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.

In this paper, we consider the stochastic Burgers' equation, which is forced by multiplicative noise in the Stratonovich sense. To get a new trigonometric and hyperbolic stochastic solutions, we apply $\exp (-\phi (\mu ))$-expansion method. In addition, we demonstrate the effect of multiplicative noise on the exact solutions of the Burgers' equation by introducing some graphic representations.

In this work, the novel iterative transformation technique and homotopy perturbation transformation technique are used to calculate the fractional-order gas dynamics equation. In this technique, the novel iteration method and homotopy perturbation method are combined with the Elzaki transformation. The current methods are implemented with four examples to show the efficacy and validation of the techniques. The approximate solutions obtained by the given techniques show that the methods are accurate and easy to apply to other linear and nonlinear problems.

This research deals with the mathematical model for the development of the peristaltic principle of the combination of the pressure and electroosmotic flow (EOF) of ionic liquid across microchannels with electrokinetic effects. For thermomechanical dynamics, the convective conditions on the boundary for mass and heat transfer at the walls of the channel are quantified. For the microchannel, a porous structure is presumed. Soret, Dufour, and Joule heating are also listed in the scope of the problem addressed. The corresponding equations for the ionic fluid flow, mass, and heat transfer along with the Poisson–Boltzmann equation within the electrical double layer (EDL) are studied. The exact solution has been obtained based on lubrication theory (i.e., low Reynolds number and long wavelength approximations). The channel height is therefore believed to be much higher than the electrical double layer (EDL) thickness. Various dimensionless pertinent parameters illustrate the important aspects of electroosmotically controlled flow and subsequent convective mass/heat transfer attributes in a microchannel. A linear dependency on the fluid flow rate is exhibited by the pressure drop. The analysis shows that the electroosmotic parameter gives a reducing effect on the channel permeability. The distribution of temperature and concentration is greatly affected by convective heat and mass parameters, respectively. In biomedical engineering, the application areas of the study proposed are for the design of the devices such as a microfluidic pump to pump a small amount of ionic liquids by regulating the variation in temperature and concentration.