Most physical phenomena are formulated in the form of non-linear fractional partial differential equations to better understand the complexity of these phenomena. This article introduces a recent attractive analytic-numeric approach to investigate the approximate solutions for nonlinear time fractional partial differential equations by means of coupling the Laplace transform operator and the fractional Taylor’s formula. The validity and the applicability of the used method are illustrated via solving nonlinear time-fractional Kolmogorov and Rosenau–Hyman models with appropriate initial data. The approximate series solutions for both models are produced in a rapid convergence McLaurin series based upon the limit of the concept with fewer computations and more accuracy. Graphs in two and three dimensions are drawn to detect the effect of time-Caputo fractional derivatives on the behavior of the obtained results to the aforementioned models. Comparative results point out a more accurate approximation of the proposed method compared with existing methods such as the variational iteration method and the homotopy perturbation method. The obtained outcomes revealed that the proposed approach is a simple, applicable, and convenient scheme for solving and understanding a variety of non-linear physical models.

In this article, an attractive numeric–analytic algorithm, called the fractional residual power series algorithm, is implemented for predicting the approximate solutions for a certain class of fractional systems of partial differential equations in terms of Caputo fractional differentiability. The solution methodology combines the residual function and the fractional Taylor’s formula. In this context, the proposed algorithm provides the unknown coefficients of the expansion series for the governed system by a straightforward pattern as well as it presents the solutions in a systematic manner without including any restrictive conditions. To enhance the theoretical framework, some numerical examples are tested and discussed to detect the simplicity, performance, and applicability of the proposed algorithm. Numerical simulations and graphical plots are provided to check the impact of the fractional order on the geometric behavior of the fractional residual power series solutions. Moreover, the efficiency of this algorithm is discussed by comparing the obtained results with other existing methods such as Laplace Adomian decomposition and Iterative methods. Simulation of the results shows that the fractional residual power series technique is an accurate and very attractive tool to obtain the solutions for nonlinear fractional partial differential equations that occur in applied mathematics, physics, and engineering.

This article employs the Laplace residual power series approach to study nonlinear systems of time-fractional partial differential equations with time-fractional Caputo derivative. The proposed technique is based on a new fractional expansion of the Maclurian series, which provides a rapid convergence series solution where the coefficients of the proposed fractional expansion are computed with the limit concept. The nonlinear systems studied in this work are the Broer-Kaup system, the Burgers’ system of two variables, and the Burgers’ system of three variables, which are used in modeling various nonlinear physical applications such as shock waves, processes of the wave, transportation of vorticity, dispersion in porous media, and hydrodynamic turbulence. The results obtained are reliable, efficient, and accurate with minimal computations. The proposed technique is analyzed by applying it to three attractive problems where the approximate analytical solutions are formulated in rapid convergent fractional Maclurian formulas. The results are studied numerically and graphically to show the performance and validity of the technique, as well as the fractional order impact on the behavior of the solutions. Moreover, numerical comparisons are made with other well-known methods, proving that the results obtained in the proposed technique are much better and the most accurate. Finally, the obtained outcomes and simulation data show that the present method provides a sound methodology and suitable tool for solving such nonlinear systems of time-fractional partial differential equations.