Most complex physical phenomena are described by non-linear Burgers’ equations, which help us understand them better. This article uses the transformation and the fractional Taylor’s formula to find approximate solutions for non-linear fractional-order partial differential equations. Solving non-linear Burgers’ equations with the right starting data shows that the method utilized is correct and can be utilized. Based on the limit of the idea, a rapid convergence McLaurin series is used to obtain close series solutions for both models with less work and more accuracy. To see how time-Caputo fractional derivatives affect how the results of the above models behave, in three dimension figures are drawn. The results showed that the proposed method is an easy, flexible, and helpful way to solve and understand a wide range of non-linear physical models.

In this study, we use a dual technique that combines the Laplace residual power series method (LRPSM) and the new iteration method, both of which are combined with the Caputo operator. Our primary goal is to solve two unique but difficult partial differential equations: the foam drainage equation and the nonlinear time-fractional Fisher’s equation. These equations, which are crucial in modeling complex processes, confront analytical complications, owing to their fractional derivatives and nonlinear behavior. We develop exact and efficient solutions by merging these unique methodologies, which are supported by thorough figures and tables that demonstrate the precision and trustworthiness of our methodology. We not only shed light on the solutions to these equations, but also demonstrate the prowess of the LRPSM and the new iteration method as powerful tools for grappling with complex mathematical and physical models, significantly contributing to advancements in various scientific domains.

A numerical study based on finite difference approximation is attempted to analyze the bulk flow, micro spin flow and heat transfer phenomenon for micropolar fluids dynamics through Darcy porous medium. The fluid flow mechanism is considered over a moving permeable sheet. The heat transfer is associated with two different sets of boundary conditions, the isothermal wall and isoflux boundary. On the basis of porosity of medium, similarity functions are utilized to avail a set of ordinary differential equations. The non-linear coupled ODE’s have been solved with a very stable and reliable numerical scheme that involves Simpson’s Rule and Successive over Relaxation method. The accuracy of the results is improved by making iterations on three different grid sizes and higher order accuracy in the results is achieved by Richardson extrapolation. This study provides realistic and differentiated results with due considerations of micropolar fluid theory. The micropolar material parameters demonstrated reduction in the bulk fluid speed, thermal distribution and skin friction coefficient but increase in local heat transfer rate and couple stress. The spin behavior of microstructures is also exhibited through microrotation vector $N(\eta )$ .

In this paper, we propose a mathematical model to describe the influence of the SARS-CoV-2 virus with correlated sources of randomness and with vaccination. The total human population is divided into three groups susceptible, infected, and recovered. Each population group of the model is assumed to be subject to various types of randomness. We develop the correlated stochastic model by considering correlated Brownian motions for the population groups. As the environmental reservoir plays a weighty role in the transmission of the SARS-CoV-2 virus, our model encompasses a fourth stochastic differential equation representing the reservoir. Moreover, the vaccination of susceptible is also considered. Once the correlated stochastic model, the existence and uniqueness of a positive solution are discussed to show the problem’s feasibility. The SARS-CoV-2 extinction, as well as persistency, are also examined, and sufficient conditions resulted from our investigation. The theoretical results are supported through numerical/graphical findings.

In this study, we presented the conformable Laplace transform iterative method to find the approximate solution of the systems of nonlinear temporal-fractional differential equations in the sense of the conformable derivative. The advantage of the suggested approach was to compute the solution without discretization and restrictive assumptions. Three distinct examples were provided to show the applicability and efficacy of the proposed approach. To examine the exact and approximate solutions, we utilized the 2D and 3D graphics. Furthermore, the outcomes produced in this study were consistent with the exact solutions; hence, this strategy efficiently and effectively determined exact and approximate solutions to nonlinear temporal-fractional differential equations.

We propose a theoretical study to investigate the spread of the SARS‐CoV‐2 virus, reported in Wuhan, China. We develop a mathematical model based on the characteristic of the disease and then use fractional calculus to fractionalize it. We use the Caputo-Fabrizio operator for this purpose. We prove that the considered model has positive and bounded solutions. We calculate the threshold quantity of the proposed model and discuss its sensitivity analysis to find the role of every epidemic parameter and the relative impact on disease transmission. The threshold quantity (reproductive number) is used to discuss the steady states of the proposed model and to find that the proposed epidemic model is stable asymptotically under some constraints. Both the global and local properties of the proposed model will be performed with the help of the mean value theorem, Barbalat’s lemma, and linearization. To support our analytical findings, we draw some numerical simulations to verify with graphical representations.