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In this article, analytical solutions of couple stress fluid flow modeled with a power law fractional differential operator are discussed. Stokes’ second problem for an incompressible couple stress fluid is studied for an horizontal plate of infinite length. The governing equations of the flow problem are expressed in terms of a partial differential operator and then converted into a non-dimensional model by using dimensional analysis. Then the integer order problem was formulated in terms of the non-integer order of three types of fractional derivatives and then solved with the help of the Laplace transform method. The obtained solutions are complex and expressed in terms of series. In order to check the memory index of the solutions obtained with three different fractional operators, we have plotted some graphs. It is found that the constant proportional operator provides us a better choice about the memory and maximum enhancement achieved in the comparison of Caputo and Caputo–Fabrizio. Furthermore, in order to check the accuracy of the present results, we have compared the obtained solutions with the existing literature and found a good agreement between them.

It is a well-known fact that functional effects like relaxation and retardation of materials, and heat transfer phenomena occur in a wide range of industrial and engineering problems. In this context, a mathematical model is developed in the view of Caputo fractional derivative for Oldroyd-B nano-fluid. Nano-sized particles of copper (Cu) are used to prepare nano-fluid taking water as the base fluid. The coupled non-linear governing equations of the problem are transformed into dimensionless form. Finite difference scheme is developed and applied successfully to get the numerical solutions of deliberated problem. Influence of different physical parameters on fluid velocity profile and temperature profile are analyzed briefly. It is observed that for increasing values of fractional parameter (α), fluid velocity increased, but opposite behavior was noticed for temperature profile. Nusselt number (Nu) decayed for advancement in values of heat source/sink parameter (Q0), radiation parameter (Nr), volume fraction parameter of nano-fluid (ϕ), and viscous dissipation parameter (Ec). Skin friction (Cf) boosts for the increase in the values of magnetic field parameter (Ha). It can also be noticed that the extended finite difference scheme is an efficient tool and gives the accurate results of discussed problem. It can be extended for more numerous type heat transfer problems arising in physical nature with complex geometry.

In this article, we are attracted to discover the multiple-optical soiltons in birefringent fibers for Radhakrishnan–Kundu–Lakshmanan equation (RKLE) by applying the Sardar-subequation method (SSM) and the new extended hyperbolic function method (EHFM). We construct the solutions in the form of exponential, trigonometric, and hyperbolic functions solitons solutions like mixed complex solitons and multiple-optical solitons solutions. In addition, singular periodic wave solutions are constructed, and the restraint conditions for the presence of soliton solutions are also defined. Moreover, the physical interpretation of the obtained solutions is disclosed in forms of 3D and 2D plots for different suitable parameters. The attained results indicate that the implemented computational scheme is straight, proficient, and brief and can be applied in more complex phenomena with the associate of representative computations. We have obtained several sorts of new solutions.

This article aims to address the exact solution of the prestigious partial differential equation, namely, a double dispersive equation. Here, we are obtaining some new traveling wave solutions of the double dispersive equation with the more general mathematical technique, which is a direct algebraic extended method. This proposed technique is more general and integrated. The obtained solutions contain dark, bright, dark–bright, singular, periodic, kink, and rational function solutions. More illustration of traveling wave solutions of the double dispersive equation is given by plotting the two- and three-dimensional graphs with the suitable selection of parameters. This graphical presentation of solutions identifies the pattern of wave propagation. The acquired consequences are new and may play a significant role to examine the physical phenomena of wave propagation, where this model is used.

In this piece of research, our aim is to investigate the novel solitons solutions of nonlinear (4+1)-dimensional Fokas equation (FE) and (2+1)-dimensional Breaking soliton equation (BSE) via new extended direct algebraic method. New acquired solutions are bright, singular, dark, periodic singular, combined dark-bright and combined dark-singular solitons along with hyperbolic and trigonometric functions solutions. We achieved different kinds of solitons solutions contain key applications in engineering and physics. By taking the appropriate values of these parameters, numerous novel structures are also plotted. These solutions define the wave performance of the governing models, actually. Furthermore, the physical understanding of the acquired solutions is revealed in forms of 3-D, 2-D and contour graphs for different appropriate parameters. From results, we conclude that the applied computational method is straight, talented and can be applied in more complex phenomena for such models.

This work deals the construction of novel soliton solutions to the Atangana–Baleanu (AB) fractional system of equations for the ion sound and Langmuir waves by using Sardar-subequation method (SSM). The outcomes are in the form of bright, singular, dark and combo soliton solutions. These solutions have wide applications in the arena of optoelectronics and wave propagation. The bright solitons will be a vast advantage in controlling the soliton disorder, dark solitons are also beneficial for soliton communication when a background wave exists and singular solitons only elaborate the shape of solitons and show a total spectrum of soliton solutions created from the model. These results would be very helpful to study and understand the physical phenomena in nonlinear optics. The performance of the SSM shows that this is powerful, talented, suitable and direct technique to discover the exact solutions for a number of nonlinear fractional models.