Different techniques to solving fuzzy time-fractional Navier-Stokes equations with fuzzy nonlinear fractional modified KdV equation

In this paper, we provide the fuzzy time-fractional Navier-Stokes equations (NSEs) by substituting the time derivative with the Katugampola fractional derivative (KFD), which is expressed in Caputo type. The simplified fuzzy generalized Laplace residual power series method (GLRPSM) is used to derive an analytical solution to the fuzzy time-fractional NSEs. The proof of convergence to the proposed solution is established. Additionally, we present a novel method that combines the residual power series approach with the Laplace transform, abbreviated as (LRPSM), to produce a series solution for the fuzzy time-fractional NSEs. The suggested technique for solving fuzzy fractional physical equations was constructed using Laurent series. The new procedure’s most notable features are its speed and precision in deriving an exact or approximate solution. Two fuzzy time-fractional NSEs that describe the motion of a pipe’s flow were analyzed by the suggested approaches. Our findings are compared to the results of previous studies that were conducted to show the technique’s accuracy and effectiveness. In addition, the fractional derivatives are used to present the fuzzy fractional modified Korteweg-de Vries equation (fmKdV) and fKdV. The technique is predicated mostly on the approximate two-dimensional differential transform method (DTM). The technique is suitable for numerous problems and has the capacity to reduce the amount of computing work. In the sense of the Caputo derivative, the fractional derivative is described. Finally, we establish the approximate analytical solutions to fmKdV; the fuzzy homotopy-perturbation method (HPM) is used. To produce the fuzzy fmKdV, replace the first-order time and space derivatives with fractional derivatives of order α and β with α > 0, β ≤ 1. The provided approaches to solving diverse nonlinear fuzzy fractional-order partial differential equations are interesting, easy, and very accurate.