Analytical and numerical investigations of the modified Camassa–Holm equation

Abdelrahman, Mahmoud A. E.; Alharbi, Abdulghani

doi:10.1007/s12043-021-02153-6

Cited by 13 publications

(3 citation statements)

References 44 publications

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“…The Riccati equation admits various kinds of solutions, presented in [32]. By using the solution to Equation (24), we obtain:…”

Section: Generalized Tanh Methodsmentioning

confidence: 99%

“…Additionally, it is essential to consider stability, convergence, and efficiency when selecting a numerical method for solving nonlinear PDEs. Some other analytical and numerical solution methodologies can be found in the literature [20][21][22][23][24][25]. Several studies have examined the coupled nonlinear Schrödinger equation for its numerical and exact solutions.…”

Section: Introductionmentioning

confidence: 99%

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A Study of Traveling Wave Structures and Numerical Investigations into the Coupled Nonlinear Schrödinger Equation Using Advanced Mathematical Techniques

Alharbi,

Alharbi

2023

Mathematics

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This article explores adapted mathematical methods to solve the coupled nonlinear Schrödinger (C-NLS) equation through analytical and numerical methods. To obtain exact solutions for the (C-NLS) equation, we utilize the improved modified, extended tanh-function method. By separating the Schrödinger equation into real and imaginary parts, we can obtain four coupled equations, which we then analyze using the generalized tanh method to extract exact solutions. This system of equations is essential for understanding the behavior of quantum systems and has various applications in quantum mechanics. We obtain an analytical solution and demonstrate numerical solutions using implicit finite difference. Studies have shown that this scheme is second-order in space and time, and the von Neumann stability analysis confirms its unconditional stability. We introduce the comparison between numerical and exact solutions.

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“…The Riccati equation admits various kinds of solutions, presented in [32]. By using the solution to Equation (24), we obtain:…”

Section: Generalized Tanh Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Study of Traveling Wave Structures and Numerical Investigations into the Coupled Nonlinear Schrödinger Equation Using Advanced Mathematical Techniques

Alharbi,

Alharbi

2023

Mathematics

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“…The modified mapping principle was applied in [21] to establish a class of periodic wave solutions in terms of the Jacobi elliptic functions. Moreover, Abdelrahman et al [22] utilized the singular manifold technique to construct two general solutions for system (1). Every obtained solution contains two arbitrary functions.…”

Section: Introductionmentioning

confidence: 99%

A Study of Traveling Wave Structures and Numerical Investigation of Two-Dimensional Riemann Problems with Their Stability and Accuracy

Alharbi¹

2023

Computer Modeling in Engineering &Amp; Sciences

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The Riemann wave system has a fundamental role in describing waves in various nonlinear natural phenomena, for instance, tsunamis in the oceans. This paper focuses on executing the generalized exponential rational function approach and some numerical methods to obtain a distinct range of traveling wave structures and numerical results of the two-dimensional Riemann problems. The stability of obtained traveling wave solutions is analyzed by satisfying the constraint conditions of the Hamiltonian system. Numerical simulations are investigated via the finite difference method to verify the accuracy of the obtained results. To extract the approximation solutions to the underlying problem, some ODE solvers in FORTRAN software are applied, and outcomes are shown graphically. The stability and accuracy of the numerical schemes using Fourier's stability method and error analysis, respectively, to increase the reassurance are investigated. A comparison between the analytical and numerical results is obtained and graphically provided. The proposed methods are effective and practical to be applied for solving more partial differential equations (PDEs).

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Neuro-swarm computational heuristic for solving a nonlinear second-order coupled Emden–Fowler model

Sabir

Raja²,

Băleanu

et al. 2022

Soft Comput

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The aim of the current study is to present the numerical solutions of a nonlinear second-order coupled Emden–Fowler equation by developing a neuro-swarming-based computing intelligent solver. The feedforward artificial neural networks (ANNs) are used for modelling, and optimization is carried out by the local/global search competences of particle swarm optimization (PSO) aided with capability of interior-point method (IPM), i.e., ANNs-PSO-IPM. In ANNs-PSO-IPM, a mean square error-based objective function is designed for nonlinear second-order coupled Emden–Fowler (EF) equations and then optimized using the combination of PSO-IPM. The inspiration to present the ANNs-PSO-IPM comes with a motive to depict a viable, detailed and consistent framework to tackle with such stiff/nonlinear second-order coupled EF system. The ANNs-PSO-IP scheme is verified for different examples of the second-order nonlinear-coupled EF equations. The achieved numerical outcomes for single as well as multiple trials of ANNs-PSO-IPM are incorporated to validate the reliability, viability and accuracy.

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Analytical and numerical investigations of the modified Camassa–Holm equation

Cited by 13 publications

References 44 publications

A Study of Traveling Wave Structures and Numerical Investigations into the Coupled Nonlinear Schrödinger Equation Using Advanced Mathematical Techniques

A Study of Traveling Wave Structures and Numerical Investigations into the Coupled Nonlinear Schrödinger Equation Using Advanced Mathematical Techniques

A Study of Traveling Wave Structures and Numerical Investigation of Two-Dimensional Riemann Problems with Their Stability and Accuracy

Neuro-swarm computational heuristic for solving a nonlinear second-order coupled Emden–Fowler model

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