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

2016 ◽  
Vol 30 (6) ◽  
pp. e2210 ◽  
Author(s):  
Maryam Kamranian ◽  
Mehdi Dehghan ◽  
Mehdi Tatari
Keyword(s):  
Josephson Junctions ◽  
Gordon Equation ◽  
Point Method ◽  
Two Dimensional ◽  
Sine Gordon Equation ◽  
Finite Point ◽  
Finite Point Method

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

2019 ◽  
Vol 2019 ◽  
pp. 1-14
Author(s):  
Xinqiang Qin ◽  
Xin Yang
Keyword(s):  
Point Method ◽  
Richards Equation ◽  
Two Dimensional ◽  
Finite Point ◽  
Basic Principles ◽  
Finite Point Method ◽  
Spatial Derivatives ◽  
Time Fractional Derivative ◽  
The Stability ◽  
The One

In this paper, we propose a numerical method for solving the time fractional Richards’ equation. We first approximate the time fractional derivative of the mentioned equations by a scheme of order O(τ2−α), 0 < a<1; then, we use the finite point method to approximate the spatial derivatives. Before the discrete spatial derivatives, we introduced the basic principles of the finite point method. We solve the one- and two-dimensional versions of these equations using the proposed method. Moreover, the stability properties of the discretized scheme related to time are theoretically analyzed. Numerical results showed the efficiency of the method presented in this paper.

scholarly journals Duality of positive and negative integrable hierarchies via relativistically invariant fields

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
S. Y. Lou ◽  
X. B. Hu ◽  
Q. P. Liu
Keyword(s):  
Integrable Hierarchies ◽  
Gordon Equation ◽  
Formal Series ◽  
Two Dimensional ◽  
Relativistic Invariance ◽  
Sine Gordon Equation ◽  
Recursion Operators ◽  
Symmetry Approach ◽  
Duality Relations ◽  
Duality Problem

Abstract It is shown that the relativistic invariance plays a key role in the study of integrable systems. Using the relativistically invariant sine-Gordon equation, the Tzitzeica equation, the Toda fields and the second heavenly equation as dual relations, some continuous and discrete integrable positive hierarchies such as the potential modified Korteweg-de Vries hierarchy, the potential Fordy-Gibbons hierarchies, the potential dispersionless Kadomtsev-Petviashvili-like (dKPL) hierarchy, the differential-difference dKPL hierarchy and the second heavenly hierarchies are converted to the integrable negative hierarchies including the sG hierarchy and the Tzitzeica hierarchy, the two-dimensional dispersionless Toda hierarchy, the two-dimensional Toda hierarchies and negative heavenly hierarchy. In (1+1)-dimensional cases the positive/negative hierarchy dualities are guaranteed by the dualities between the recursion operators and their inverses. In (2+1)-dimensional cases, the positive/negative hierarchy dualities are explicitly shown by using the formal series symmetry approach, the mastersymmetry method and the relativistic invariance of the duality relations. For the 4-dimensional heavenly system, the duality problem is studied firstly by formal series symmetry approach. Two elegant commuting recursion operators of the heavenly equation appear naturally from the formal series symmetry approach so that the duality problem can also be studied by means of the recursion operators.

Numerical modeling of long Josephson junctions in the frame of double sine-gordon equation

2011 ◽  
Vol 3 (3) ◽  
pp. 389-398 ◽  
Author(s):  
P. Kh. Atanasova ◽  
T. L. Boyadjiev ◽  
Yu. M. Shukrinov ◽  
E. V. Zemlyanaya
Keyword(s):  
Numerical Modeling ◽  
Josephson Junctions ◽  
Gordon Equation ◽  
Sine Gordon Equation

scholarly journals Real-Time Computer Simulation of Three Dimensional Elastostatics Using the Finite Point Method

2011 ◽  
Vol 110-116 ◽  
pp. 2740-2745
Author(s):  
Kirana Kumara P. ◽  
Ashitava Ghosal
Keyword(s):  
Real Time ◽  
Graphics Processing Unit ◽  
Numerical Technique ◽  
Three Dimensional ◽  
Point Method ◽  
Processing Unit ◽  
Finite Point ◽  
Deformable Solids ◽  
Finite Point Method ◽  
Linear Elastostatics

Real-time simulation of deformable solids is essential for some applications such as biological organ simulations for surgical simulators. In this work, deformable solids are approximated to be linear elastic, and an easy and straight forward numerical technique, the Finite Point Method (FPM), is used to model three dimensional linear elastostatics. Graphics Processing Unit (GPU) is used to accelerate computations. Results show that the Finite Point Method, together with GPU, can compute three dimensional linear elastostatic responses of solids at rates suitable for real-time graphics, for solids represented by reasonable number of points.

Two Implicit Meshless Finite Point Schemes for the Two-Dimensional Distributed-Order Fractional Equation

2019 ◽  
Vol 19 (4) ◽  
pp. 813-831
Author(s):  
Rezvan Salehi
Keyword(s):  
Reproducing Kernel ◽  
Point Method ◽  
Least Square ◽  
Computationally Efficient ◽  
Finite Point ◽  
Moving Least Square ◽  
Point Collocation ◽  
Finite Point Method ◽  
Distributed Order ◽  
Fractional Equation

AbstractIn this paper, the distributed-order time fractional sub-diffusion equation on the bounded domains is studied by using the finite-point-type meshless method. The finite point method is a point collocation based method which is truly meshless and computationally efficient. To construct the shape functions of the finite point method, the moving least square reproducing kernel approximation is employed. Two implicit discretisation of order{O(\tau)}and{O(\tau^{1+\frac{1}{2}\sigma})}are derived, respectively. Stability and{L^{2}}norm convergence of the obtained difference schemes are proved. Numerical examples are provided to confirm the theoretical results.

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