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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>β</mml:mi> <mml:mo>,</mml:mo></mml:mrow> </mml:math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>b</mml:mi> <mml:mn>1</mml:mn></mml:msub> </mml:math> , and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow></mml:mrow> <mml:mfenced><mml:mi>β</mml:mi></mml:mfenced> </mml:mrow> </mml:math> . 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.