Study of the two‐dimensional sine‐Gordon equation arising in Josephson junctions using meshless finite point method

Abstract

In this paper, the finite point method is discussed for solving the initial‐boundary value problem associated with the sine‐Gordon equation in two‐dimensional domains arising in the Josephson junctions problem. The resulting nonlinear system is solved using an appropriate predictor‐corrector scheme. The proposed scheme is simple and efficient. The collisional properties for cases involving the most known from the bibliography, line, and ring solitons are studied in numerical results. Also the birth of a single Josephson vortex in a Josephson transmission line at a T‐shaped junction is studied.

Structure-preserving algorithms for the two-dimensional sine-Gordon equation with Neumann boundary conditions

“…Following [17,22] for this experiment it is considered φ(x, y) = 1 with initial conditions f 1 (x, y) = 4 tan −1 (exp(3 − x 2 + y 2 )),…”

“…There have many efficient numerical schemes concerning the two-dimensional sine-Gordon equation (1.1) with Neumann boundary conditions on both damped and undamped circumstances, which are based on several different methodologies, such as the finite difference schemes [17,18,19], the finite element schemes [20], the meshless methods [21,22] and so on. Although these proposed methods can well resolve the Neumann boundary condition, none of them are strictly SPAs and thereby cannot guarantee a long-time stability as well as the preservation of conservative quantities, for example, the system energy.…”

Pseudospectral meshless radial point interpolation for generalized biharmonic equation in the presence of Cahn–Hilliard conditions

“…• Meshless methods based on weak forms such as the element free Galerkin (EFG) method (and its developments like for example improved EFG (IEFG), complex variable (EFG) (CVEFG) and improved complex variable (EFG) (ICVEFG)), meshless local Petrov-Galerkin (MLPG) method (Belytschko et al 1994;Fili et al 2010;Peng et al 2011;Shivanian et al 2015;Dai and Cheng 2010;Bai et al 2012), MLPG based on the particular solutions (MLPG-PS), and the method of approximate particular solutions (MAPS) (Abbasbandy et al 2014). • Meshless techniques based on collocation schemes (strong forms), such as the meshless collocation method based on radial basis functions (RBFs) (Parand et al 2011;Kansa 1990;Jakobsson et al 2009;Abbasbandy et al 2012Abbasbandy et al , 2013Kamranian et al 2016;Moradipour and Yousefi 2018). • Meshless techniques based on the combination of weak forms and collocation method (Santin and Schaback 2016;Schaback 2015;Young et al 2008;Assari and Dehghan 2018;Shirzadi 2010, 2011;Dehghan and Ghesmati 2010;Liu et al 2002;Liu and Gu 2001;Shivanian 2013Shivanian , 2014Khodabandehlo 2014, 2016;Hosseini et al 2015Hosseini et al , 2016.…”

A Finite Point Method for Solving the Time Fractional Richards’ Equation

“…Shojaei et al [15] used the meshless finite point method to solve elastodynamic problems through an explicit velocity. In the same year, Kamranian et al [16] discussed the two-dimensional initial boundary value problem associated with the sine-Gordon equation using the finite point method. Li and Qin [17] used the meshless finite point method to solve the time fractional convectiondiffusion equations, and the numerical results show that this method has higher computational accuracy than the finite difference method.…”