Exploring new traveling wave solutions by solving the nonlinear space–time fractal Fornberg−Whitham equation

Abstract

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\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document}, β,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\upbeta ,$$\\end{document}b1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b_{1}$$\\end{document}, and k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k$$\\end{document} 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\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${ }\\left( \\upbeta \\right)$$\\end{document}. 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.