Time-dependent Robin Boundary Condition Research Articles

Discrete-coordinate crypto-Hermitian quantum system controlled by time-dependent Robin boundary conditions

A family of exactly solvable quantum square wells with discrete coordinates and with certain non-stationary Hermiticity-violating Robin boundary conditions is proposed and studied. Manifest non-Hermiticity of the model in conventional Hilbert space Hfriendly is required to coexist with the unitarity of system in another, ad hoc Hilbert space Hphysical . Thus, quantum mechanics in its non-Hermitian interaction picture (NIP) representation is to be used. We must construct the time-dependent states (say, ψ(t)) as well as the time-dependent observables (say, Λ(t)). Their evolution in time is generated by the operators denoted, here, by the respective symbols G(t) (a Schrödinger-equation generator) and Σ(t) (a Heisenberg-equation generator, a.k.a. quantum Coriolis force). The unitarity of evolution in Hphysical is then guaranteed by the reality of spectrum of the energy observable alias Hamiltonian H(t) = G(t) + Σ(t). The applicability of these ideas is illustrated via an N by N matrix model. At N = 2, closed formulae are presented not only for the measurable instantaneous energy spectrum but also for all of the eligible time-dependent physical inner-product metrics Θ(N=2)(t), for the related Dyson maps Ω(N=2)(t), for the Coriolis force Σ(N=2)(t) as well as, in the very ultimate step of the construction, for the truly nontrivial Schrödinger-equation generator G (N=2)(t).

Dynamical Casimir effect enhanced by decreasing the mirror reflectivity

In the present paper, we show that a partially reflecting static mirror with time-dependent properties can produce, via dynamical Casimir effect in the context of a massless scalar field in $1+1$ dimensions, a larger number of particles than a perfectly reflecting one. As particular limits, our results recover those found in the literature for a perfect static mirror imposing a generalized or an usual time-dependent Robin boundary condition.

Analysis of a Delayed Free Boundary Problem with Application to a Model for Tumor Growth of Angiogenesis

In this paper, we consider a time-delayed free boundary problem with time dependent Robin boundary conditions. The special case where n=3 is a mathematical model for the growth of a solid nonnecrotic tumor with angiogenesis. In the problem, both the angiogenesis and the time delay are taken into consideration. Tumor cell division takes a certain length of time, thus we assume that the proliferation process leg behind as compared to the process of apoptosis. The angiogenesis is reflected as the time dependent Robin boundary condition in the model. Global existence and uniqueness of the nonnegative solution of the problem is proved. When c>0 is sufficiently small, the stability of the steady state solution is studied, where c is the ratio of the time scale of diffusion to the tumor doubling time scale. Under some conditions, the results show that the magnitude of the delay does not affect the final dynamic behavior of the solutions. An application of our results to a mathematical model for tumor growth of angiogenesis is given and some numerical simulations are also given.

A note on the norm-continuity for evolution families arising from non-autonomous forms

We consider evolution equations of the form $$\begin{aligned} \dot{u}(t)+{\mathcal {A}}(t)u(t)=0,\ \ t\in [0,T],\ \ u(0)=u_0, \end{aligned}$$where $${\mathcal {A}}(t),\ t\in [0,T],$$ are associated with a non-autonomous sesquilinear form $${\mathfrak {a}}(t,\cdot ,\cdot )$$ on a Hilbert space H with constant domain $$V\subset H.$$ In this note we continue the study of fundamental operator theoretical properties of the solutions. We give a sufficient condition for norm-continuity of evolution families on each spaces V, H and on the dual space $$V'$$ of V. The abstract results are applied to a class of equations governed by time dependent Robin boundary conditions on exterior domains and by Schrodinger operator with time dependent potentials.

Accurate solution method for the Maxey–Riley equation, and the effects of Basset history

The Maxey–Riley equation has been extensively used by the fluid dynamics community to study the dynamics of small inertial particles in fluid flow. However, most often, the Basset history force in this equation is neglected. Analytical solutions have almost never been attempted because of the difficulty in handling an integro-differential equation of this type. Including the Basset force in numerical solutions of particulate flows involves storage requirements which rapidly increase in time. Thus the significance of the Basset history force in the dynamics has not been understood. In this paper, we show that the Maxey–Riley equation in its entirety can be exactly mapped as a forced, time-dependent Robin boundary condition of the one-dimensional diffusion equation, and solved using the unified transform method. We obtain the exact solution for a general homogeneous time-dependent flow field, and apply it to a range of physically relevant situations. In a particle coming to a halt in a quiescent environment, the Basset history force speeds up the decay as a stretched exponential at short time while slowing it down to a power-law relaxation, ${\sim}t^{-3/2}$, at long time. A particle settling under gravity is shown to relax even more slowly to its terminal velocity (${\sim}t^{-1/2}$), whereas this relaxation would be expected to take place exponentially fast if the history term were to be neglected. An important mechanism for the growth of raindrops is by the gravitational settling of larger drops through an environment of smaller droplets, and repeatedly colliding and coalescing with them. Using our solution we estimate that the rate of growth rate of a raindrop can be grossly overestimated when history effects are not accounted for. We solve exactly for particle motion in a plane Couette flow and show that the location (and final velocity) to which a particle relaxes is different from that due to Stokes drag alone. For a general flow, our approach makes possible a numerical scheme for arbitrary but smooth flows without increasing memory demands and with spectral accuracy. We use our numerical scheme to solve an example spatially varying flow of inertial particles in the vicinity of a point vortex. We show that the critical radius for caustics formation shrinks slightly due to history effects. Our scheme opens up a method for future studies to include the Basset history term in their calculations to spectral accuracy, without astronomical storage costs. Moreover, our results indicate that the Basset history can affect dynamics significantly.

Analytical Solutions of One-Dimensional Contaminant Transport in Soils with Source Production-Decay

An analytical solution in closed form of the advection-dispersion equation in one-dimensional contaminated soils is proposed in this paper. This is valid for non-conservative solutes with first order reaction, linear equilibrium sorption, and a time-dependent Robin boundary condition. The Robin boundary condition is expressed as a combined production-decay function representing a realistic description of the source release phenomena in time. The proposed model is particularly useful to describe sources as the contaminant release due to the failure in underground tanks or pipelines, Non Aqueous Phase Liquid pools, or radioactive decay series. The developed analytical model tends towards the known analytical solutions for particular values of the rate constants.

Regularity properties for evolution families governed by non-autonomous forms

Evolution families are the non-autonomous counterpart of operator semigroups in the well-posedness theory of non-autonomous evolution equations. This note is devoted to fundamental operator theoretical properties, beginning with norm continuity—which we regard as a fingerprint of analyticity. While no known analogue of analytic semigroups is known in the non-autonomous case, we give a sufficient condition for norm-continuity of evolution families. Furthermore, we develop a theory of compact and trace class evolution families. The abstract results are applied to the Laplace operator with time dependent Robin boundary conditions.

Meshless and analytical solutions to the time-dependent advection-diffusion-reaction equation with variable coefficients and boundary conditions

• The unsteady ADR equation is formulated and solved by the meshless generalised RKPM. • A general Robin BC is formulated and precisely enforced in a novel approach. • An analytical solution is derived for 1D problem with temporally changing Robin BC. • A normalisation procedure is introduced to interpret the results to relevant applications. A variety of physical problems in science may be expressed using the advection-diffusion-reaction (ADR) equation that covers heat transfer and transport of mass and chemicals into a porous or a nonporous media. In this paper, the meshless generalised reproducing kernel particle method (RKPM) is utilised to numerically solve the time-dependent ADR problem in a general n -dimensional space with variable coefficients and boundary conditions. A time-dependent Robin boundary condition is formulated and precisely enforced in a novel approach. The accuracy and robustness of the meshless solution is verified against finite element simulations and a general one-dimensional analytical solution obtained in this study.

TIME-DEPENDENT ROBIN BOUNDARY CONDITIONS IN THE DYNAMICAL CASIMIR EFFECT

Motivated by experiments in which moving boundaries are simulated by time-dependent properties of static systems, we discuss the model of a massless scalar field submitted to a time-dependent Robin boundary condition (BC) at a static mirror in 1 + 1 dimensions. Using a perturbative approach, we compute the spectral distribution of the created particles and the total particle creation rate, considering a thermal state as the initial field state.

Simple model for the dynamical Casimir effect for a static mirror with time-dependent properties

We consider a real massless scalar field in 1+1 dimensions satisfying time-dependent Robin boundary condition at a static wall. This condition can simulate moving reflecting mirrors whose motions are determined by the time-dependence of the Robin parameter. We show that particles can be created from vacuum, characterizing in this way a dynamical Casimir effect.