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This article is dealing with the analytical solution of Fornberg–Whitham equations in fractional view of Caputo operator. The effective method among the analytical techniques, natural transform decomposition method, is implemented to handle the solutions of the proposed problems. The approximate analytical solutions of nonlinear numerical problems are determined to confirm the validity of the suggested technique. The solution of the fractional-order problems are investigated for the suggested mathematical models. The solutions-graphs are then plotted to understand the effectiveness of fractional-order mathematical modeling over integer-order modeling. It is observed that the derived solutions have a closed resemblance with the actual solutions. Moreover, using fractional-order modeling various dynamics can be analyzed which can provide sophisticated information about physical phenomena. The simple and straight-forward procedure of the suggested technique is the preferable point and thus can be used to solve other nonlinear fractional problems.

The complete classification of solutions to the defocusing complex modified KdV equation with step-like initial condition is studied by the finite-gap integration approach and Whitham modulation theory. All kinds of combination solutions consisting of genus-0 regions, genus-1 regions, or genus-2 regions are found by classifying the Riemann invariants. The behaviors of wave breaking in Riemann problem of the defocusing complex modified KdV equation are much richer and more complicated than those in the nonlinear Schrödinger equation. It is demonstrated that a large oscillating region can be composed of four basic genus-1 dispersive shock waves, a case of solution may be consisted of up to six regions, and the plateau, vacuum, rarefaction wave, and dispersive shock wave can coexist in the same solution region. Moreover, the genus-2 region, produced from the collision of two dispersive shock waves, is described detailedly by the genus-2 Whitham equations. The direct numerical simulations on the defocusing complex modified KdV equation show remarkable agreement with the results from Whitham modulation theory.

This study presents semi-analytical solutions for the fractional Fornberg–Whitham (FFW) equation using the Fractional Temimi–Ansari Method (FTAM). Fractional derivatives are defined in the senses of Atangana–Baleanu–Caputo (ABC) and Caputo. The existence and uniqueness of solutions are rigorously examined. Furthermore, a detailed discussion of the FTAM framework, including convergence analysis is provided. The accuracy of the obtained solutions is validated through comparisons with exact solutions when the fractional order is . Another key objective of the study is to compare the two fractional derivative definitions to assess the extent to which each captures memory effects from the past. Owing to its Mittag-Leffler-type kernel, the ABC derivative enhances memory effects and promotes faster stabilization.

In this article, we also introduced two well-known computational techniques for solving the time-fractional Fornberg–Whitham equations. The methods suggested are the modified form of the variational iteration and Adomian decomposition techniques by ρ-Laplace. Furthermore, an illustrative scheme is introduced to verify the accuracy of the available methods. The graphical representation of the exact and derived results is presented to show the suggested approaches reliability. The comparative solution analysis via graphs also represented the higher reliability and accuracy of the current techniques.

Dispersive shock waves (DSWs), also termed undular bores in fluid mechanics, governed by the non-local Whitham equation are studied in order to investigate short wavelength effects that lead to peaked and cusped waves within the DSW. This is done by combining the weak nonlinearity of the Korteweg–de Vries equation with full linear dispersion relations. The dispersion relations considered are those for surface gravity waves, the intermediate long wave equation and a model dispersion relation introduced by Whitham to investigate the 120° peaked Stokes wave of highest amplitude. A dispersive shock fitting method is used to find the leading (solitary wave) and trailing (linear wave) edges of the DSW. This method is found to produce results in excellent agreement with numerical solutions up until the lead solitary wave of the DSW reaches its highest amplitude. Numerical solutions show that the DSWs for the water wave and Whitham peaking kernels become modulationally unstable and evolve into multi-phase wavetrains after a critical amplitude which is just below the DSW of maximum amplitude.

Complex and nonlinear fractal equations are ubiquitous in natural phenomena. This research employs the fractal Euler−Lagrange and semi-inverse methods to derive the nonlinear space–time fractal Fornberg–Whitham equation. This derivation provides an in-depth comprehension of traveling wave propagation. Consequently, the nonlinear space–time fractal Fornberg–Whitham equation is pivotal in elucidating fundamental phenomena across applied sciences. A novel analytical technique, the generalized Kudryashov method, is presented to address the space–time fractal Fornberg–Whitham equation. This method combines the fractional complex approach with the modified Kudryashov method to enhance its effectiveness. We derive an analytical solution for the space–time fractal Fornberg–Whitham equation to elucidate how various parameters influence the propagation of new traveling wave solutions. Furthermore, Figures 1 through 6 analyze the impact of parameters α , β , b 1 , and k on these new traveling wave solutions. Our results show that the solitary wave solutions remain intact for both case 1 and case 2, regardless of the time fractional orders β . At the end, the manuscript discusses the implications of these findings for understanding complex wave phenomena, paving the way for further exploration and applications in wave propagation studies.

The complete classification of solutions to the defocusing complex modified Korteweg-de Vries (cmKdV) equation with the step-like initial condition is given by Whitham theory. The process of studying the solution of cmKdV equation can be reduced to explore four quasi-linear equations, which predicts the evolution of dispersive shock wave. The results obtained here are quite different from the defocusing nonlinear Schrödinger equation: the bidirectionality of defocusing nonlinear Schrödinger equation determines that there are two basic rarefaction and shock structures while in the cmKdV case three basic rarefaction structures and four basic dispersive shock structures are constructed which lead to more complicated classification of step-like initial condition, and wave patterns even consisted of six different regions while each of wave patterns is consisted of five regions in the defocusing nonlinear Schrödinger equation. Direct numerical simulations of cmKdV equation are agreed well with the solutions corresponding to Whitham theory.

Fractional Fornberg-Whitham equation with He?s fractional derivative is studied in a fractal process. The fractional complex transform is adopted to convert the studied fractional equation into a differential equation, and He's homotopy perturbation method is used to solve the equation. nema

The genus-1 Kadomtsev–Petviashvili (KP)-Whitham system is derived for both variants of the KP equation; namely the KPI and KPII equations. The basic properties of the KP-Whitham system, including symmetries, exact reductions and its possible complete integrability, together with the appropriate generalization of the one-dimensional Riemann problem for the Korteweg–de Vries equation are discussed. Finally, the KP-Whitham system is used to study the linear stability properties of the genus-1 solutions of the KPI and KPII equations; it is shown that all genus-1 solutions of KPI are linearly unstable, while all genus-1 solutions of KPII are linearly stable within the context of Whitham theory.

In this note, we study the wave breaking phenomena for the Fornberg-Whitham equation. By virtue of L 2 -conservation law of solutions, we establish a new wave-breaking criterion for this equation.