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Keywords = MRLW equation

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12 pages, 1623 KB  
Article
Numerical Analysis of the Discrete MRLW Equation for a Nonlinear System Using the Cubic B-Spline Collocation Method
by Xingxia Liu, Lijun Zhang and Jianan Sun
Symmetry 2024, 16(4), 438; https://doi.org/10.3390/sym16040438 - 5 Apr 2024
Cited by 3 | Viewed by 1331
Abstract
By employing the cubic B-spline functions, a collocation approach was devised in this study to address the Modified Regularized Long Wave (MRLW) equation. Then, we derived the corresponding nonlinear system and easily solved it using Newton’s iterative approach. It was established that the [...] Read more.
By employing the cubic B-spline functions, a collocation approach was devised in this study to address the Modified Regularized Long Wave (MRLW) equation. Then, we derived the corresponding nonlinear system and easily solved it using Newton’s iterative approach. It was established that the cubic B-spline collocation technique exhibits unconditional stability. The dynamics of solitary waves, including their pairwise and triadic interactions, were meticulously investigated utilizing the proposed numerical method. Additionally, the transformation of the Maxwellian initial condition into solitary wave formations is presented. To validate the current work, three distinct scenarios were compared against the analytical solution and outcomes from alternative methods under both L2- and L-error norms. Primarily, the key strength of the suggested scheme lies in its capacity to yield enhanced numerical resolutions when employed to solve the MRLW equation, and these conservation laws show that the solitary waves have time and space translational symmetry in the propagation process. Finally, this paper concludes with a summary of our findings. Full article
(This article belongs to the Section Physics)
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15 pages, 931 KB  
Article
Two Reliable Computational Techniques for Solving the MRLW Equation
by Kamel Al-Khaled and Haneen Jafer
Axioms 2023, 12(2), 174; https://doi.org/10.3390/axioms12020174 - 8 Feb 2023
Cited by 2 | Viewed by 1845
Abstract
In this paper, a numerical solution of the modified regularized long wave (MRLW) equation is obtained using the Sinc-collocation method. This approach approximates the space dimension of the solution with a cardinal expansion of Sinc functions. First, discretizing the time derivative of the [...] Read more.
In this paper, a numerical solution of the modified regularized long wave (MRLW) equation is obtained using the Sinc-collocation method. This approach approximates the space dimension of the solution with a cardinal expansion of Sinc functions. First, discretizing the time derivative of the MRLW equation by a classic finite difference formula, while the space derivatives are approximated by a θweighted scheme. For comparison purposes, we also find a soliton solution using the Adomian decomposition method (ADM). The Sinc-collocation method was were found to be more accurate and efficient than the ADM schemes. Furthermore, we show that the number of solitons generated can be approximated using the Maxwellian initial condition. The proposed methods’ results, analytical solutions, and numerical methods are compared. Finally, a variety of graphical representations for the obtained solutions makes the dynamics of the MRLW equation visible and provides the mathematical foundation for physical and engineering applications. Full article
(This article belongs to the Special Issue Special Topics in Differential Equations with Applications)
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14 pages, 648 KB  
Article
First Integral Technique for Finding Exact Solutions of Higher Dimensional Mathematical Physics Models
by Shumaila Javeed, Sidra Riaz, Khurram Saleem Alimgeer, M. Atif, Atif Hanif and Dumitru Baleanu
Symmetry 2019, 11(6), 783; https://doi.org/10.3390/sym11060783 - 12 Jun 2019
Cited by 22 | Viewed by 4203
Abstract
In this work, we establish the exact solutions of some mathematical physics models. The first integral method (FIM) is extended to find the explicit exact solutions of high-dimensional nonlinear partial differential equations (PDEs). The considered models are: the space-time modified regularized long wave [...] Read more.
In this work, we establish the exact solutions of some mathematical physics models. The first integral method (FIM) is extended to find the explicit exact solutions of high-dimensional nonlinear partial differential equations (PDEs). The considered models are: the space-time modified regularized long wave (mRLW) equation, the (1+2) dimensional space-time potential Kadomtsev Petviashvili (pKP) equation and the (1+2) dimensional space-time coupled dispersive long wave (DLW) system. FIM is a powerful mathematical tool that can be used to obtain the exact solutions of many non-linear PDEs. Full article
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