Analysis of Fuzzy-Fractional Fornberg–Whitham Models Using Extended He-Laplace Methodology

The Fornberg–Whitham equations (FWEs) are crucial models for capturing complicated wave dynamics, ranging from the ocean to plasma, and emphasizing the rich-interplay of nonlinearity and dispersion. These equations represent wave propagation in scenarios where linear approximations fail due to high nonlinearity and dispersion. These equations have applications in analyzing surface water waves, shallow water flows, and other physical frameworks where waves with both dispersion and nonlinearity relate over long distances. In this manuscript, fuzzy modeling of general fractional-order FWE is performed, and He-Laplace algorithm is proposed for the solution and analysis of general as well as modified FWEs. Triangular fuzzy numbers (TFNs) are used to encapsulate the uncertain wave behaviors by incorporating fuzzy parameter in the initial conditions. The resulting model is then solved via He-Laplace algorithm, and its precision is investigated by comparing its results with ADTM, VITM, q-HAShTM, and mVIM. In addition to crisp form, accuracy of the fuzzy solution at upper and lower bounds is also examined by calculating the residual errors. These numerical results are presented in the form of tables; moreover, various 2D, 3D, and contour plots are illustrated to analyze the obtained solution. It is discovered that the errors observed are negligible to verify the plausibility of the obtained solution. This study can help scholars in conducting fuzzy-fractional modeling of other phenomenon, as well as solving highly nonlinear fuzzy-fractional models through proposed algorithm.