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In this study, we investigate the traveling wave solutions of the Gilson-Pickering equation using two different approaches: F-expansion and (1/G')-expansion. To carry out the analysis, we perform a numerical study using the implicit finite difference approach on a uniform mesh and the parabolic-Monge-Ampère (PMA) method on a moving mesh. We examine the truncation error, stability, and convergence of the difference scheme implemented on a fixed mesh. MATLAB software generates accurate representations of the solution based on specified parameter values by creating 3D and 2D graphs. Numerical simulations with the finite difference scheme demonstrate excellent agreement with the analytical solutions, further confirming the validity of our approaches. Convergence analysis confirms the stability and high accuracy of the implemented scheme. Notably, the PMA method performs better in capturing intricate wave interactions and dynamics that are not readily achievable with a fixed mesh. [ABSTRACT FROM AUTHOR]

In this research, we apply some new mathematical methods to the study of solving couplebreaking soliton equations in two dimensions. Soliton solutions for equations with free parameters like the wave number, phase component, nonlinear coeffcient and dispersion coeffcient can be obtained analytically by adding trigonometric, rational and hyperbolic functions. We will also look into how two-dimensional diagrams are affected by the wave phenomena, illustrating the answers with a mix of two- and three-dimensional graphs. The proposed system will be transformed into a numerical system by using the finite difference method to simulate Black-Scholes equations numerically. Furthermore, we will evaluate the stability and accuracy of the numerical findings by making analytical and graphical comparisons with precise solutions and we will talk about the error analysis of the numerical scheme. All forms of nonlinear evolutionary systems can benefit from the methods utilized in this work because they are suffcient and acceptable. [ABSTRACT FROM AUTHOR]

This study explores adapted mathematical methods to solve the couple-breaking soliton (BS) equations in two-dimensional spatial domains. Using these methods, we obtained analytical soliton solutions for the equations involving free parameters such as the wave number, phase component, nonlinear coefficient, and dispersion coefficient. The solutions are expressed as hyperbolic, rational, and trigonometric functions. We also examined the impact of wave phenomenon on two-dimensional diagrams and used composite two-dimensional and three-dimensional graphs to represent the solutions. We used the finite difference method to transform the proposed system into a numerical system to obtain numerical simulations for the Black–Scholes equations. Additionally, we discuss the stability and error analysis of numerical schemes. We compare the validity and accuracy of the numerical results with the exact solutions through analytical and graphical comparisons. The methodologies presented in this research can be applied to various forms of nonlinear evolutionary systems because they are appropriate and acceptable. [ABSTRACT FROM AUTHOR]

The extraction of traveling wave solutions for nonlinear evolution equations is a challenge in various mathematics, physics, and engineering disciplines. This article intends to analyze several traveling wave solutions for themodified regularized long-wave (MRLW) equation using several approaches, namely, the generalized algebraic method, the Jacobian elliptic functions technique, and the improved Q-expansion strategy. We successfully obtain analytical solutions consisting of rational, trigonometric, and hyperbolic structures. The adaptive moving mesh technique is applied to approximate the numerical solution of the proposed equation. The adaptive moving mesh method evenly distributes the points on the high error areas. This method perfectly and strongly reduces the error. We compare the constructed exact and numerical results to ensure the reliability and validity of the methods used. To better understand the considered equation's physical meaning, we present some 2D and 3D figures. The exact and numerical approaches are efficient, powerful, and versatile for establishing novel bright, dark, bell-kink-type, and periodic traveling wave solutions for nonlinear PDEs. [ABSTRACT FROM AUTHOR]

This article presents the Parabolic-Monge-Ampere (PMA) method for numerical solutions of two-dimensional fourth-order parabolic thin film equations with constant flux boundary conditions. We track the PMA technique, which employs special functions to acclimate and force the mesh moving associated with the physical PDE representing the thin liquid film equation. The accuracy and convergence of the PMA approach are investigated numerically using a one two-dimensional problem. Comparing the results of this method to the uniform mesh finite difference scheme, the computing effort is reduced. [ABSTRACT FROM AUTHOR]

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). [ABSTRACT FROM AUTHOR]

In this paper, the numerical solution to the regularized long wave (GRLW) equation is investigated numerically applying two different techniques a Parabolic Monge Ampere (PMA) moving mesh and uni- form mesh. The PMA moving mesh is a method for generating a moving mesh which moves as the solution moves with time. The gen- erated mesh is obtained by having the gradient of a grid potential function. I use here the centred finite differences for both schemes. The comparison between these schemes is shown in the last table and figure. The analytical solution is derived in Appendix A. I find a so- lution when it has only one solitary wave. Next, I show when it has two or three solitary waves. Figures 2D and 3D shows all of these nu- merical results using the mentioned schemes compared with the exact solution. [ABSTRACT FROM AUTHOR]