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New Efficient Computations with Symmetrical and Dynamic Analysis for Solving Higher-Order Fractional Partial Differential Equations

Due to the rapid development of theoretical and computational techniques in the recent years, the role of nonlinearity in dynamical systems has attracted increasing interest and has been intensely investigated. A study of nonlinear waves in shallow water is presented in this paper. The classic form of the Korteweg–de Vries (KdV) equation is based on oceanography theory, shallow water waves in the sea, and internal ion-acoustic waves in plasma. A shallow fluid assumption is shown in the framework by a sequence of nonlinear fractional partial differential equations. Indeed, the primary purpose of this study is to use a semi-analytical technique based on Fractional Taylor Series to achieve numerical results for nonlinear fifth-order KdV models of non-integer order. Caputo is the operator used for dealing with fractional derivatives. The generated solutions of nonlinear fifth-order KdV models of non-integer order for modeling turbulence processes in the field of ocean engineering are compared analytically and numerically, to demonstrate the behaviors of several parameters of the current model. We verified the method’s convergence analysis and provided an error estimate by showing 2D and 3D graphs to further confirm its efficacy.