Exploring new traveling wave solutions by solving the nonlinear space-time fractal Fornberg-Whitham equation.

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 , , and 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 . 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.