of the initial concentration, to a universal profile for large times in parabolic similarity coordinates. For the concentration function, the notion of convergence appears to be similar to attraction to a stable equilibrium point in phase space. The reaction term, however, is discontinuous so that it can only convergence in a much weaker, averaged sense. This also means that most of the traditional analytical tools for studying the long-time behavior fail on this problem. In this paper, we identify the candidate limit profile as the solution of a certain one-dimensional boundary value problem which can be solved explicitly. We distinguish two nontrivial regimes. In the first, the transitional regime, precipitation is restricted to a bounded region in space. We prove that the concentration converges to a single asymptotic profile. In the second, the supercritical regime, we show that the concentration converges to one of a one-parameter family of asymptotic profiles, selected by a solvability condition for the one-dimensional boundary value problem. Here, our convergence result is only conditional: we prove that if convergence happens, either pointwise for the concentration or in an averaged sense for the precipitation function, then the other field converges likewise; convergence in concentration is uniform, and the asymptotic profile is indeed the profile selected by the solvability condition. A careful numerical study suggests that the actual behavior of the equation is indeed the one suggested by the theorem.

Entanglement phase transitions in random stabilizer tensor networks

We explore a class of random tensor network models with "stabilizer" local tensors which we name Random Stabilizer Tensor Networks (RSTNs). For RSTNs defined on a two-dimensional square lattice, we perform extensive numerical studies of entanglement phase transitions between volume-law and area-law entangled phases of the one-dimensional boundary states. These transitions occur when either (a) the bond dimension $D$ of the constituent tensors is varied, or (b) the tensor network is subject to random breaking of bulk bonds, implemented by forced measurements. In the absence of broken bonds, we find that the RSTN supports a volume-law entangled boundary state with bond dimension $D\geq3$ where $D$ is a prime number, and an area-law entangled boundary state for $D=2$. Upon breaking bonds at random in the bulk with probability $p$, there exists a critical measurement rate $p_c$ for each $D\geq 3$ above which the boundary state becomes area-law entangled. To explore the conformal invariance at these entanglement transitions for different prime $D$, we consider tensor networks on a finite rectangular geometry with a variety of boundary conditions, and extract universal operator scaling dimensions via extensive numerical calculations of the entanglement entropy, mutual information and mutual negativity at their respective critical points. Our results at large $D$ approach known universal data of percolation conformal field theory, while showing clear discrepancies at smaller $D$, suggesting a distinct entanglement transition universality class for each prime $D$. We further study universal entanglement properties in the volume-law phase and demonstrate quantitative agreement with the recently proposed description in terms of a directed polymer in a random environment.

Tailoring the Energy Landscape of Graphene Nanostructures on Graphene and Manipulating Them Using Tilt Grain Boundaries

In two-dimensional van der Waals (vdW) materials, the relative twist angle between adjacent layers not only controls their electronic properties but also determines their stacking energy. This effect makes it difficult to stabilize the vdW materials with twist angles. Here, we demonstrate that we can controllably stabilize the system with custom-designed twist angles, for example, magic angle, which is realized by tailoring the adhesive-energy landscape of graphene nanostructures on graphene by using a one-dimensional tilt grain boundary (GB). Our result demonstrates that the flat band of the custom-built nanoscale magic-angle-twisted bilayer graphene is still robust, even when its size is comparable to a single moir\'e spot. Moreover, in our experiments, the area ratio with different stacking orders separated by the tilt GB can be continuously tuned by using the scanning-tunneling-microscope tip, and we can repeatedly fold and unfold the graphene nanostructure along the one-dimensional GB, demonstrating the ability to manipulate the graphene nanostructure at the atomic scale.

Local existence of a solution to a free boundary problem describing migration into rubber with a breaking effect

<abstract><p>We consider a one-dimensional free boundary problem describing the migration of diffusants into rubber. In our setting, the free boundary represents the position of the front delimitating the diffusant region. The growth rate of this region is described by an ordinary differential equation that includes the effect of breaking the growth of the diffusant region. In this specific context, the breaking mechanism is should be perceived as a non-dissipative way of describing eventual hyperelastic response to a too fast diffusion penetration. In recent works, we considered a similar class of free boundary problems modeling diffusants penetration in rubbers, but without attempting to deal with the possibility of breaking or accelerating the occurring free boundaries. For simplified settings, we were able to show the global existence and uniqueness as well as the large time behavior of the corresponding solutions to our formulations. Since here the breaking effect is contained in the free boundary condition, our previous results are not anymore applicable. The main mathematical obstacle in ensuring the existence of a solution is the non-monotonic structure of the free boundary. In this paper, we establish the existence and uniqueness of a weak solution to the free boundary problem with breaking effect and give explicitly the maximum value that the free boundary can reach.</p></abstract>

One-Dimensional Boundary Layer in the Problem of Negel-Free Zone Melting in a Magnetic Field

One-Dimensional Boundary Layer in the Problem of Negel-Free Zone Melting in a Magnetic Field

РЕЗОНАНСНІ ВЛАСТИВОСТІ ШАРУВАТОГО ҐРУНТОВОГО МАСИВУ, ЩО МІСТИТЬ ПРОШАРОК З КОЛИВНИМИ ВКЛЮЧЕННЯМИ

The paper is devoted to modeling of the seismic reaction of an inhomogeneous soil massif consisting of three layers that can incorporate oscillating inclusions. The description of the behavior of soil strata is carried out on the basis of the continual approach, within which the dynamics of the medium is determined by a system of equations of motion and equations of state, where the elastic or visco-elastic properties of the medium are indicated. Classical linear models are quite simple and well verified experimentally, but due to the complex structure of natural geomedia, these models cannot always adequately describe wave processes in them. This requires improvement of models in such a way as to take into account the internal structure of materials. In this paper, to describe the shear dynamics of an inhomogeneous soil massif, the equations of motion for a continuous medium in the form of mutually penetrating continua are used, one of which coincides with the Kelvin-Voigt carrier medium, and the other one is described by a set of noninteracting partial oscillators. Various inclusions, cracks, or cavities filled with gases and/or liquids can act as oscillators. The equations of motion are supplemented by the compatibility conditions at the boundaries of the layers. The set of layers is considered, among which only one contains oscillating inclusions. The one-dimensional boundary value problem with a free surface and the harmonic law of deformation of massif's rigid bedrock is solved. Using the exact solution of the boundary value problem, the amplification factor of strains for massifs with characteristics close to natural is calculated. Moreover, the iterative procedure, which makes it possible to write down the final formula for the amplification factor in the case of a layered medium with layers of more than 3 is developed. It is shown that the incorporation of inclusions in one of the layers leads to the appearance of an additional resonance frequency, a shift of resonances to the low-frequency region, the appearance of zones with significant damping of resonance peaks. The results obtained make it possible to improve the computational methods for determining the quantitative parameters of seismic hazard when work on seismic micro zoning of construction and operational sites in the seismic regions of Ukraine is carried out.

Enhanced piezotronics by single-crystalline ferroelectrics for uniformly strengthening the piezo-photocatalysis of electrospun BaTiO3@TiO2 nanofibers.

Turning the built-in electric field by modulating the morphology and microstructure of ferroelectric materials is considered a viable approach to enhancing the piezo-photocatalytic activity of the ferroelectric/oxide semiconductor heterojunctions. Here, hydrothermally synthesized single-crystalline BaTiO3 nanoparticles are employed to construct BaTiO3@TiO2 hybrid nanofibers by sol-gel assisted electrospinning of TiO2 nanofibers and annealing. Because of the obvious enhancement of the synergetic piezo-photocatalytic effect under both ultrasonic and ultraviolet (UV) light irradiation, the piezo-photocatalytic degradation rate constant (k) of BaTiO3@TiO2 hybrid nanofibers on methyl orange (MO) reaches 14.84 × 10-2 min-1, which is approximately seven fold that for piezocatalysis and six fold that for photocatalysis. Moreover, BaTiO3@TiO2 core-shell nanoparticles are also synthesized for comparison purposes to assess the influence of microstructure on the piezo-photocatalysis by a wet-chemical coating of TiO2 on BaTiO3 nanoparticles. Such a high piezo-photocatalytic activity is attributed to the enhancement of the piezotronic effect by the single-crystalline ferroelectric nanoparticles and the nanoconfinement effect caused by the one-dimensional boundary of nanofibers with high specific surface areas. The mechanically induced uniform local built-in electric fields originated from the single-crystalline ferroelectric nanoparticles can enhance the separation of photogenerated electron and hole pairs and promote the formation of free hydroxyl radicals, resulting in a strong piezotronic effect boosted photochemical degradation of organic dye. This work introduces the single-crystalline ferroelectrics to construct ferroelectric/oxide semiconductor heterojunctions, and the enhanced local piezotronic effect uniformly strengthens the photochemical reactivity, which offers a new option to design high-efficiency piezo-photocatalysts for pollutant treatment.

Near-field spectroscopic imaging of exciton quenching at atomically sharp MoS2/WS2 lateral heterojunctions.

Heterojunctions made by laterally stitching two different transition metal dichalcogenide monolayers create a unique one-dimensional boundary with intriguing local optical properties that can only be characterized by nanoscale-spatial-resolution spectral tools. Here, we use near-field photoluminescence (NF-PL) to reveal the narrowest region (105 nm) ever reported of photoluminescence quenching at the junction of a laterally stitched WS2/MoS2monolayer. We attribute this quenching to the atomically sharp band offset that generates a strong electric force at the junction to easily dissociate excitons. Besides the sharp heterojunction, a model considering various widths of the alloying interfacial region under low or high optical pumping is presented. With a spatial resolution six times better than that of confocal microscopy, NF-PL provides an unprecedented spectral tool for non-scalable 1D lateral heterojunctions.

H-Adaptive Finite Element Methods for 1D Stationary High Gradient Boundary Value Problems

This article considered the traditional finite element method (FEM) and adaptive finite element method (FEM) for the numerical solution of the one-dimensional boundary value problems. We established the preference or the superiority of the h-adaptive FEM to traditional FEM in high gradient problems in terms of accuracy and cost of computation. Numerical examples which confirm the performance and adaptability of the h-adaptive method over the traditional finite element method and the high accuracy of the numerical solution are presented. Detailed error analysis of linear elements was also discussed. In conclusion, h-adaptive FEM is recommended for complex systems with high gradient problems.

Modulation of electric dipoles inside electrospun BaTiO3@TiO2 core-shell nanofibers for enhanced piezo-photocatalytic degradation of organic pollutants

Ferroelectric materials can generate a built-in electric field under strain, which can produce reactive oxygen species (ROS) for in-situ piezocatalysis via a series of redox reactions, or enhance photocatalytic activities through the separation of the photoinduced electron and hole pairs. In this study, the synergistic piezo-phototronic effect is enhanced by increasing the inner piezopotential of ferroelectric nanofibers in fibrous piezoelectric/oxide semiconductor heterostructures. Ferroelectric BaTiO3 nanofibers are fabricated by a sol-gel assisted electrospinning and subsequent annealing at 700 ºC for 2 h, 4 h, and 6 h, respectively (denoted as BT2H, BT4H, and BT6H). Piezoelectric force microscopy results reveal that the necklace-like BT6H nanofibers demonstrate the highest local piezoelectric coefficient d33 (42.7 pm V−1) than BT2H (26.3 pm V−1) and BT4H (37.8 pm V−1) nanofibers because BT6H nanofibers are composed of highly crystallized electric dipoles with single domain nanostructures. Additionally, BT6H@TiO2 core-shell nanofibers are synthesized by a wet-chemical coating of TiO2 on BT6H nanofibers. Under both ultrasound and UV light irradiations, BT6H@TiO2 core-shell nanofibers exhibit a high piezo-photocatalytic degradation rate constant of 6 × 10−2 min−1 because of the enhancement of the piezotronic effect by the high piezoelectricity of electric dipoles that promote the separation of the photoinduced electron and hole pairs. The enhancement of the synergistic piezo-phototronic effect is ascribed to the highly active reaction sites that are confined into the high specific surface area of the one-dimensional fibrous boundary of BT6H@TiO2 core-shell nanofibers, beneficial for the migration of the interfacial charge carriers. This study presents a simple approach to improve the piezo-photocatalysis through modulating the crystallization and the size of electric dipoles, providing an efficient pathway for underpinning the coupling mechanism between the enhancement of local ferro/piezoelectricity and piezo-phototronic effect.

From d’Alembert to Bloch and back: A semi-analytical solution of 1D boundary value problems governed by the wave equation in periodic media

We propose a simple semi-analytical model for solving one-dimensional (1D) boundary value problems governed by the wave equation in periodic media at arbitrary frequency. When the boundary value problem is well-posed in that the excitation frequency does not match any of its resonant frequencies, any solution that satisfies the field equation and boundary conditions is the (unique) solution of the problem. This motivates us to seek the solution in the form of Bloch waves that by design solve the wave equation with periodic coefficients. For a given excitation frequency this assumption leads to a quadratic eigenvalue problem in terms of the wavenumber k which, in turn, produces a d’Alembert-type solution in terms of the “right”- and “left”- propagating (or evanescent) Bloch waves. In this vein, the solution of a boundary value problem is obtained by computing the amplitudes of the two Bloch waves from the prescribed boundary conditions. For completeness, we consider situations when the driving frequency is: (i) within a passband, (ii) inside a band gap, (iii) at the edge of a band gap (the so-called exceptional points), and (iv) at band crossing (repeated eigenfrequency). From simulations, we find that the semi-analytical solution – expressed in terms of either propagating, evanescent, or standing Bloch waves – is computationally effective and reproduces the numerical results with high fidelity. Consistent with related studies, we also find that the solution of the boundary value problem undergoes sharp transition in a frequency neighborhood of exceptional points. One of the most intriguing results of this study is our finding that near exceptional points, the global resonance of a periodic medium could be induced by “microscopically” adjusting the support locations so they coincide with a shared node of the germane Bloch eigenfunctions. This opens a door toward the design of adaptive wave control systems whose performance can be tuned by adjusting the boundary conditions – as opposed to altering the properties of a periodic medium.

Большие упругие деформации эластомерной торовой оболочки (резиновой муфты) при совместном действии крутящих моментов и центробежных сил

The article considers solving the problem of large axisymmetric deformations of elastomeric torus shells of revolution, loaded with jointly acting torques, axial and centrifugal forces. The task is posed due to the calculation of rubber elements of couplings. The calculations are performed according to the momentless shell theory by solving a nonlinear one-dimensional boundary value problem using the shooting method, as well as in a three-dimensional formulation using the finite element method. The calculation results are presented both for convex and concave torus shells. The load characteristics are compared for free and constrained torsion. The dependence of axial reactions in supports on torque and centrifugal forces has been investigated.

Un análisis comparativo de los métodos: miméticos, diferencias finitas y elementos finitos para problemas estacionarios 1-dimensional

Numerical methods are useful for solving differential equations that model physical problems, for example, heat transfer, fluid dynamics, wave propagation, among others; especially when these cannot be solved by means of exact analysis techniques, since such problems present complex geometries, boundary or initial conditions, or involve non-linear differential equations. Currently, the number of problems that are modeled with partial differential equations are diverse and these must be addressed numerically, so that the results obtained are more in line with reality. In this work, a comparison of the classical numerical methods such as: the finite difference method (FDM) and the finite element method (FEM), with a modern technique of discretization called the mimetic method (MIM), or mimetic finite difference method or compatible method, is approached. With this comparison we try to conclude about the efficiency, order of convergence of these methods. Our analysis is based on a model problem with a one-dimensional boundary value, that is, we will study convection-diffusion equations in a stationary regime, with different variations in the gradient, diffusive coefficient and convective velocity.

A general solution procedure for the scaled boundary finite element method via shooting technique

The scaled boundary finite element method (SBFEM) is known for its inherent ability to simulate unbounded domains and singular fields, and its flexibility in the meshing procedure. Keeping the analytical form of the field variables along one coordinate intact, it transforms the governing partial differential equations of the problem into a system of one-dimensional (initial–)boundary value problems. However, closed-form solution of the said system is not available for most cases (e.g. transient heat transfer, acoustics, ultrasonics, etc.) since the system cannot be diagonalized in general. This paper aims to establish a numerical tool within the context of the shooting technique to evaluate the coefficient matrices of the subdomains without a priori knowledge of the analytical solution of the semi-discretized system. With proper choice of boundary conditions, the technique uses the strong form of the scaled boundary finite element equations to pass the required information and with the desired accuracy from one boundary to another. Due to generality of the technique, its procedure can be adjusted for any field equations. Since this technique is presented here for the first time, linear elastostatics, for which the closed-form solution is well-established, is formulated to provide valid comparisons. In addition, any direct solution method can be used for integrating the scaled boundary equations. Thus, without loss of generality, a Nyström extension of the classical fourth-order Runge–Kutta method is employed. A quantitative sensitivity analysis is also conducted, and efficiency of the classical and proposed solution techniques is compared in terms of computational time. Finally, some numerical examples, including bounded and unbounded domains, as well as singular stress fields are simulated based on the classical and proposed solution techniques.

The dynamical problem on acting concentrated load on the elastic quarter space

The wave field of an elastic quarter space is constructed when one face is rigidly fixed and a dynamic normal compressive load acts on the other along a rectangular section at the initial moment of time. Integral Laplace and Fourier transforms are applied sequentially to the equations of motion and boundary conditions in contrast to traditional approaches when integral transforms are applied to solutions' representations through harmonic functions. This leads to a one-dimensional vector homogeneous boundary value problem with respect to unknown displacement's transformants. The problem was solved using matrix differential calculus. The original displacement field was found after applying inverse integral transforms. For the case of stationary vibrations a method of calculating integrals in the solution in the near loading zone was indicated. For the analysis of oscillations in a remote zone the asymptotic formulas were constructed. The amplitude of vertical vibrations was investigated depending on the shape of the load section, natural frequencies of vibrations and the material of the medium.

A novel design of Gaussian Wavelet Neural Networks for nonlinear Falkner-Skan systems in fluid dynamics

In this study, a design of integrated computational intelligent paradigm has been presented for numerical treatment of the one-dimensional boundary value problems represented with Falkner-Skan equations (FSE) by exploitation of Gaussian wavelet neural networks (GWNNs), genetic algorithms (GAs) and sequential quadratic programming (SQP), i.e., GWNN-GA-SQP. The GWNNs is used for mathematical modeling of the problem by constructing mean squared error based objective function while optimization of the cost function is initially conducted with efficacy of GAs as a global search and while fine tuning is performed with efficiency local search with SQP. The numerical results are obtained by proposed GWNN-GA-SQP for different FSEs arising in nonlinear regimes of computation fluid mechanics studies. A comparison of the results of proposed GWNN-GA-SQP stochastic numerical solver with reference state of the art solutions of Adams method establishes the accuracy, convergence and stability, which further endorsed through statistics on multiples runs. The T-Paired test is also applied to validate the effectiveness of the proposed GWNN-GA-SQP algorithm for solving nonlinear FSEs.