In this paper, the Elzaki transform decomposition method is implemented to solve the time-fractional Swift–Hohenberg equations. The presented model is related to the temperature and thermal convection of fluid dynamics, which can also be used to explain the formation process in liquid surfaces bounded along a horizontally well-conducting boundary. In the Caputo manner, the fractional derivative is described. The suggested method is easy to implement and needs a small number of calculations. The validity of the presented method is confirmed from the numerical examples. Illustrative figures are used to derive and verify the supporting analytical schemes for fractional-order of the proposed problems. It has been confirmed that the proposed method can be easily extended for the solution of other linear and non-linear fractional-order partial differential equations.

In the present article, we related the analytical solution of the fractional-order dispersive partial differential equations, using the Laplace–Adomian decomposition method. The Caputo operator is used to define the derivative of fractional-order. Laplace–Adomian decomposition method solutions for both fractional and integer orders are obtained in series form, showing higher convergence of the proposed method. Illustrative examples are considered to confirm the validity of the present method. The fractional order solutions that are convergent to integer order solutions are also investigated.

In this paper, we find the solution of the fractional-order Kaup–Kupershmidt (KK) equation by implementing the natural decomposition method with the aid of two different fractional derivatives, namely the Atangana–Baleanu derivative in Caputo manner (ABC) and Caputo–Fabrizio (CF). When investigating capillary gravity waves and nonlinear dispersive waves, the KK equation is extremely important. To demonstrate the accuracy and efficiency of the proposed technique, we study the nonlinear fractional KK equation in three distinct cases. The results are given in the form of a series, which converges quickly. The numerical simulations are presented through tables to illustrate the validity of the suggested technique. Numerical simulations in terms of absolute error are performed to ensure that the proposed methodologies are trustworthy and accurate. The resulting solutions are graphically shown to ensure the applicability and validity of the algorithms under consideration. The results that we obtain confirm that the proposed method is the best tool for handling any nonlinear problems arising in science and technology.

This article applies the homotopy perturbation transform technique to analyze fractional-order nonlinear fifth-order Korteweg–de-Vries-type (KdV-type)/Kawahara-type equations. This method combines the Zain Ul Abadin Zafar-transform (ZZ-T) and the homotopy perturbation technique (HPT) to show the validation and efficiency of this technique to investigate three examples. It is also shown that the fractional and integer-order solutions have closed contact with the exact result. The suggested technique is found to be reliable, efficient, and straightforward to use for many related models of engineering and several branches of science, such as modeling nonlinear waves in different plasma models.

This article investigates different nonlinear systems of fractional partial differential equations analytically using an attractive modified method known as the Laplace residual power series technique. Based on a combination of the Laplace transformation and the residual power series technique, we achieve analytic and approximation results in rapid convergent series form by employing the notion of the limit, with less time and effort than the residual power series method. Three challenges are evaluated and simulated to validate the suggested method’s practicability, efficiency, and simplicity. The analysis of the acquired findings demonstrates that the method mentioned above is simple, accurate, and appropriate for investigating the solutions to nonlinear applied sciences models.