These days, heat transfer plays a significant role in the fields of engineering and energy, particularly in the biological sciences. Ordinary fluid is inadequate to transfer heat in an efficient manner, therefore, several models were considered for the betterment of heat transfer. One of the most prominent models is a single-phase nanofluid model. The present study is devoted to solving the problem of micropolar fluid with a single-phase model in a channel numerically. The governing partial differential equations (PDEs) are converted into nonlinear ordinary differential equations (ODEs) by introducing similarity transformation and then solved numerically by the finite difference method. Response surface methodology (RSM) together with sensitivity analysis are implemented for the optimization analysis. The study reveals that sensitivity of the skin friction coefficient (Cf_x) to the Reynolds number (R) and magnetic parameter (M) is positive (directly proportional) and negative (inversely proportional) for the micropolar parameter.

Fractional calculus is at this time an area where many models are still being developed, explored, and used in real-world applications in many branches of science and engineering where non-locality plays a key role. Although many wonderful discoveries have already been reported by researchers in important monographs and review articles, there is still a great deal of non-local phenomena that have not been studied and are only waiting to be explored. As a result, we can continually learn about new applications and aspects of fractional modelling. In this study, a precise and analytical method with non-singular kernel derivatives is used to solve the Caudrey–Dodd–Gibbon (CDG) model, a modification of the fifth-order KdV equation (fKdV). The fractional derivative is taken into account by the Caputo–Fabrizio (CF) derivative and the Atangana–Baleanu derivative in the Caputo sense (ABC). This model illustrates the propagation of magneto-acoustic, shallow-water, and gravity–capillary waves in a plasma medium. The dynamic behaviour of the acquired solutions has been represented in a number of two- and three-dimensional figures. A number of simulations are also performed to demonstrate how the resulting solutions physically behave with respect to fractional order. The significance of the current research is that new solutions are obtained by using a strong analytical approach. Utilizing a fractional derivative operator to solve equivalent models is another benefit of this approach. The results of the present work have similar aspects to the symmetry of partial differential equations.

The object of this paper is to bring out the space r^q_B(g,p) of non-absolute patterns. Also, we will structure its completeness property. Also, the Köthe duals will be determined. Moreover, the Schauder basis for it will be constructed.

In this work, the exact solutions of nonlinear wave equations are studied in the framework of fractional gas dynamics. Two powerful methods, such as the generalized projective Riccati equation method (GPREM) and modified extended simple equation method (MESM), are applied to solve the fractional gas dynamics equations, which are used to model complex wave phenomena, exactly and analytically. A central role in this approach is played by the Riccati equation, which we can use to obtain the soliton solutions, periodic waves, and other nonlinear wave patterns. We show that both GPREM and MESM give a systematic way to find exact and soliton solutions that are essential to understanding the dynamics of fractional systems and facilitate the introduction of a novel order of the gas dynamics equations, which poses a unique set of challenges. The physical interpretation of these solutions is discussed in terms of wave propagation, stability, and interaction of nonlinear waves. The findings demonstrate the applicability of the Riccati‐based techniques to the solution of fractional nonlinear evolution equations and will help in the development of theoretical models in fractional dynamics. This paper sets the groundwork for additional research into the use of fractional calculus in the study of fluid dynamics and wave propagation and could prove useful in other branches of physics and engineering.