scholarly journals ON STABILITY ANALYSIS OF FINITE DIFFERENCE SCHEMES FOR GENERALIZED KURAMOTO-TSUZUKI EQUATION WITH NONLOCAL BOUNDARY CONDITIONS

2016 ◽  
Vol 21 (5) ◽  
pp. 630-643 ◽  
Author(s):  
Teresė Leonavičienė ◽  
Andrej Bugajev ◽  
Gerda Jankevičiūtė ◽  
Raimondas Čiegis
Keyword(s):  
Boundary Condition ◽  
Stability Analysis ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Finite Difference Schemes ◽  
Discrete Approximations ◽  
Integral Boundary ◽  
Non Local ◽  
Problem Stability ◽  
The Stability

A general methodology for the stability analysis of discrete approximations of nonstationary PDEs is applied to solve the Kuramoto-Tsuzuki equation, including also the Schr¨odinger problem. Stability regions are constructed for the explicit, backward and symmetrical Euler schemes. The obtained results are applied to solve the Kuramoto-Tsuzuki problem with a non-local integral boundary condition. Results of computational experiments are provided.

scholarly journals Relations between Spectrum Curves of Discrete Sturm-Liouville Problem with Nonlocal Boundary Conditions and Graph Theory. II

2021 ◽  
Vol 62 ◽  
pp. 1-8
Author(s):  
Jonas Vitkauskas ◽  
Artūras Štikonas
Keyword(s):  
Boundary Condition ◽  
Graph Theory ◽  
Boundary Conditions ◽  
Critical Points ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Nonlocal Boundary Conditions ◽  
Integral Boundary Condition ◽  
Integral Boundary ◽  
Sturm Liouville

In this paper, relations between discrete Sturm--Liouville problem with nonlocal integral boundary condition characteristics (poles, critical points, spectrum curves) and graphs characteristics (vertices, edges and faces) were found. The previous article was devoted to the Sturm--Liouville problem in the case two-points nonlocal boundary conditions.

scholarly journals Investigation of Negative Critical Points of the Characteristic Function for Problems with Nonlocal Boundary Conditions

2008 ◽  
Vol 13 (4) ◽  
pp. 467-490 ◽  
Author(s):  
S. Pečiulytė ◽  
O. Štikonienė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Characteristic Function ◽  
Critical Points ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Nonlocal Boundary Conditions ◽  
The Other ◽  
Integral Boundary ◽  
Sturm Liouville

In this paper the Sturm-Liouville problem with one classical and the other nonlocal two-point or integral boundary condition is investigated. Critical points of the characteristic function are analyzed. We investigate how distribution of the critical points depends on nonlocal boundary condition parameters. In the first part of this paper we investigate the case of negative critical points.

scholarly journals Investigation of complex eigenvalues for a stationary problem with two-point nonlocal boundary condition

2011 ◽  
Vol 52 ◽  
pp. 303-308
Author(s):  
Kristina Skučaitė-Bingelė ◽  
Artūras Štikonas
Keyword(s):  
Boundary Condition ◽  
Finite Difference ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Nonlocal Boundary Conditions ◽  
Stationary Problem ◽  
Finite Difference Schemes ◽  
Complex Eigenvalues ◽  
Sturm Liouville

The Sturm–Liouville problem with one classical and another two-point nonlocal boundary condition is considered in this paper. These problems with nonlocal boundary condition are not self-adjoint, so the spectrum has complex points. We investigate how the spectrum in the complex plane of these problems (and for the Finite-Difference Schemes) depends on parameters γ  and ξ  of the nonlocal boundary conditions.

scholarly journals ON CONSTRUCTION AND ANALYSIS OF FINITE DIFFERENCE SCHEMES FOR PSEUDOPARABOLIC PROBLEMS WITH NONLOCAL BOUNDARY CONDITIONS

2014 ◽  
Vol 19 (2) ◽  
pp. 281-297 ◽  
Author(s):  
Raimondas Čiegis ◽  
Natalija Tumanova
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Two Dimensional ◽  
The Stability

In this paper the one- and two-dimensional pseudoparabolic equations with nonlocal boundary conditions are approximated by the Euler finite difference scheme. In the case of classical boundary conditions the stability of all schemes is investigated by the spectral method. Stability regions of finite difference schemes approximating pseudoparabolic problem are compared with the stability regions of the classical discrete parabolic problem. These results are generalized for problems with nonlocal boundary conditions if a matrix of the finite difference scheme can be diagonalized. For the two-dimensional problem an efficient algorithm is constructed, which is based on the combination of the FFT method and the factorization algorithm. General stability results, known for the three level finite difference schemes, are applied to investigate the stability of some explicit approximations of the two-dimensional pseudoparabolic problem with classical boundary conditions. A connection between the energy method stability conditions and the spectrum Hurwitz stability criterion is shown. The obtained results can be applied for pseudoparabolic problems with nonlocal boundary conditions, if a matrix of the finite difference scheme can be diagonalized.

scholarly journals SPECTRUM CURVES FOR STURM–LIOUVILLE PROBLEM WITH INTEGRAL BOUNDARY CONDITION

2015 ◽  
Vol 20 (6) ◽  
pp. 802-818 ◽  
Author(s):  
Agnė Skučaitė ◽  
Artūras Štikonas
Keyword(s):  
Boundary Condition ◽  
Characteristic Function ◽  
Critical Points ◽  
Nonlocal Condition ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Nonlocal Boundary Conditions ◽  
Integral Boundary ◽  
Sturm Liouville

We consider Sturm–Liouville problem with one integral type nonlocal boundary condition depending on three parameters γ (multiplier in nonlocal condition), ξ1, ξ2 ([ξ1, ξ2] is a domain of integration). The distribution of zeroes, poles, and constant eigenvalue points of Complex Characteristic Function is presented. We investigate how Spectrum Curves depend on the parameters of nonlocal boundary conditions. In this paper we describe the behaviour of Spectrum Curves and classify critical points of Complex-Real Characteristic function. Phase Trajectories of critical points in Phase Space of the parameters ξ1, ξ2 are investigated. We present the results of modelling and computational analysis and illustrate those results with graphs.

scholarly journals Existence and Uniqueness of Generalized Solutions to a Telegraph Equation with an Integral Boundary Condition via Galerkin's Method

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
Assia Guezane-Lakoud ◽  
Jaydev Dabas ◽  
Dhirendra Bahuguna
Keyword(s):  
Boundary Condition ◽  
Existence And Uniqueness ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Generalized Solution ◽  
Telegraph Equation ◽  
Generalized Solutions ◽  
Galerkin’S Method ◽  
Integral Boundary ◽  
Galerkin's Method

We consider a telegraph equation with nonlocal boundary conditions, and using the application of Galerkin's method we established the existence and uniqueness of a generalized solution.

scholarly journals An Approximate Grid Solution of a Nonlocal Boundary Value Problem with Integral Boundary Condition for Laplace's Equation

2018 ◽  
Vol 22 ◽  
pp. 01016 ◽  
Author(s):  
Adıgüzel A. Dosiyev ◽  
Rifat Reis
Keyword(s):  
Boundary Condition ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Integral Boundary Condition ◽  
Unknown Boundary ◽  
Laplace’S Equation ◽  
Nonlocal Boundary Value Problem ◽  
Integral Boundary ◽  
Laplace's Equation

A new method for the solution of a nonlocal boundary value problem with integral boundary condition for Laplace's equation on a rectangular domain is proposed and justified. The solution of the given problem is defined as a solution of the Dirichlet problem by constructing the approximate value of the unknown boundary function on the side of the rectangle where the integral boundary condition was given. Further, the five point approximation of the Laplace operator is used on the way of finding the uniform estimation of the error of the solution which is order of 0(h2), where hi s the mesh size. Numerical experiments are given to support the theoretical analysis made.

scholarly journals Rothe time-discretization method for the semilinear heat equation subject to a nonlocal boundary condition

2006 ◽  
Vol 2006 ◽  
pp. 1-20 ◽  
Author(s):  
Nabil Merazga ◽  
Abdelfatah Bouziani
Keyword(s):  
Boundary Condition ◽  
Heat Equation ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Time Discretization ◽  
Neumann Condition ◽  
Integral Boundary ◽  
Semilinear Heat Equation ◽  
Rothe Method ◽  
The Given

This paper is devoted to prove, in a nonclassical function space, the weak solvability of a mixed problem which combines a Neumann condition and an integral boundary condition for the semilinear one-dimensional heat equation. The investigation is made by means of approximation by the Rothe method which is based on a semidiscretization of the given problem with respect to the time variable.

A Normal Mode Stability Analysis of Numerical Interface Conditions for Fluid/Structure Interaction

2011 ◽  
Vol 10 (2) ◽  
pp. 279-304 ◽  
Author(s):  
J. W. Banks ◽  
B. Sjögreen
Keyword(s):  
Stability Analysis ◽  
Normal Mode ◽  
Compressible Fluid ◽  
Elastic Solid ◽  
Interface Condition ◽  
Finite Difference Schemes ◽  
Interface Conditions ◽  
Characteristic Boundary ◽  
Mode Stability ◽  
The Stability

AbstractIn multi physics computations where a compressible fluid is coupled with a linearly elastic solid, it is standard to enforce continuity of the normal velocities and of the normal stresses at the interface between the fluid and the solid. In a numerical scheme, there are many ways that velocity- and stress-continuity can be enforced in the discrete approximation. This paper performs a normal mode stability analysis of the linearized problem to investigate the stability of different numerical interface conditions for a model problem approximated by upwind type finite difference schemes. The analysis shows that depending on the ratio of densities between the solid and the fluid, some numerical interface conditions are stable up to the maximal CFL-limit, while other numerical interface conditions suffer from a severe reduction of the stable CFL-limit. The paper also presents a new interface condition, obtained as a simplified characteristic boundary condition, that is proved to not suffer from any reduction of the stable CFL-limit. Numerical experiments in one space dimension show that the new interface condition is stable also for computations with the non-linear Euler equations of compressible fluid flow coupled with a linearly elastic solid.

Sign in / Sign up

Export Citation Format

Share Document