Efficient approximate analytical technique to solve nonlinear coupled Jaulent–Miodek system within a time-fractional order

In this article, we considered the nonlinear time-fractional Jaulent–Miodek model (FJMM), which is applied to modeling many applications in basic sciences and engineering, especially physical phenomena such as plasma physics, fluid dynamics, electromagnetic waves in nonlinear media, and many other applications. The Caputo fractional derivative (CFD) was applied to express the fractional operator in the mathematical formalism of the FJMM. We implemented the modified generalized Mittag-Leffler method (MGMLFM) to show the analytical approximate solution of FJMM, which is represented by a set of coupled nonlinear fractional partial differential equations (FPDEs) with suitable initial conditions. The suggested method produced convergent series solutions with easily computable components. To demonstrate the accuracy and efficiency of the MGMLFM, a comparison was made between the solutions obtained by MGMLFM and the known exact solutions in some tables. Also, the absolute error was compared with the absolute error provided by some of the other famous methods found in the literature. Our findings confirmed that the presented method is easy, simple, reliable, competitive, and did not require complex calculations. Thus, it can be extensively applied to solve more linear and nonlinear FPDEs that have applications in various areas such as mathematics, engineering, and physics

An efficient approximate-analytical method to solve time-fractional KdV and KdVB equations

In this article, we present the modified generalized Mittag-Leffler function method (MGMLFM) as an approximate- analytical method to give a proper solution of time-fractional Korteweg-de Vries (KdV) and Korteweg-de Vries-Burger’s (KdVB) equations, which have various applications in physics and applied mathematics. The time-fractional partial derivatives are based on Caputo sense. The obtained solution is constructed in a rapidly convergent power series. By comparing the approximate MGMLFM solutions when the fractional operator equal one with known exact solutions we have an appropriate agreement. The advantage of the article is to apply the suggested method to solve linear and nonlinear time-fractional partial differential equations, where it needs less computational effort which saves time and effort. The convergence of absolute error be controlled on by the parameters in the time- fractional KdV and KdVB equations were found. The simulation of the obtained results is presented in the forms of graphs to illustrate the reliability and efficiency of our method

Efficient Approaches for Solving Systems of Nonlinear Time-Fractional Partial Differential Equations

In this work, we present a modified generalized Mittag–Leffler function method (MGMLFM) and Laplace Adomian decomposition method (LADM) to get an analytic-approximate solution for nonlinear systems of partial differential equations (PDEs) of fractional-order in the Caputo derivative. We apply the MGMLFM and LADM on systems of nonlinear time-fractional PDEs. Precisely, we consider some important fractional-order nonlinear systems, namely Broer–Kaup (BK) and Burgers, which have found major significance because they arise in many physical applications such as shock wave, wave processes, vorticity transport, dispersal in porous media, and hydrodynamic turbulence. The analysis of these methods is implemented on the BK, Burgers systems and solutions have been offered in a simple formula. We show our results in figures and tables to demonstrate the efficiency and reliability of the used methods. Furthermore, our outcome converges rapidly to the given exact solutions