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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]

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]

The Kudryashov technique is employed to extract several classes of solitary wave solutions for the Joseph–Egri equation. The stability of the achieved solutions is tested. The numerical solution of this equation is also investigated. We also present the accuracy and the stability of the numerical schemes. Some two- and three-dimensional figures are shown to present the solutions on some specific domains. The used methods are found useful to be applied on other nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

This paper points out several exact travelling wave solutions for (1 + 1) -dimensional Ito integro-differential equation via the Riccati–Bernoulli sub-ODE approach. We also aim to develop a numerical solution of the respective equation using central finite difference formulas. The stability of the presented exact and numerical solutions is also deduced using the Hamiltonian system and Von Neumann's concept, respectively. Moreover, the numerical schemes are studied in terms of their accuracy. The relative error arising from executing the numerical method is exhibited. We compare our results with others published in some articles. The accomplished numerical solutions are successfully compared with the analytical ones. The used processes can be extended to solve more integrable problems as well as non-integrable ones. [ABSTRACT FROM AUTHOR]

Tsunamis are strong waves, arisen from volcanic eruptions, landslides, or earthquakes sweeping across oceans. The geophysical Korteweg-de Vries (gKdV) equation which governs the tsunami wave propagation in oceans is investigated in this work using an improved exp (- F (η)) -expansion method. Shooting and adaptive moving approaches are taken into account. We retrieve several new solitary solutions for the gKdV equation. The obtained solution from implementing shooting method is successfully used as an initial value for the adaptive approach which is utilized to construct the numerical solution of the proposed problem. The constructed exact solutions coincide with the obtained numerical solutions. The accuracy of the presented numerical approximations is discussed. We apply Fourier concept to explore the accuracy and stability of the numerical schemes which is unconditionally stable. A clear comparison between the analytic and numerical outcomes is presented via some 2D and 3D sketches which are depicted under special selections of some parameters. Moreover, we illustrate the relative error and CPU time for the numerical technique. The proposed approaches can be easily utilized to deal with other partial differential equations. [ABSTRACT FROM AUTHOR]

• This work focuses on developing new exact and numerical solutions for (1 + 1)-dimensional Ito integro-differential equation. • Trigonometric, hyperbolic and rational solutions are successfully presented. • The stability and accuracy of the obtained numerical scheme are discussed. • L 2 error which illustrates the effectiveness of the used numerical approach is comprehensively studied. • The applied methods can be effectively invoked to solve more nonlinear PDEs. Constructing exact and numerical solutions of partial differential equations (PDEs) has become an active area in recent years. This work focuses on developing new exact and numerical solutions for (1 + 1) - dimensional Ito integro-differential equation by applying exp (- f (ζ)) -expansion and finite difference methods, respectively. Trigonometric, hyperbolic and rational solutions are successfully presented. The stability and accuracy of the obtained numerical simulation are discussed. The presented graphical comparison shows that the exact and numerical solutions nearly coincide with each other. L 2 error which illustrates the effectiveness of the used numerical approach is comprehensively studied. The applied methods can be effectively invoked to solve more nonlinear PDEs. [ABSTRACT FROM AUTHOR]