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Keywords = moving mesh PDEs (MMPDEs)

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13 pages, 707 KiB  
Article
Fundamental Solutions for the Coupled KdV System and Its Stability
by Mahmoud A. E. Abdelrahman, M. B. Almatrafi and Abdulghani Alharbi
Symmetry 2020, 12(3), 429; https://doi.org/10.3390/sym12030429 - 7 Mar 2020
Cited by 28 | Viewed by 3541
Abstract
In this paper, we establish exact solutions for the non-linear coupled KdV equations. The exp-function method is used to construct the solitary travelling wave solutions for these equations. The numerical adaptive moving mesh PDEs (MMPDEs) method is also implemented in order to solve [...] Read more.
In this paper, we establish exact solutions for the non-linear coupled KdV equations. The exp-function method is used to construct the solitary travelling wave solutions for these equations. The numerical adaptive moving mesh PDEs (MMPDEs) method is also implemented in order to solve the proposed coupled KdV equations. The achieved results may be applicable to some plasma environments, such as ionosphere plasma. Some numerical simulations compared with the exact solutions are provided to illustrate the validity of the proposed methods. Furthermore, the modulational instability is analyzed based on the standard linear-stability analysis. The depiction of the techniques are straight, powerful, robust and can be applied to other nonlinear systems of partial differential equations. Full article
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11 pages, 405 KiB  
Article
Moving Mesh Strategies of Adaptive Methods for Solving Nonlinear Partial Differential Equations
by Qinjiao Gao and Shenggang Zhang
Algorithms 2016, 9(4), 86; https://doi.org/10.3390/a9040086 - 15 Dec 2016
Cited by 11 | Viewed by 4724
Abstract
This paper proposes moving mesh strategies for the moving mesh methods when solving the nonlinear time dependent partial differential equations (PDEs). Firstly we analyse Huang’s moving mesh PDEs (MMPDEs) and observe that, after Euler discretion they could be taken as one step of [...] Read more.
This paper proposes moving mesh strategies for the moving mesh methods when solving the nonlinear time dependent partial differential equations (PDEs). Firstly we analyse Huang’s moving mesh PDEs (MMPDEs) and observe that, after Euler discretion they could be taken as one step of the root searching iteration methods. We improve Huang’s MMPDE by adding one Lagrange speed term. The proposed moving mesh PDE could draw the mesh to equidistribution quickly and stably. The numerical algorithm for the coupled system of the original PDE and the moving mesh equation is proposed and the computational experiments are given to illustrate the validity of the new method. Full article
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