An evaluation of boundary conditions for one-dimensional solute transport: 1. Mathematical development

AbstractInitial-boundary value problems for the one-dimensional linear advection–dispersion equation with decay (LAD) are studied by utilizing a unified method, known as the Fokas method. The method takes advantage of the spectral analysis of both parts of Lax pair and the global algebraic relation coupling all initial and boundary values. We present the explicit analytical solution of the LAD equation posed on the half line and a finite interval with general initial and boundary conditions. In addition, for the case of periodic boundary conditions, we show that the solution of the LAD equation is asymptotically t-periodic for large t if the Dirichlet boundary datum is periodic in t. Furthermore, it can be shown that if the Dirichlet boundary value is asymptotically periodic for large t, then so is the unknown Neumann boundary value, which is uniquely characterized in terms of the given asymptotically periodic Dirichlet boundary datum. The analytical predictions for large t are compared with numerical results showing the excellent agreement.

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Pair-correlation ansatz for the ground state of interacting bosons in an arbitrary one-dimensional potential

Scientific Reports ◽

10.1038/s41598-021-92556-7 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Przemysław Kościk ◽

Arkadiusz Kuroś ◽

Adam Pieprzycki ◽

Tomasz Sowiński

Keyword(s):

Boundary Conditions ◽

Ground State ◽

Pair Correlation ◽

Contact Interactions ◽

One Dimensional ◽

Particle Correlations ◽

Variational Scheme ◽

Full Control ◽

Arbitrary Shapes

AbstractWe derive and describe a very accurate variational scheme for the ground state of the system of a few ultra-cold bosons confined in one-dimensional traps of arbitrary shapes. It is based on assumption that all inter-particle correlations have two-body nature. By construction, the proposed ansatz is exact in the noninteracting limit, exactly encodes boundary conditions forced by contact interactions, and gives full control on accuracy in the limit of infinite repulsions. We show its efficiency in a whole range of intermediate interactions for different external potentials. Our results manifest that for generic non-parabolic potentials mutual correlations forced by interactions cannot be captured by distance-dependent functions.

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On a stochastic Burgers equation with Dirichlet boundary conditions

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171203211121 ◽

2003 ◽

Vol 2003 (43) ◽

pp. 2735-2746 ◽

Cited By ~ 1

Author(s):

Ekaterina T. Kolkovska

Keyword(s):

Weak Solution ◽

Boundary Conditions ◽

Dirichlet Boundary ◽

Burgers Equation ◽

Martingale Problem ◽

Weak Limit ◽

Dirichlet Boundary Conditions ◽

One Dimensional ◽

Stochastic Burgers Equation ◽

The One

We consider the one-dimensional Burgers equation perturbed by a white noise term with Dirichlet boundary conditions and a non-Lipschitz coefficient. We obtain existence of a weak solution proving tightness for a sequence of polygonal approximations for the equation and solving a martingale problem for the weak limit.

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Order parameter profiles in a system with Neumann – Neumann boundary conditions

MATEC Web of Conferences ◽

10.1051/matecconf/201814501009 ◽

2018 ◽

Vol 145 ◽

pp. 01009 ◽

Cited By ~ 1

Author(s):

Vassil M. Vassilev ◽

Daniel M. Dantchev ◽

Peter A. Djondjorov

Keyword(s):

Boundary Conditions ◽

Order Parameter ◽

Neumann Boundary ◽

Elliptic Functions ◽

Thermodynamic System ◽

Neumann Boundary Conditions ◽

Ginzburg Landau ◽

One Dimensional ◽

The One ◽

Field Plane

In this article we consider a critical thermodynamic system with the shape of a thin film confined between two parallel planes. It is assumed that the state of the system at a given temperature and external ordering field is described by order-parameter profiles, which minimize the one-dimensional counterpart of the standard ϕ4 Ginzburg–Landau Hamiltonian and meet the so-called Neumann – Neumann boundary conditions. We give analytic representation of the extremals of this variational problem in terms ofWeierstrass elliptic functions. Then, depending on the temperature and ordering field we determine the minimizers and obtain the phase diagram in the temperature-field plane.

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