Unbounded Domains Research Articles

Time-dependent electromagnetic scattering from dispersive materials

Abstract This paper studies time-dependent electromagnetic scattering from obstacles that are described by dispersive material laws. We consider the numerical treatment of a scattering problem in which a dispersive material law, for a causal and passive homogeneous material, determines the wave–material interaction in the scatterer. The resulting problem is nonlocal in time inside the scatterer and is posed on an unbounded domain. Well-posedness of the scattering problem is shown using a formulation that is fully given on the surface of the scatterer via a time-dependent boundary integral equation. Discretizing this equation by convolution quadrature in time and boundary elements in space yields a provably stable and convergent method that is fully parallel in time and space. Under regularity assumptions on the exact solution we derive error bounds with explicit convergence rates in time and space. Numerical experiments illustrate the theoretical results and show the effectiveness of the method.

A Novel Approach to the Fractional Laplacian via Generalized Spherical Means

Although at least ten equivalent definitions of the fractional Laplacian exist in unbounded domains, we introduce an additional equivalent definition based on the generalized spherical mean-value operator—a Fourier multiplier operator involving the normalized Bessel function. Specifically, we demonstrate that this new definition allows us to reduce any n-dimensional fractional Laplacian to a one-dimensional operator, which simplifies computation and enhances efficiency. We propose two methods for computing the generalized spherical means of a given function: one by solving standard wave equations and the other by solving Darboux’s equations.

Dynamics of Double Time-Delayed Newton–Boussinesq Equations on Unbounded Domains

Dynamics of Double Time-Delayed Newton–Boussinesq Equations on Unbounded Domains

Collective phase transitions in confined fish schools

The collective patterns that emerge in schooling fish are often analyzed using models of self-propelled particles in unbounded domains. However, while schooling fish in both field and laboratory settings interact with domain boundaries, these effects are typically ignored. Here, we propose a model that incorporates geometric confinement, by accounting for both flow and wall interactions, into existing data-driven behavioral rules. We show that new collective phases emerge where the school of fish "follows the tank wall" or "double mills." Importantly, confinement induces repeated switching between two collective states, schooling and milling. We describe the group dynamics probabilistically, uncovering bistable collective states along with unintuitive bifurcations driving phase transitions. Our findings support the hypothesis that collective transitions in fish schools could occur spontaneously, with no adjustment at the individual level, and opens venues to control and engineer emergent collective patterns in biological and synthetic systems that operate far from equilibrium.

English

english

A closed-form solution for thermally induced affine deformation in unbounded domains with a temporally accelerated anomalous thermal conductivity

Abstract One of the successful mathematical tools in the 21st century is the distributed-order fractional derivative, particularly, the bi-fractional diffusion equation of natural form (Chechkin, Gorenflo, and Sokolov, Phys. Rev. E 2002) and the bi-fractional diffusion equation of the modified form (Sokolov, Chechkin, and Klafter, Acta Phys. Pol. B 2004). The present study adopts the bi-fractional diffusion heat equation of the modified form that uses two Riemann-Liouville time-fractional derivatives with different fractional orders and the Riesz space-fractional derivative, and bears the property of variable thermal conductivity with temporal acceleration; i.e., the material thermal conduction transits from a low value at the small values of time ($t\to 0$) to a larger value at the long-time domain ($t\to \infty$). The corresponding fractional thermoelasticity theory, that uses the bi-fractional diffusion heat equation of the modified form, is exclusively considered in this work. For a quasi-static unbounded Hookean domain, exact solutions for the temperature, hydrostatic stress, and the displacement fields are derived in both short-time and long-time domains, in terms of the Fox \textit{H}-function. The exact solutions for the linearized problem are derived using the inversion of problem variables from the Laplace-Fourier space. Adding a zero initial condition on the normal stress of the unbounded space transforms the conventional Cauchy problem to a mixed initial-boundary value problem in order to derive a generalized form of the displacement. Additionally, thermal energy is not conserved in the presence of space fractality. The effect of such an accelerating transition in the thermal conduction on the displacement component is focused on, and it is found that the displacement inherits a similar transition behavior.

Adaptive hyperbolic-cross-space mapped Jacobi method on unbounded domains with applications to solving multidimensional spatiotemporal integrodifferential equations

In this paper, we develop a new adaptive hyperbolic-cross-space mapped Jacobi (AHMJ) method for solving multidimensional spatiotemporal integrodifferential equations in unbounded domains. By devising adaptive techniques for sparse mapped Jacobi spectral expansions defined in a hyperbolic cross space, our proposed AHMJ method can efficiently solve various spatiotemporal integrodifferential equations such as the anomalous diffusion model with reduced numbers of basis functions. Our analysis of the AHMJ method gives a uniform upper error bound for solving a class of spatiotemporal integrodifferential equations, leading to effective error control.

A novel numerical method for solving the Kirchhoff-Helmholtz integral equation based on the sound field representation using the reproducing kernel in the frequency domain

The paper introduces a unique numerical solution method for the Kirchhoff-Helmholtz integral equation. This method is based on the representation of the sound field using the reproducing kernel in the solution space of the homogeneous Helmholtz equation. When compared to the conventional boundary element method, the proposed method holds numerous advantages. This paper presents the analysis theory of the proposed method and demonstrates its validity by calculating bounded and unbounded sound fields. The accuracy of the calculation is assessed by computing the sound field in a rectangular room under various conditions. Furthermore, although some theoretical issues remain, the proposed method effectively avoids the non-uniqueness of solutions in unbounded domain problems.

Bayesian Inference for Estimating Heat Sources through Temperature Assimilation

Abstract This paper utilizes a Bayesian inference framework to address the two-dimensional steady-state heat conduction problem, focusing on the estimation of unknown distributed heat sources in a thermally-conducting medium with uniform conductivity. The goal is to infer the locations, strength, and shape of heaters by assimilating temperature data in Euclidean space, employing a Fourier series to represent each heater's shape. The Markov Chain Monte Carlo method, incorporating the random-walk Metropolis-Hasting algorithm and parallel tempering, is utilized for posterior distribution exploration in both unbounded and wall-bounded domains. It is found that multiple solutions arise in cases where the number of temperature sensors is less than the number of unknown states. Moreover, smaller heaters introduce greater uncertainty in estimated strength. To address the challenge of estimating the heater's strength and shape simultaneously due to their strong correlation, our method incorporates sharp priors on one to ensure accurate and feasible solutions of the other. The diffusive nature of heat conduction smooths out any deformations in the temperature contours, especially in the presence of multiple heaters positioned near each other, impacting convergence. In wall-bounded domains with Neumann boundary conditions, the inference of heater parameters tends to be more accurate than in unbounded domains.

Sobolev compact embeddings in unbounded domains and its applications to elliptic equations

We give a necessary and sufficient condition for the compact embedding of the Sobolev space W0m,p(Ω) for unbounded domains Ω. Applying this condition, we can decide whether the compact embedding holds or not. We give several examples of unbounded domains Ω satisfying the compact embedding. Using our condition, we study a semilinear elliptic equation in unbounded domains and prove the existence of a positive solution and infinitely many solutions.

Verification ofHarmonic Functions on Unbounded Domains in Ahlfors Regular Spaces using Sphericalization

Verification ofHarmonic Functions on Unbounded Domains in Ahlfors Regular Spaces using Sphericalization

On thermally driven fluid flows arising in astrophysics

We investigate a potential model for an unbounded celestial bodies of finite mass composed of a solid core and a gaseous atmosphere. The system is governed by the Navier–Stokes–Fourier–Poisson equations, incorporating no-slip boundary conditions for velocity and a specified temperature distribution on the surface of the solid core. Additionally, a positive far-field condition is imposed on the temperature. This manuscript extends the mathematical theory of open fluid systems to unbounded exterior domains addressing these physically motivated yet highly challenging combination of boundary conditions. Notably, we establish the existence of global-in-time weak solutions and demonstrate the weak-strong uniqueness principle

Asymptotic analysis of the 2D and 3D convective Brinkman–Forchheimer equations in unbounded domains

In this work, we carry out the asymptotic analysis of two- and three-dimensional convective Brinkman–Forchheimer (CBF) equations, which characterize the motion of incompressible fluid flows in a saturated porous medium. We establish the existence of a global attractor in both bounded (using compact embedding) and unbounded Poincaré domains (using asymptotic compactness property). In Poincaré domains, the estimates for Hausdorff as well as fractal dimensions of the global attractors are also obtained. We then show an upper semicontinuity of global attractors with respect to domains for CBF equations. We consider an expanding sequence of simply connected, bounded and smooth subdomains [Formula: see text] of the Poincaré domain [Formula: see text] such that [Formula: see text] as [Formula: see text]. If [Formula: see text] and [Formula: see text] are the global attractors of CBF equations corresponding to [Formula: see text] and [Formula: see text], respectively, then we show that for large enough [Formula: see text], the global attractor [Formula: see text] enters into any neighborhood [Formula: see text] of [Formula: see text]. The presence of the Darcy term in CBF equations helps us to obtain the above mentioned results in general unbounded domains also. Finally, we discuss the quasi-stability property of the semigroup associated with CBF equations in bounded domains and establish the existence of finite fractal dimensional global as well as exponential attractors.

Efficient spectral element method for the Euler equations on unbounded domains

Mitigating the impact of waves leaving a numerical domain has been a persistent challenge in numerical modeling. Reducing wave reflection at the domain boundary is crucial for accurate simulations. Absorbing layers, while common, often incur significant computational costs. This paper introduces an efficient application of a Legendre-Laguerre basis for absorbing layers for two-dimensional non-linear compressible Euler equations. The method couples a spectral-element bounded domain with a semi-infinite region, employing a tensor product of Lagrange and scaled Laguerre basis functions. Semi-infinite elements are used in the absorbing layer with Rayleigh damping. In comparison to existing methods with similar absorbing layer extensions, this approach, a pioneering application to the Euler equations of compressible and stratified flows, demonstrates substantial computational savings. The study marks the first application of semi-infinite elements to mitigate wave reflection in the solution of the Euler equations, particularly in nonhydrostatic atmospheric modeling. A comprehensive set of tests demonstrates the method's versatility for general systems of conservation laws, with a focus on its effectiveness in damping vertically propagating mountain gravity waves, a benchmark for atmospheric models. Across all tests, the model presented in this paper consistently exhibits notable performance improvements compared to a traditional Rayleigh damping approach.

Exponential attractors for a nonlocal delayed reaction-diffusion equation on an unbounded domain

The main objective of this paper is to investigate exponential attractors for a nonlocal delayed reaction-diffusion equation on an unbounded domain. We first obtain the existence of a globally attractive absorbing set for the dynamical system generated by the equation under the assumption that the nonlinear term is bounded. Then, we construct exponential attractors of the equation directly in its natural phase space, i.e., a Banach space with explicit fractal dimension by combining squeezing properties of the system as well as a covering lemma of finite dimensional subspaces of a Banach space. Our result generalizes the methods established in Hilbert spaces and weighted spaces, and the fractal dimension of the obtained exponential attractor does not depend on the entropy number but only depends on some inner characteristic of the studied equation.

Existence and asymptotical behavior of solutions of a class of parabolic systems with homogeneous nonlinearity

In this paper we investigate the global existence and asymptotical stability of solutions to a class of parabolic systems with homogeneous nonlinearity for both bounded and unbounded domains. First we prove both global existence and finite time blow-up of solutions of the system for different initial conditions by using the potential well method, and the asymptotic behavior of the solutions are also considered. On the other hand, we also obtain global existence and finite time blow-up of solutions for both Sobolev subcritical and critical cases. We use a method of comparing least energy levels with that of semitrivial solutions to overcome the difficulties here.

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

Solvability of a Hadamard fractional boundary value problem with multi‐term integral and Hadamard fractional derivative boundary conditions

In the present paper, we construct the existence of nontrivial solutions to a new kind of Hadamard fractional boundary value problem on an unbounded domain. With the contribution of some fixed point theorems in cone and the corresponding Green function, we ensure sufficient conditions for the Hadamard fractional boundary value problem. Also, the paper concludes with two examples to demonstrate our results.

Invariant Measures for Stochastic Reaction–Diffusion Problems on Unbounded Thin Domains Driven by Nonlinear Noise

Invariant Measures for Stochastic Reaction–Diffusion Problems on Unbounded Thin Domains Driven by Nonlinear Noise

Boundary Domain Integral Equations for Variable Coefficient Mixed BVP in 2D Unbounded Domain

In this paper, the direct segregated Boundary Domain Integral Equations (BDIEs) for the Mixed Boundary Value Problems (MBVPs) for a scalar second order elliptic Partial Differential Equation (PDE) with variable coefficient in unbounded (exterior) 2D domain is considered. Otar Chkadua, Sergey Mikhailov and David Natroshvili formulated both the interior and exterior 3D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order elliptic PDE with a variable coefficients. On the other hand Sergey Mikhailov and Tamirat Temesgen formulated only the interior 2D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order divergent elliptic PDE with a variable coefficients. However, in this paper we formulated the exterior 2D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order divergent elliptic PDE with a variable coefficients. The aim of this work is to reduce the MBVPs to some direct segregated BDIEs with the use of an appropriate parametrix (Levi function). We examine the characteristics of corresponding parametrix-based integral volume and layer potentials in some weighted Sobolev spaces, as well as the unique solvability of BDIEs and their equivalence to the original MBVPs. This analysis is based on the corresponding properties of the MBVPs in weighted Sobolev spaces that are proved as well.