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.

scholarly journals An efficient numerical method for one-dimensional hyperbolic interface problems

2019 ◽  
Vol 29 ◽  
pp. 01002
Author(s):  
Chartese Jones ◽  
Xu Zhang
Keyword(s):  
Finite Element ◽  
Numerical Experiments ◽  
Numerical Scheme ◽  
Method Of Lines ◽  
Interface Problems ◽  
Uniform Mesh ◽  
One Dimensional ◽  
Immersed Finite Element ◽  
New Methods ◽  
The Method Of Lines

In this paper, we develop an efficient numerical scheme for solving one-dimensional hyperbolic interface problems. The immersed finite element (IFE) method is used for spatial discretization, which allows the solution mesh to be independent of the interface. Consequently, a fixed uniform mesh can be used throughout the entire simulation. The method of lines is used for temporal discretization. Numerical experiments are provided to show the features of these new methods.

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

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