Boundary Problem Research Articles

Studying a cylindrical moving boundary problem in ductal carcinoma in situ

In this paper, a mathematical model of ductal carcinoma in situ (DCIS) is considered as a moving boundary problem in cylindrical coordinates. Using the front-fixing technique, first, we transform the problem into a nonlinear parabolic problem on a fixed domain, and then some essential properties of the solution, such as the uniqueness of the solution, are shown. A numerical approach based on the finite difference method is established for solving the fixed domain problem. The solvability of the proposed finite difference method is proved, and the stability and convergence of the numerical solution are shown. For a test case problem, analytical and numerical results are compared to demonstrate the ability and effectiveness of the numerical method.

Sharp estimates of solutions of free boundary problems with nonlocal diffusion

Sharp estimates of solutions of free boundary problems with nonlocal diffusion

Closed-form expressions for magnetic polarizabilities of circular apertures excited by nonuniform magnetic fields

Abstract The problem of electromagnetic penetration through an aperture in a perfectly electrically conducting plane is a classical problem in electromagnetic theory and applications. This work aims to investigate the behavior of an electrically small aperture when excited by nonuniform magnetic fields. Based on the Taylor expansion of the exact solution in integral form of the corresponding mixed boundary problem, we derive the equivalent magnetic dipole moment of a circular aperture. Closed-form expressions of dipole moments are derived. Also, for the special case of axisymmetric excitation where the dipole moment vanishes, we develop a quadrupole model to describe the aperture behavior. The effectiveness of the obtained expressions is verified by comparisons with accurate analytical solutions in special cases and the exact solutions in integral form.

Triple junction benchmark for multiphase-field models combining capillary and bulk driving forces

Abstract A benchmark problem is formulated which is well suited for the validation of mesoscopic phase-field models for grain-boundary migration in polycrystals. First, an analytical steady-state solution of the sharp moving boundary problem is derived for a symmetric lamellar structure, which is valid for arbitrary bulk driving forces and triple junction angles. Characteristic quantities are identified to reduce the parameter space which in turn allows a systematic comparison of simulations and analytical results. Various multiphase-field (MPF) formulations are compared which approximate the sharp interface problem in terms of a diffuse regularization. An interfacial thickness convergence study reveals that the model error is largely dependent on the ratio of bulk to interfacial stabilizing force as well as the underlying model formulation. An additional grid convergence study highlights the efficiency of a more advanced discretization scheme. The results can be used to guide the selection of appropriate models and to estimate the interface thickness and spatial resolution required to achieve a given accuracy target. The post-processing framework consists of a fully automated determination of well-defined metrics from the phase field simulation data, eliminating human bias and facilitating reproducibility. The corresponding code is made openly available to assist the materials science and engineering community in validating MPF, multi-order parameter and similar model developments. We believe that this work provides a reliable benchmark procedure to better understand the potentials and limitations of current multiphase-field models as well as alternative approaches.

Influence of density variations of ionosphere plasma on the conditions of electromagnetic whistler waves propagation in the ionosphere

Background. Plasma density variations caused by infrasonic wave can significantly affect conditions of propagation and reflection of whistler electromagnetic wave incident on the ionosphere from above. Aim. In that work, relationship between the coefficient of wave energy reflection from ionosphere, electromagnetic wave field near the ground surface and parameters of infrasonic wave are studied. Methods. The collocation method for solving the boundary problem for a plane-layered ionosphere and the perturbation theory method are used to find the electromagnetic wave field. Results. The greatest modulation of the reflection coefficient is associated with density perturbations at altitudes 80–110 km where whistler decay increases by an order of magnitude at the local altitude region (less than 15–20 km). In that case, the reflection coefficient variation can achieve 40 %. Conclusion. The results obtained are important for understanding the interconnection of magnetosphere radiation processes of different nature. The study of the modulation of coefficient of whistler reflection from the ionosphere is relevant for explaining the operation modes of a plasma magnetospheric maser.

Optimal approximations for the free boundary problems of the space-time fractional Black-Scholes equations using a combined physics-informed neural network

The combined physics-informed neural network is employed to deal with the free boundary problems of fractional Black-Scholes equations. The solution assumption and the loss function are determined, the transfer learning is borrowed, the combined neural network with data enhancement layer is designed, then the classical Black-Scholes model is numerically solved and the comparative analysis of numerical results under different neural networks is made. For further insight into the long-term memory of fluctuation, the free boundary problems of the space-time Black-Scholes equations under Caputo, Caputo-Fabrizio and Atangana-Baleanu-Caputo fractional derivatives are studied. The corresponding empirical analyses are presented and the optimal exercise boundaries of American put option are simulated. The market analysis shows that introducing fractional calculus tools and neural network algorithms into American put option pricing can yield more realistic prediction results. The work provides a viable method for subsequent researchers to study American option pricing using fractional calculus and neural networks combined with true market data and to deal with the free boundary problems in other research fields.

Well-posedness and stability for the two-phase periodic quasistationary Stokes flow

The two-phase horizontally periodic quasistationary Stokes flow in \mathbb{R}^{2} , describing the motion of two immiscible fluids with equal viscosities that are separated by a sharp interface, parameterized as the graph of a function f=f(t) , is considered in the general case when both gravity and surface tension effects are included. Using potential theory, the moving boundary problem is formulated as a fully nonlinear and nonlocal parabolic problem for the function f . Based on abstract parabolic theory, it is shown that the problem is well-posed in all subcritical spaces \mathrm{H}^{r}(\mathbb{S}) , with r\in(3/2,2) . Moreover, the stability properties of the flat equilibria are analyzed in dependence on the physical properties of the fluids.

“We Got Tired”

In this commentary, I raise the issue of how well we can distinguish between religious systems and those presided over by messianic “quasi-divine” leaders. This problematizes the degree to which all such systems can be described as ecologically functional when they can also be directed by the idiosyncratic whims of unstable charismatic individuals. Aside from such issues, I also raise the issue of how we can be sure that the “religious system” is reliably adaptive, if we may not have an accurate tally of such systems in the first place, particularly given the fact that religious studies scholars are incapable of providing a distinction between a “cult” that may contain but a few individuals and quickly “fail” due to its esoteric demands, and millennia-old traditions commanding millions of adherents and enforcing prosocial norms. I also raise two issues that have long bedevilled functionalist approaches but that do not make much of an appearance in Religion Evolving: Cultural, Cognitive, and Ecological Dynamics (Purzycki & Sosis 2022): the boundary problem and the Pangloss problem. Despite these reservations, I applaud the book both for its rigour and for providing a sorely needed template for analyzing religious systems within the cognitive and evolutionary study of religion (CESR). I close by tentatively suggesting that religion’s ubiquity lies not with its adaptiveness per se, but with its capacity to accelerate cultural evolution by causing societies to lock in values that may either be adaptive or maladaptive, making sacralization ubiquitous even when its outcomes can be destructive.

A duality and free boundary approach to adverse selection

Adverse selection is a version of the principal-agent problem that includes monopolist nonlinear pricing, where a monopolist with known costs seeks a profit-maximizing price menu facing a population of potential consumers whose preferences are known only in the aggregate. For multidimensional spaces of agents and products, Rochet and Choné (Econometrica, 1998) reformulated this problem as a concave maximization over the set of convex functions, by assuming agent preferences combine bilinearity in the product and agent parameters with a quasilinear sensitivity to prices. We characterize solutions to this problem by identifying a dual minimization problem. This duality allows us to reduce the solution of the square example of Rochet–Choné to a novel free boundary problem, giving the first analytical description of an overlooked market segment.

Approximate Solution of Fuzzy Sedimentary Ocean Basin Boundary Value Problem Using Variational Iteration Method

The problem that is considered in this paper is developed from the migration of the coastline over time in sedimentary ocean basins that may be considered as a moving boundary problem with a variable latent heat transfer equation as a governing equation if one of the boundaries is unknown, with boundary and initial conditions are given. This model discusses the shoreline movement within sedimentary basins. The main objective of the paper is to use fuzzy logic to formulate this problem when modelling sand particles migration in saltwater by utilizing fuzzy numbers whenever replacing some of the problem's parameters. In reality, the proposed model is more general than the nonfuzzy or crisp models. By employing the variational iteration method, the approximate numerical solution of the proposed problem has been found, and computer software written in Mathematica 11 have been used to obtain the results.

On the vacuum free boundary problem of the viscous Saint-Venant system for shallow water in two dimensions

On the vacuum free boundary problem of the viscous Saint-Venant system for shallow water in two dimensions

A Risk Assessment Model of Gas Pipeline Leakage Based on a Fuzzy Hybrid Analytic Hierarchy Process

Given the rising urban demand for gas, it has emerged as a primary energy source for urban activities and daily life. However, China’s urban gas pipeline network has witnessed a surge in accidents, leading to significant losses and disasters. Therefore, it is particularly necessary to study the disaster risk assessment model caused by urban gas pipeline leakage. There are some problems in the previous evaluation methods, such as less consideration of the influence relationships between disaster factors. To redress this issue, a novel fuzzy hybrid analytic hierarchy process evaluation methodology is proposed. First, a hybrid hierarchical risk assessment model is developed by combining the analytic hierarchy process and the network analytic hierarchy process. Membership matrices and impact matrices are utilized to calculate comprehensive factor weights. This approach enhances the understanding of relationships between risk factors within the hierarchical structure model. Subsequently, employing a fuzzy evaluation method, the risk level matrix is derived by using multiplication and bounded operators to ascertain the risk level state. This solves the problem of the fuzzy boundaries when measuring the index factors of the gas pipeline network. Finally, experimental analysis is carried out on the gas pipeline network in the central area of a city and validates the model’s accuracy in practical applications.

A weakly coupled system of p-Laplace type in a heat conduction problem

Abstract Given is a bounded domain Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} , and a vector-valued function defined on ∂ ⁡ Ω {\partial\Omega} (depicting temperature distributions from different sources), our objective is to study the mathematical model of a physical problem of enclosing ∂ ⁡ Ω {\partial\Omega} with a specific volume of insulating material to reduce heat loss in a stationary scenario. Mathematically, this task involves identifying a vector-valued function 𝐮 = ( u 1 , … , u m ) {\mathbf{u}=(u^{1},\dots,u^{m})} ( m ≥ 1 {m\geq 1} ) that represents the temperature within Ω and gives rise to a free boundary, somehow reminiscent of, but not equivalent to, the Bernoulli free boundary problem.

Well-posedness and stability of a stochastic neural field in the form of a partial differential equation

A system of partial differential equations representing stochastic neural fields was recently proposed with the aim of modelling the activity of noisy grid cells when a mammal travels through physical space. The system was rigorously derived from a stochastic particle system and its noise-driven pattern-forming bifurcations have been characterised. However, due to its nonlinear and non-local nature, standard well-posedness theory for smooth time-dependent solutions of parabolic equations does not apply. In this article, we transform the problem through a suitable change of variables into a nonlinear Stefan-like free boundary problem and use its representation formulae to construct local-in-time smooth solutions under mild hypotheses. Then, we prove that fast-decaying initial conditions and globally Lipschitz modulation functions imply that solutions are global-in-time. Last, we improve previous linear stability results by showing nonlinear asymptotic stability of stationary solutions under suitable assumptions.

On the two nonzero boundary problems of the AB system with multiple poles

The Riemann–Hilbert method is used to study AB systems with two nonzero boundary conditions. In order to reconstruct the potentials, spectral analysis of Lax pair and the corresponding Riemann–Hilbert problem are discussed. For these two nonzero boundary problems, we consider the cases where the sectional analytic functions have double and triple poles, respectively. The derived scattering data has multiple zeros or pairs of multiple zeros, from which the relationship between the eigenfunctions at the corresponding poles can be obtained. By solving algebraic equations, we can derive the formulas of N-order solutions in different cases. As an application, we obtain different types of solitons and breathers in the reflectionless case and the dynamic analysis of these solutions is revealed.

Stabilization in a chemotaxis system modelling T-cell dynamics with simultaneous production and consumption of signals

Abstract In a smoothly bounded domain $\Omega \subset \mathbb{R}^n$ , $n\ge 1$ , this manuscript considers the homogeneous Neumann boundary problem for the chemotaxis system \begin{eqnarray*} \left \{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v), \\[5pt] v_t = \Delta v + u - \alpha uv, \end{array} \right . \end{eqnarray*} with parameter $\alpha \gt 0$ and with coincident production and uptake of attractants, as recently emphasized by Dallaston et al. as relevant for the understanding of T-cell dynamics. It is shown that there exists $\delta _\star =\delta _\star (n)\gt 0$ such that for any given $\alpha \ge \frac{1}{\delta _\star }$ and for any suitably regular initial data satisfying $v(\cdot, 0)\le \delta _\star$ , this problem admits a unique classical solution that stabilizes to the constant equilibrium $(\frac{1}{|\Omega |}\int _\Omega u(\cdot, 0), \, \frac{1}{\alpha })$ in the large time limit.

Phase-field formulated meshless simulation of axisymmetric Rayleigh-Taylor instability problem

A formulation of the immiscible Newtonian two-liquid system with different densities and influenced by gravity is based on the Phase-Field Method (PFM) approach. The solution of the related governing coupled Navier-Stokes (NS) and Cahn-Hillard (CH) equations is structured by the meshless Diffuse Approximate Method (DAM) and Pressure Implicit with Splitting of Operators (PISO). The variable density is involved in all the terms. The related moving boundary problem is handled through single-domain, irregular, fixed node arrangement in Cartesian and axisymmetric coordinates. The meshless DAM uses weighted least squares approximation on overlapping subdomains, polynomial shape functions of second-order and Gaussian weights. This solution procedure has improved stability compared to Chorin's pressure-velocity coupling, previously used in meshless solutions of related problems. The Rayleigh-Taylor instability problem simulations are performed for an Atwood number of 0.76. The DAM parameters (shape parameter of the Gaussian weight function and number of nodes in a local subdomain) are the same as in the authors’ previous studies on single-phase flows. The simulations did not need any upwinding in the range of the simulations. The results compare well with the mesh-based finite volume method studies performed with the open-source code Gerris, Open-source Field Operation and Manipulation (OpenFOAM®) code and previously existing results.

A Free Boundary Problem in an Unbounded Domain and Subsonic Jet Flows from Divergent Nozzles

A Free Boundary Problem in an Unbounded Domain and Subsonic Jet Flows from Divergent Nozzles

Erosion of surfaces by trapped vortices

Two two-dimensional free boundary problems describing the erosion of solid surfaces by the flow of inviscid fluid in the presence of trapped vortices are considered. The first problem tackles an initially flat, infinite fluid-solid interface with uniform flow at infinity and a vortex in equilibrium above the surface. The second involves flow around a finite body with a trailing Föppl-type vortex pair. The conformal invariance of the complex potential permits both problems to be formulated as a Polubarinova–Galin (PG) type equation in which the time-dependent eroding surface in the physical z-plane is mapped to the fixed boundary of the ζ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta $$\\end{document}-disk. The Hamiltonian governing the equilibrium position of the vortex (or vortex pair in the second problem) is also found from the same map. In each problem, the PG equation giving the conformal map is found numerically and the time-dependent evolution of the interface and vortex location is determined. Different models governing the erosion of the interface are investigated in which the normal velocity of the boundary depends on some given function of the fluid flow velocity at the boundary. Typically, in the infinite surface case, erosion leads to the formation of a symmetric valley beneath the vortex which, in turn, moves downward toward the interface. A finite body undergoes erosion which is asymmetric in the flow direction leading to a flattening of the lee surface of the body so displaying some similarity to the experiments and associated viscous theory of Ristroph et al, Moore et al (Proc Natl Acad Sci 109(48):19606–19609, 2012, Phys Fluids 25(11):116602, 2013).

Convergence to the planar interface for a nonlocal free‐boundary evolution

AbstractWe capture optimal decay for the Mullins–Sekerka evolution, a nonlocal, parabolic free boundary problem from materials science. Our main result establishes convergence of BV solutions to the planar profile in the physically relevant case of ambient space dimension three. Far from assuming small or well‐prepared initial data, we allow for initial interfaces that do not have graph structure and are not connected, hence explicitly including the regime of Ostwald ripening. In terms only of initially finite (not small) excess mass and excess surface energy, we establish that the surface becomes a Lipschitz graph within a fixed timescale (quantitatively estimated) and remains trapped within this setting. To obtain the graph structure, we leverage regularity results from geometric measure theory. At the same time, we extend a duality method previously employed for one‐dimensional PDE problems to higher dimensional, nonlocal geometric evolutions. Optimal algebraic decay rates of excess energy, dissipation, and graph height are obtained.