Two-dimensional Spatial Domain Research Articles

Effective description of the impact of inhomogeneities on the movement of the kink front in 2+1 dimensions.

In the present work we explore the interaction of a quasi-one-dimensional line kink of the sine-Gordon equation moving in two-dimensional spatial domains. We develop an effective equationdescribing the kink motion, characterizing its center position dynamics as a function of the transverse variable. The relevant description is valid both in the Hamiltonian realm and in the nonconservative one bearing gain and loss. We subsequently examine a variety of different scenarios, without and with a spatially dependent heterogeneity. The latter is considered both to be one dimensional (y independent) and genuinely two dimensional. The spectral features and the dynamical interaction of the kink with the heterogeneity are considered and comparison with the effective quasi-one-dimensional description (characterizing the kink center as a function of the transverse variable) is also provided. Generally, good agreement is found between the analytical predictions and the computational findings in the different cases considered.

New Numerical and Analytical Solutions for Nonlinear Evolution Equations Using Updated Mathematical Methods

This study explores adapted mathematical methods to solve the couple-breaking soliton (BS) equations in two-dimensional spatial domains. Using these methods, we obtained analytical soliton solutions for the equations involving free parameters such as the wave number, phase component, nonlinear coefficient, and dispersion coefficient. The solutions are expressed as hyperbolic, rational, and trigonometric functions. We also examined the impact of wave phenomenon on two-dimensional diagrams and used composite two-dimensional and three-dimensional graphs to represent the solutions. We used the finite difference method to transform the proposed system into a numerical system to obtain numerical simulations for the Black–Scholes equations. Additionally, we discuss the stability and error analysis of numerical schemes. We compare the validity and accuracy of the numerical results with the exact solutions through analytical and graphical comparisons. The methodologies presented in this research can be applied to various forms of nonlinear evolutionary systems because they are appropriate and acceptable.

Spatiotemporal dynamics of a multi-delayed prey-predator system with variable carrying capacity.

This paper presents the temporal and spatiotemporal dynamics of a delayed prey-predator system with a variable carrying capacity. Prey and predator interact via a Holling type-II functional response. A detailed dynamical analysis, including well-posedness and the possibility of coexistence equilibria, has been performed for the temporal system. Local and global stability behavior of the co-existence equilibrium is discussed. Bistability behavior between two coexistence equilibria is demonstrated. The system undergoes a Hopf bifurcation with respect to the parameter β, which affects the carrying capacity of the prey species. The delayed system exhibits chaotic behavior. A maximal Lyapunov exponent and sensitivity analysis are done to confirm the chaotic dynamics. In the spatiotemporal system, the conditions for Turing instability are derived. Furthermore, we analyzed the Turing pattern formation for different diffusivity coefficients for a two-dimensional spatial domain. Moreover, we investigated the spatiotemporal dynamics incorporating two discrete delays. The effect of the delay parameters in the transition of the Turing patterns is depicted. Various Turing patterns, such as hot-spot, coldspot, patchy, and labyrinth, are obtained in the case of a two-dimensional spatial domain. This study shows that the parameter β and the delay parameters significantly instigate the intriguing system dynamics and provide new insights into population dynamics. Furthermore, extensive numerical simulations are carried out to validate the analytical findings. The findings in this article may help evaluate the biological revelations obtained from research on interactions between the species.

The effect of nonlocal interaction on chaotic dynamics, Turing patterns, and population invasion in a prey-predator model.

Pattern formation is a central process that helps to understand the individuals' organizations according to different environmental conditions. This paper investigates a nonlocal spatiotemporal behavior of a prey-predator model with the Allee effect in the prey population and hunting cooperation in the predator population. The nonlocal interaction is considered in the intra-specific prey competition, and we find the analytical conditions for Turing and Hopf bifurcations for local and nonlocal models and the spatial-Hopf bifurcation for the nonlocal model. Different comparisons have been made between the local and nonlocal models through extensive numerical investigation to study the impact of nonlocal interaction. In particular, a legitimate range of nonlocal interaction coefficients causes the occurrence of spatial-Hopf bifurcation, which is the emergence of periodic patterns in both time and space from homogeneous periodic solutions. With an increase in the range of nonlocal interaction, the whole Turing pattern suppresses after a certain threshold, and no pure Turing pattern exists for such cases. Specifically, at low diffusion rates for the predators, nonlocal interaction in the prey population leads to the extinction of predators. As the diffusion rate of predators increases, impulsive wave solutions emerge in both prey and predator populations in a one-dimensional spatial domain. This study also includes the effect of nonlocal interaction on the invasion of populations in a two-dimensional spatial domain, and the nonlocal model produces a patchy structure behind the invasion where the local model predicts only the homogeneous structure for such cases.

Multiple-scattering frequency-time hybrid solver for the wave equation in interior domains

This paper proposes a frequency-time hybrid solver for the time-dependent wave equation in two-dimensional interior spatial domains. The approach relies on four main elements, namely, (1) A multiple scattering strategy that decomposes a given interior time-domain problem into a sequence of limited-duration time-domain problems of scattering by overlapping open arcs, each one of which is reduced (by means of the Fourier transform) to a sequence of Helmholtz frequency-domain problems; (2) Boundary integral equations on overlapping boundary patches for the solution of the frequency-domain problems in point (1); (3) A smooth “Time-windowing and recentering” methodology that enables both treatment of incident signals of long duration and long time simulation; and, (4) A Fourier transform algorithm that delivers numerically dispersionless, spectrally-accurate time evolution for given incident fields. By recasting the interior time-domain problem in terms of a sequence of open-arc multiple scattering events, the proposed approach regularizes the full interior frequency domain problem—which, if obtained by either Fourier or Laplace transformation of the corresponding interior time-domain problem, must encapsulate infinitely many scattering events, giving rise to non-uniqueness and eigenfunctions in the Fourier case, and ill conditioning in the Laplace case. Numerical examples are included which demonstrate the accuracy and efficiency of the proposed methodology.

Enhanced delivery of amendments in contaminated low-permeability soils by hydraulic fracturing

In-situ remediation of contaminated low-permeability sites remains to be a critical challenge due to severe mass transfer limitations. This work developed an analytical model for enhanced in-situ remediation by hydraulic fracturing in low permeability sites, considering the mechanisms of convection, diffusion, adsorption, and degradation of amendments. In this model, amendment-filled fractures were conceptualized as mass release sources, with the release process controlled by a dissolution-diffusion equation. Combining the Laplace and Fourier-cosine transformation techniques, the semi-analytic solutions in the two-dimensional spatial domain were obtained and then validated by COMSOL Multiphysics 6.0. Based on this model, the influence range, effective longevity, degradability, amendment reserves, and fracture spacing design of reactive fractures were investigated. Results show that (1) three reaction zones with different degradability will potentially form around the reactive fractures; (2) a larger reservoir of amendments effectively increases the influence range and effective longevity of reactive fractures; (3) the synergy of multiple fracture systems outweigh the sum of individual fracture properties, but the reasonable spacing design is decisive; (4) ignoring adsorption in low-permeability site remediation may overestimate the influence range of reactive fractures, leading to remediation failure. This work comprehensively analyzed the properties of reactive fractures, providing practitioners with a reference for the enhanced remediation of low-permeability contaminated sites.

Exponential convergence of hp-time-stepping in space-time discretizations of parabolic PDES

For linear parabolic initial-boundary value problems with self-adjoint, time-homogeneous elliptic spatial operator in divergence form with Lipschitz-continuous coefficients, and for incompatible, time-analytic forcing term in polygonal/polyhedral domains D, we prove time-analyticity of solutions. Temporal analyticity is quantified in terms of weighted, analytic function classes, for data with finite, low spatial regularity and without boundary compatibility. Leveraging this result, we prove exponential convergence of a conforming, semi-discrete hp-time-stepping approach. We combine this semi-discretization in time with first-order, so-called “h-version’’ Lagrangian Finite Elements with corner-refinements in space into a tensor-product, conforming discretization of a space-time formulation. We prove that, under appropriate corner- and corner-edge mesh-refinement of D, error vs. number of degrees of freedom in space-time behaves essentially (up to logarithmic terms), to what standard FEM provide for one elliptic boundary value problem solve in D. We focus on two-dimensional spatial domains and comment on the one- and the three-dimensional case.

Spatially varying earthquake ground motion: artificial generation and effects on earth-fill dam behavior

ABSTRACT The observed amplitude and phase differences of seismic ground motions recorded at different locations over extended areas are termed as spatially varying earthquake ground motion (SVEGM). Earthquake-resistant design and seismic risk assessment of extended structures require three-dimensional dynamic analyses for which the excitation is needed in the form of three-directional SVEGM components, where each component needs to be consistent with the coherency model. In this paper, the principal features of the performed analyses are as follows: (1) discretisation of the spatial domain at the foundation base, (2) the selection of coherency model and (3) propagation angle of the seismic waves. In the first part of the paper, three cases for the generation of three-directional SVEGM are presented. First, the two-dimensional spatial domain, representing the foundation base of the extended structure, is discretised. Second, for case 1, the three-directional SVEGM is generated using the Liu and Hong coherency model at the cells of the grid. In this case, the loss of coherency is considered independent of the direction of the wave propagation (case general). Then, for the case 2, the two different propagation angles of 30 and 60 degrees of seismic waves and the Liu and Hong coherency model are considered. Finally, using the anisotropic coherency model of Hao et al. and the propagation angle 45 degree, the three-directional SVEGM is generated, in case 3. Results show that each component of the generated SVEGM records is affected by propagation angle and directional dependency of the coherency model. The second part of the paper is devoted to two-dimensional analyses of Marun earthfill dam and three-dimensional analyses of earthfill dams with different length-to-height ratios subjected to uniform input motion and artificially generated SVEGM.

A model for the collective synchronization of flashing in Photinus carolinus.

Recent empirical investigations have characterized the synchronized flashing behaviours of male Photinus carolinus fireflies in their natural habitat in Great Smoky Mountain National Park as well as in controlled environments. We develop a model for the flash dynamics of an individual firefly based on a canonical elliptic burster, a slow-fast dynamical system that produces a repeating pattern of multiple flashes followed by a quiescent period. We show that a small amount of noise renders that oscillation very irregular, but when multiple model fireflies interact through their flashes, the behaviour becomes much more periodic. We show that the aggregate behaviour is qualitatively similar to the experimental findings. We next distribute the fireflies in a two-dimensional spatial domain and vary the interaction range. In addition to synchronization, various spatio-temporal patterns involving propagation of activity emerge spontaneously. Finally, we allow a certain number of fireflies to move and demonstrate how their speed affects the rate and degree of synchronization.

FC-based shock-dynamics solver with neural-network localized artificial-viscosity assignment

This paper presents a spectral scheme for the numerical solution of nonlinear conservation laws in non-periodic domains under arbitrary boundary conditions. The approach relies on the use of the Fourier Continuation (FC) method for spectral representation of non-periodic functions in conjunction with smooth localized artificial viscosity assignments produced by means of a Shock-Detecting Neural Network (SDNN). Like previous shock capturing schemes and artificial viscosity techniques, the combined FC-SDNN strategy effectively controls spurious oscillations in the proximity of discontinuities. Thanks to its use of a localized but smooth artificial viscosity term, whose support is restricted to a vicinity of flow-discontinuity points, the algorithm enjoys spectral accuracy and low dissipation away from flow discontinuities, and, in such regions, it produces smooth numerical solutions—as evidenced by an essential absence of spurious oscillations in level set lines. The FC-SDNN viscosity assignment, which does not require use of problem-dependent algorithmic parameters, induces a significantly lower overall dissipation than other methods, including the Fourier-spectral versions of the previous entropy viscosity method. The character of the proposed algorithm is illustrated with a variety of numerical results for the linear advection, Burgers and Euler equations in one and two-dimensional non-periodic spatial domains.

Oscillatory wave patterns and spiral breakup in the Brusselator model using numerical bifurcation analysis

This article addresses the core breakup analysis of the spiral dynamics of an oscillatory system of chemical reactions, the so-called Brusselator model. We first obtain stable oscillatory periodic solutions corresponding to the stable limit cycle near a Hopf bifurcation point when the diffusion terms are neglected. Then, we investigate the occurrence of periodic traveling wave solutions of the model and perform the stability analysis of these solutions on the parameter plane. A stability boundary is also identified on the parameter plane. In the two-dimensional spatial domain, we illustrate rotating spiral waves and their instability that leads to a spiral breakup from the center of rotation.

Subordinated Gaussian random fields in elliptic partial differential equations

To model subsurface flow in uncertain heterogeneous or fractured media an elliptic equation with a discontinuous stochastic diffusion coefficient—also called random field—may be used. In case of a one-dimensional parameter space, Lévy processes allow for jumps and display great flexibility in the distributions used. However, in various situations (e.g. microstructure modeling), a one-dimensional parameter space is not sufficient. Classical extensions of Lévy processes on two parameter dimensions suffer from the fact that they do not allow for spatial discontinuities [see for example Barth and Stein (Stoch Part Differ Equ Anal Comput 6(2):286–334, 2018)]. In this paper a new subordination approach is employed [see also Barth and Merkle (Subordinated gaussian random fields. ArXiv e-prints, arXiv:2012.06353 [math.PR], 2020)] to generate Lévy-type discontinuous random fields on a two-dimensional spatial parameter domain. Existence and uniqueness of a (pathwise) solution to a general elliptic partial differential equation is proved and an approximation theory for the diffusion coefficient and the corresponding solution provided. Further, numerical examples using a Monte Carlo approach on a Finite Element discretization validate our theoretical results.

Dynamics of delayed neural field models in two-dimensional spatial domains

Delayed neural field models can be viewed as a dynamical system in an appropriate functional analytic setting. On two dimensional rectangular space domains, and for a special class of connectivity and delay functions, we describe the spectral properties of the linearized equation. We transform the characteristic integral equation for the delay differential equation (DDE) into a linear partial differential equation (PDE) with boundary conditions. We demonstrate that finding eigenvalues and eigenvectors of the DDE is equivalent with obtaining nontrivial solutions of this boundary value problem (BVP). When the connectivity kernel consists of a single exponential, we construct a basis of the solutions of this BVP that forms a complete set in L2. This gives a complete characterization of the spectrum and is used to construct a solution to the resolvent problem. As an application we give an example of a Hopf bifurcation and compute the first Lyapunov coefficient.

Bayesian neural network priors for edge-preserving inversion

<p style='text-indent:20px;'>We consider Bayesian inverse problems wherein the unknown state is assumed to be a function with discontinuous structure a priori. A class of prior distributions based on the output of neural networks with heavy-tailed weights is introduced, motivated by existing results concerning the infinite-width limit of such networks. We show theoretically that samples from such priors have desirable discontinuous-like properties even when the network width is finite, making them appropriate for edge-preserving inversion. Numerically we consider deconvolution problems defined on one- and two-dimensional spatial domains to illustrate the effectiveness of these priors; MAP estimation, dimension-robust MCMC sampling and ensemble-based approximations are utilized to probe the posterior distribution. The accuracy of point estimates is shown to exceed those obtained from non-heavy tailed priors, and uncertainty estimates are shown to provide more useful qualitative information.</p>

Comparative analysis of 2D-PCA based dimensionality reduction and feature extraction

The complex spectral and spatial characteristics of hyperspectral remote sensing images (HSI) lead to higher time-consuming in classification task. To address this question, we introduced the 2D-PCA dimensionality reduction method of linear mapping in the two-dimensional spatial domain on the basis of linear dimensionality reduction in the spectral domain, thereby compressing the complex spatial structure information of HSI into a limited low-dimensional space, and realizing space-spectrum dimensionality reduction and information fusion. The experimental results on three classic data sets of Salinas, Tea Farm, and Indian Pines show that 2D-PCA has a strong ability to condense and compress spatial structure characteristics. Compared with popular deep learning frameworks such as CNN and Mixer-MLP, conventional machine learning models based on 2D-PCA have significant advantages in terms of computing time under the premise of controllable accuracy loss, which makes 2D-PCA a promising method for dimensionality reduction and feature expression in hyperspectral pixel-wise classification.

A space-time discretization of a nonlinear peridynamic model on a 2D lamina

Peridynamics is a nonlocal theory for dynamic fracture analysis consisting in a second order in time partial integro-differential equation. In this paper, we consider a nonlinear model of peridynamics in a two-dimensional spatial domain. We implement a spectral method for the space discretization based on the Fourier expansion of the solution while we consider the Newmark-β method for the time marching. This computational approach takes advantages from the convolutional form of the peridynamic operator and from the use of the discrete Fourier transform. We show a convergence result for the fully discrete approximation and study the stability of the method applied to the linear peridynamic model. Finally, we perform several numerical tests and comparisons to validate our results and provide simulations implementing a volume penalization technique to avoid the limitation of periodic boundary conditions due to the spectral approach.

Bifurcation study and pattern formation analysis of a tritrophic food chain model with group defense and Ivlev-like nonmonotonic functional response

In this work, a tritrophic food chain model has been proposed by incorporating Ivlev-like nonmonotonic functional response, where prey is equipped with defense ability. We have performed a detailed dynamical study and pattern formation analysis to obtain complex dynamics of the proposed system. Stability and bifurcation analysis have been performed in the model system. Persistence and permanence are discussed. Bifurcations of codimension-1, in particular, saddle-node, transcritical and Hopf bifurcation are observed. The model system also exhibits bifurcations of codimension-2 such as cusp, Bogdanov-Takens and generalized Hopf bifurcation. Interestingly, it is observed that the middle and top predator population become extinct due to defense ability of prey. Chaotic dynamics is observed via a period-doubling route to chaos with the change in the value of parameter β. The quantification of chaotic dynamics is done, using Lyapunov spectrum and sensitivity analysis. Diffusion induced chaos is studied in the spatiotemporal model system. Hopf bifurcation is seen in the case of a spatially extended system. Further, conditions for Turing instability have been obtained. Pattern formation study is done. In the two-dimensional spatial domain, various non-Turing patterns such as hot-spot, cold-spot, labyrinth patterns are obtained. Ripple and stripe Turing patterns are obtained in case of one-dimensional spatial domain. Also, labyrinth and patchy Turing patterns are obtained in the two-dimensional spatial domain. The spatial distribution of the species shows Turing patterns at the low cost of β, while the increased cost of β changes Turing patterns to non-Turing patterns. Throughout the study, we observe that the parameter β plays an important role in group defense mechanism and is the most sensitive parameter leading to vital change in system dynamics. A wide range of Turing and non-Turing patterns obtained in this work has not been reported so far in literature in any model with group defense.

A Novel Coupled Reaction-Diffusion System for Explainable Gene Expression Profiling.

Machine learning (ML)-based algorithms are playing an important role in cancer diagnosis and are increasingly being used to aid clinical decision-making. However, these commonly operate as ‘black boxes’ and it is unclear how decisions are derived. Recently, techniques have been applied to help us understand how specific ML models work and explain the rational for outputs. This study aims to determine why a given type of cancer has a certain phenotypic characteristic. Cancer results in cellular dysregulation and a thorough consideration of cancer regulators is required. This would increase our understanding of the nature of the disease and help discover more effective diagnostic, prognostic, and treatment methods for a variety of cancer types and stages. Our study proposes a novel explainable analysis of potential biomarkers denoting tumorigenesis in non-small cell lung cancer. A number of these biomarkers are known to appear following various treatment pathways. An enhanced analysis is enabled through a novel mathematical formulation for the regulators of mRNA, the regulators of ncRNA, and the coupled mRNA–ncRNA regulators. Temporal gene expression profiles are approximated in a two-dimensional spatial domain for the transition states before converging to the stationary state, using a system comprised of coupled-reaction partial differential equations. Simulation experiments demonstrate that the proposed mathematical gene-expression profile represents a best fit for the population abundance of these oncogenes. In future, our proposed solution can lead to the development of alternative interpretable approaches, through the application of ML models to discover unknown dynamics in gene regulatory systems.

Energy-based Control and Observer Design for higher-order infinite-dimensional Port-Hamiltonian Systems

Abstract In this paper, we present a control-design strategy based on the energy-Casimir method for infinite-dimensional, boundary-actuated port-Hamiltonian systems with two-dimensional spatial domain and second-order Hamiltonian. The resulting control law depends on distributed system states that cannot be measured, and therefore, we additionally design an infinite-dimensional observer by exploiting the port-Hamiltonian system representation. A Kirchhoff-Love plate serves as an example in order to demonstrate the proposed approaches.

A computational study of density-dependent individual movement and the formation of population clusters in two-dimensional spatial domains

The patterns of collective behaviour in a population emerging from individual animal movement have long been of interest to ecologists, as has the emergence of heterogeneous patterns among a population. In this paper we will consider these phenomena by using an individual-based modelling approach to simulate a population whose individuals undergo density-dependent movement in 2D spatial domains. We first show that the introduction of density-dependent movement in the form of two parameters, a perception radius and a probability of directed movement, leads to the formation of clusters. We then show that the properties of the clusters and their stability over time are different between populations of Brownian and non-Brownian walkers and are also dependent on the choice of parameters. Finally, we consider the effect of the probability of directed movement on the temporal stability of clusters and show that while clusters formed by Brownian and non-Brownian walkers may have similar properties with certain parameter sets, the spatio-temporal dynamics remain different.