Numerical schemes for general Klein–Gordon equations with Dirichlet and nonlocal boundary conditions

Martín-Vaquero, J.; Encinas, Ascensión Hernández; Dios, Araceli Queiruga; Martinez, Victor H. Gonzalez; Rey, Ángel Martín del

doi:10.15388/na.2018.1.5

Cited by 6 publications

(5 citation statements)

References 31 publications

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“…By substituting this expression into nonlocal conditions (15), we get system (11). A further proof of the lemma is coincident with that of Lemma 2.…”

Section: Eigenvalue Problem Of a Difference Operatormentioning

confidence: 56%

“…Eigenvalue problems of difference operators with nonlocal conditions usually arise when solving boundary problems by the finite difference method. The spectrum properties of difference operators with various nonlocal boundary conditions were explored for investigation of the stability of difference schemes [1,2,4,7,8,10,11]. Another sphere of such a spectrum analysis application is convergence of iterative methods for the systems of difference equations [17,19,22], in particular, for nonlinear elliptic equations with integral boundary conditions [22,23].…”

Section: Introduction and Problem Statementmentioning

confidence: 99%

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A new eigenvalue problem for the difference operator with nonlocal conditions

Sapagovas

Čiupaila

Jakubėlienė

et al. 2019

NAMC

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In the paper, the spectrum structure of one-dimensional differential operator with nonlocal conditions and of the difference operator, corresponding to it, has been exhaustively investigated. It has been proved that the eigenvalue problem of difference operator is not equivalent to that of matrix eigenvalue problem Au = λu, but it is equivalent to the generalized eigenvalue problem Au = λBu with a degenerate matrix B. Also, it has been proved that there are such critical values of nonlocal condition parameters under which the spectrum of both the differential and difference operator are continuous. It has been established that the number of eigenvalues of difference problem depends on the values of these parameters. The condition has been found under which the spectrum of a difference problem is an empty set. An elementary example, illustrating theoretical expression, is presented.

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“…By substituting this expression into nonlocal conditions (15), we get system (11). A further proof of the lemma is coincident with that of Lemma 2.…”

Section: Eigenvalue Problem Of a Difference Operatormentioning

confidence: 56%

Section: Introduction and Problem Statementmentioning

confidence: 99%

A new eigenvalue problem for the difference operator with nonlocal conditions

Sapagovas

Čiupaila

Jakubėlienė

et al. 2019

NAMC

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“…We tried to develop explicit finite-difference methods with one level less than the one given by equation ( 36) in a similar way to those methods developed and analyzed in [68,69], but there n c was established as 1, and for stability purposes now we utilized n c < 1. Other numerical schemes proposed for similar nonlinear Klein-Gordon (or similar) hyperbolic partial differential equations are described in [70][71][72] and references therein, they are mostly based on finite difference algorithms in time and finite difference or spectral methods in space.…”

Section: Conclusion and Further Commentsmentioning

confidence: 99%

Kink scattering in a generalized Wess-Zumino model

Alonso-Izquierdo,

Leon,

Vaquero

et al. 2021

Preprint

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In this paper, kink scattering in the dimensional reduction of the bosonic sector of a one-parameter family of generalized Wess-Zumino models with three vacuum points is discussed. The value of the model parameter determines the specific location of the vacua. The influence of the vacuum arrangements (evolving from three collinear vacua to three vacua placed at the vertices of an equilateral triangle) on the kink scattering is investigated. Two different regimes can be distinguished: in the first one, two symmetric BPS kinks/antikinks arise whereas in the second one a new different BPS kink/antikink emerges, with the exception of a three-fold rotational symmetry case, where the three topological defects are identical. The scattering between the two symmetric kinks is thoroughly analyzed. Two different scattering channels have been found: kink-kink reflection and kink-kink hybridization. In the last case, the collision between the two symmetric kinks gives rise to the third different kink. Resonance phenomena also appear allowing a vibrating kink to split into two symmetric kinks moving away.

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“…The integral conditions in the form of (2), (3) are used by many authors considering nonlocal problems, also for differential equations of other types: for hyperbolic equations [16], equations with fractional derivatives [17], problems with complex parameter in the equation or nonlocal condition [23].…”

Section: Introduction and Problem Formulationmentioning

confidence: 99%

On the numerical solution for nonlinear elliptic equations with variable weight coefficients in an integral boundary conditions

Čiupaila

Pupalaigė

Sapagovas

2021

NAMC

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In the paper the two-dimensional elliptic equation with integral boundary conditions is solved by finite difference method. The main aim of the paper is to investigate the conditions for the convergence of the iterative methods for the solution of system of nonlinear difference equations. With this purpose, we investigated the structure of the spectrum of the difference eigenvalue problem. Some sufficient conditions are proposed such that the real parts of all eigenvalues of the corresponding difference eigenvalue problem are positive. The proof of convergence of iterative method is based on the properties of the M-matrices not requiring the symmetry or diagonal dominance of the matrices. The theoretical statements are supported by the results of the numerical experiment.

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Numerical schemes for general Klein–Gordon equations with Dirichlet and nonlocal boundary conditions

Cited by 6 publications

References 31 publications

A new eigenvalue problem for the difference operator with nonlocal conditions

A new eigenvalue problem for the difference operator with nonlocal conditions

Kink scattering in a generalized Wess-Zumino model

On the numerical solution for nonlinear elliptic equations with variable weight coefficients in an integral boundary conditions

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