Identification of memory kernels in one-dimensional heat flow with boundary conditions of the third kind

2001 ◽  
Vol 9 (2) ◽  
pp. 175-198 ◽  
Author(s):  
J. Janno ◽  
L. v. wolfersdorf
Keyword(s):  
Heat Flow ◽  
Boundary Conditions ◽  
One Dimensional ◽  
The Third

scholarly journals Method of lines for solving linear equations of mathematical physics with the third and first types boundary conditions

2021 ◽  
Vol 2131 (3) ◽  
pp. 032041
Author(s):  
M Kh Eshmurodov ◽  
K M Shaimov ◽  
I Khujaev ◽  
J Khujaev
Keyword(s):  
Boundary Conditions ◽  
Characteristic Equation ◽  
Transition Matrix ◽  
Method Of Lines ◽  
Mathematical Physics ◽  
Sweep Method ◽  
Discontinuous Solutions ◽  
One Dimensional ◽  
The Third ◽  
The Method Of Lines

Abstract The use of the method of lines in solving multidimensional problems of mathematical physics makes it possible to eliminate the discrepancies caused by the use of the sweep method in certain coordinates. As a result, the solution of the Poisson equation, for example, is obtained without using the relaxation method. In the article, the problem on the eigenvalues and vectors of the transition matrix is solved for boundary conditions of the third and first types, used to solve a one-dimensional equation of parabolic type by the method of lines. Due to the features of boundary conditions of the third type for determining the eigenvalues, a mixed method was proposed based on the Vieta theorem and the representation of the characteristic equation in trigonometric form typical for the method of lines. To solve the eigenvector problem, a simple sweep method was used with the algebraic compliments to the transition matrix. Discontinuous solutions of a one-dimensional parabolic equation were presented for various values of complex 1 -αl; the method for solving the characteristic equation was selected based on these values. The calculation results are in good agreement with the analytical solution.

Heat Kernel Expansions in the Case of Conic Singularities

2003 ◽  
Vol 18 (12) ◽  
pp. 2197-2203 ◽  
Author(s):  
R. Seeley
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Zeta Function ◽  
Differential Operators ◽  
Smooth Boundary ◽  
General Nature ◽  
Dimensional Manifold ◽  
One Dimensional ◽  
The Third ◽  
Elliptic Differential Operators

For positive elliptic differential operators Δ, the asymptotic expansion of the heat trace tr(e-tΔ) and its related zeta function ζ(s, Δ) = tr(Δ-s) have numerous applications in geometry and physics. This article discusses the general nature of the boundary conditions that must be considered when there is a singular stratum, and presents three examples in which a choice of boundary conditions at the singularity must be made. The first example concerns the signature operator on a manifold with a singular stratum of conic type. The second concerns the "Zaremba problem" for a nonsingular manifold with smooth boundary, posing Dirichlet conditions on part of the boundary and Neumann conditions on the complement; the intersection of these two regions can be viewed as a singular stratum of conic type, and a boundary condition must be imposed along this stratum. The third example is a one-dimensional manifold where the operator at one end has a singularity like that in conic problems, and the choice of boundary conditions affects not just the residues at the poles of the zeta function, but also the very location of the poles

scholarly journals Heat flow in a one-dimensional semi-infinite harmonic lattice with an absorbing boundary

2020 ◽  
Vol 20 (1) ◽  
pp. 38-51
Author(s):  
A.I. Gudimenko ◽  
Keyword(s):  
Heat Flow ◽  
Boundary Conditions ◽  
Numerical Approximation ◽  
Mechanical Characteristics ◽  
Analytical Approximation ◽  
Heat Propagation ◽  
Absorbing Boundary Conditions ◽  
Absorbing Boundary ◽  
One Dimensional ◽  
Infinite Domains

Traditionally, absorbing boundary conditions are used to limit the domains of numerical approximation of partial differential equations in infinite domains. In the present paper, the simplest of these conditions is used to obtain an analytical approximation of the solution to the problem of heat propagation in a one-dimensional infinite harmonic lattice consisting of two semi-infinite homogeneous sublattices with different mechanical characteristics.

Errors in a Finite-Difference Solution of the Heat Flow Equation

1964 ◽  
Vol 86 (4) ◽  
pp. 561-562 ◽  
Author(s):  
Alan Kardas
Keyword(s):  
Heat Flow ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Flow Equation ◽  
The Third ◽  
Finite Difference Solution ◽  
Discretization Errors ◽  
Difference Solution

This note gives magnitudes of discretization errors incurred in a finite difference solution of the heat flow equation in a symmetric slab with the boundary conditions of the third kind.

Differential Approximations to the Radiation Transfer Equation by Chapman–Enskog Expansion

2011 ◽  
Vol 133 (8) ◽  
Author(s):  
A. V. Gusarov
Keyword(s):  
Heat Flow ◽  
Boundary Conditions ◽  
Optical Thickness ◽  
Transfer Equation ◽  
Radiation Transfer ◽  
Radiation Transfer Equation ◽  
Differential Approximation ◽  
Third Order ◽  
The Third ◽  
In Series

Direct numerical solution of the radiation transfer equation is often easier than implementation of its differential approximations with their cumbersome boundary conditions. Nevertheless, these approximations are still used, for example, in theoretical analysis. The existing approach to obtain a differential approximation based on expansion in series of the spherical harmonics is revised and expansion in series of the eigenfunctions of the scattering integral is proposed. A system of eigenfunctions is obtained for an arbitrary phase function, and explicit differential approximations are built up to the third Chapman–Enskog order. The results are tested by its application to the problem of a layer. The third-order Chapman–Enskog approximation is found to match the boundary conditions better than the first-order one and gives considerably more accurate value for the heat flow. The accuracy of the both first- and third-order heat flows generally increases with the optical thickness. In addition, the third-order heat flow tends to the rigorous limit value when the optical thickness tends to zero.

Initial-boundary value problems for the one-dimensional linear advection–dispersion equation with decay

2020 ◽  
Vol 75 (8) ◽  
pp. 713-725 ◽  
Author(s):  
Guenbo Hwang
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
One Dimensional ◽  
Advection Dispersion ◽  
Asymptotically Periodic ◽  
The One ◽  
Initial Boundary Value

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.

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

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.

scholarly journals On a stochastic Burgers equation with Dirichlet boundary conditions

2003 ◽  
Vol 2003 (43) ◽  
pp. 2735-2746 ◽  
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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