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Copper nanoparticles are widely used in many sectors and research endeavors owing to their unique properties, including a large surface area, catalytic capabilities, and high thermal and electrical conductivity. The selection of the base fluid for copper nanoparticles should be contingent upon the anticipated application requirements since various fluids exhibit distinct characteristics that could potentially impact the mobility of the nanoparticles. The present investigation analyzes heat transfer phenomena occurring across a radially stretched surface. The research explores the effects of different states of Cu nanoparticles when combined with base fluids, such as water and silicone oil, on the heat transfer process. The momentum and energy equations are transformed into nonlinear ordinary differential equations by applying the similarity transformation. The boundary value problem-fourth-order (BVP4C) method numerically solves the governing ordinary differential equation for the modeled problem. In addition, the influence of various factors such as the slip parameter, solid volume fraction, Eckert number, Prandtl number, and unsteadiness parameter are examined. It has been discovered that blade-shaped nanoparticles transfer heat as quickly as possible via silicone oil and water. However, for platelet-shaped nanoparticles, a minimum heat transfer rate has been noted. The maximum skin friction coefficient is observed in platelet-shaped nanoparticles, while blade-shaped nanoparticles have the lowest skin friction coefficient.

This article presents a comprehensive literature survey on fractional-order optimal control problems. Fractional-order differential equation is extensively used nowadays to model real-world systems accurately, which exhibit fractal dimensions, memory effects, as well as chaotic behaviour. These versatile features attract engineers to concentrate more on this, and it is widely used in the broad domain of science and technology. The mentioned numerical tools take the necessary optimal conditions into account, which makes it a two-point boundary value problem of non-integer order. In this review article, some numerical approaches for the approximation have been stated for obtaining the solution to fractional optimal control problems (FOCPs). Here, few numerical approaches including Grunwald-Letnikov approximation, Adams type predictor-corrector method, generalized Euler’s method, Caputo-Fabrizio method Bernoulli and Legendre polynomials method, Legendre operational method, and Ritz’s and Jacobi’s method are treated as an advanced method to obtain the solution of FOCP. Fractional delayed optimal control is selected for our investigation. It refers to a type of control problem where the control action is delayed by a fractional amount of time. In other words, the control input at a given time depends not only on the current state of the system but also on its past state at fractional times. The fractional delayed optimal control problem is formulated as an optimization problem that seeks to minimize a cost function subject to a set of constraints that represent the dynamics of the system and the fractional delay in the control input. The solution to this problem typically involves the use of fractional polynomials types, i.e. Chebyshev and Bassel polynomials.

The principal focal point of the current review is the second-grade nanofluid (SGNF) stream with slanted magnetohydrodynamics and viscous disintegration impacts across a moving level flat surface with entropy investigation. Here, we have done a comparative study on copper–methanol and iron–methanol second-grade nanoliquids. The governing conditions of the SGNF model are changed into ordinary differential equations (ODEs) by using supportive changes. To tackle the non-straight ODEs, the Runge-Kutta Fehlberg-45 procedure is utilized. The result reveals that the velocity gradient of copper–methanol second-grade nanoliquid is strongly affected by suction, magnetic, and second-grade fluid parameters and declines faster when compared to iron–methanol second-grade nanoliquid. Copper–methanol SGNF shows improved heat transfer than iron–methanol SGNF for improved values of Eckert and Biot numbers.

In this article, we investigate the mechanics of breathing performed by a ventilator with different kernels by an effective integral transform. We mainly obtain the solutions of the fractional respiratory mechanics model. Our goal is to give the underlying model flexibly by making use of the advantages of the non-integer order operators. The big advantage of fractional derivatives is that we can formulate models describing much better the systems with memory effects. Fractional operators with different memories are related to different types of relaxation process of the non-local dynamical systems. Additionally, since we consider the utilisation of different kinds of fractional derivatives, most often having benefit in the implementation, the similarities and differences can be obviously seen between these derivatives.

The aim of this study is to investigate the numerical analysis of an innovative model containing, bioconvection phenomena with a gyrotactic motile microorganism of magnetohydrodynamics Williamson nanofluids flow along with heat and mass transfer past a stretched surface. The effect of thickness variation and thermal conductivity feature is employed in the model. Bioconvection in nanofluid helps in bioscience such as in blood flow, drug delivery, micro-enzyme, biosensors, nanomedicine, for content detection, etc. For simulation procedure, the mathematical partial differential equations are converted into dimensionless systems owing to dimensionless variations such as magnetic field, power index velocity, Williamson parameter, wall thickness parameter, thermal conductivity variation, Prandtl number, thermal radiation, Brownian motion, Lewis number, Peclet number, and different concentration parameter, etc. For numerical simulation, New Iterative Technique (NIM) numerical algorithm is adopted and employed for the linear regression planned for the proposed model. For comparison purposes, the homotopy technique is employed on the flow model. Close agreement is seen between both methods revealing the accuracy and consistency of NIM numerical technique. Many features of no-scale constraints are evaluated through graphical data for a key profile of the flow model. Results show that microorganism concentration is heavy due to the magnetic effect and Hall current.

The mathematical modeling of hybrid nanofluid flow and heat transfer with entropy generation toward parabolic trough surface collector (PTSC) inside the solar-powered ship (SPS) is performed. The mathematical model used non-Newtonian Oldroyd-B model amidst a constant inclined magnetic field influence is being considered. The mathematical model is then reduced by adopting appropriate similarity transformation into a higher-order nonlinear differential equations system. The reduced model is computed using the well-known technique called the Keller Box scheme. Physical parameters effectiveness, for instance, thermal radiation, viscous dissipation, hybrid nanoparticles, and Joule heating, is displayed in graphs. The silver-ethylene glycol (Ag-EG) characteristic performance outperformed the silver-magnetite-ethylene glycol (Ag-Fe3O4/EG). The maximum efficiency of Ag-EG is about 26.3%, while the minimum is at least 5.6%.

In this research, we analyze the magnetohydrodynamics heat act of a viscous incompressible Jeffrey nanoliquid, which passed in the neighborhood of a linearly extending foil. As a process, we employ alumina (Al2O3)\left({{\rm{Al}}}_{2}{{\rm{O}}}_{3}) as nanoparticles, assuming that the base fluid is ethylene glycol. In this involvement, we consider the heating by Joule effect and viscous dissipation. We select the passable transformations, motion, and temperature formulas converting into non-linear differential equation arrangement. We solved the system by using a Keller-box method. Then, we provide a graphical description of outcomes according to the selected control parameters. Higher values of dissipation parameter cause a surge in temperature field as well as strengthen width of the heat boundary layer. The velocity, drag coefficient, and heat transfer (HT) rate for the base fluid are comparatively greater than that of the Al2O3{{\rm{Al}}}_{2}{{\rm{O}}}_{3}–ethylene glycol nanofluid, although the temperature is embellished by the inclusion of nanoparticles. Moreover, we report depreciation in surface drag as well as HT by the virtue of amplification in the Deborah number. The proclaimed outcomes are advantageous to boost the incandescent light bulb’s, cooling and heating processes, filament emitting light, energy generation, multiple heating devices, etc.

The purpose of the present investigation is to examine the heat, mass and microorganism concentration transfer rates in the magnetohydrodynamics (MHD) stratified boundary layer flow of tangent hyperbolic nanofluid past a linearly, uniform stretching surface comprising gyrotactic microorganisms as well as nanoparticles. The governing PDEs with relevant end point conditions are molded into a non-dimensional ordinary differential equation (ODE) form by means of the similarity transformation. The numerical solution of dimensionless problem is acquired within the frame of robust Keller-Box technique. The velocity, temperature, mass and motile microorganism density are investigated graphically within the context of different significant parameters. Numerical results have been inspected via plots and table (namely as the local Nusselt number, the local wall mass flux and the local microorganisms wall flux). This article proves that the energy, concentration and motile microorganism density reduce with increase in thermal, solutal and motile density stratification parameters. The asserted outcomes are beneficial to enhance the cooling and heating processes, energy generation, thermal machines, solar energy systems, industrial processes etc.

In nanofluids, the effect of convection in the presence of double diffusivity on a magneto couple stress fluid with the peristaltic flow of a model in a non-uniform channel (MCSFM) is reviewed in this article. This research discusses MCSF in a non-uniform channel by applying the Levenberg–Marquardt procedure via an artificial backpropagated neural network (LMP-ABNN). For two-dimensional and two-directional flows, mathematical formulations of double-diffusivity convection of a magneto couple stress fluid in nanofluids are developed. The partial differential equations are reduced to ordinary differential equations by using appropriate transformations. The assessment of the Hartmann number, thermophoresis parameter, Dufour parameter, Soret parameter, and magnetic Reynolds number over concentration profiles and temperature profiles is made by generating a dataset for LMP-ABNN based on the ND solve method for different variations of MSCFM. To examine the approximate solution validation, training and testing procedures are interpreted, and the performance is verified through error histogram and mean square error results. The extremely nonlinear equations are reduced by employing a long-wavelength approximation and a low but finite Reynolds number. To describe the behavior of flow quantities, graphical representations of a variety of physical characteristics of importance are shown. The impact of the Hartmann number and magnetic Reynolds number over axial magnetic field and current density is also studied. The concentration increases as the thermophoresis parameter and Dufour parameter values increase. This occurs because the concentration and both these parameters have a direct relationship. We observed opposite behavior for both the magnetic Reynolds number and the Hartman number. The behavior of current density J z increases with increasing values of R m. Both the temperature distribution and solute concentration increase. The final outcome of this study is to provide the potential for these techniques to provide new insights and solutions to challenging problems in nanofluids and other areas of fluid mechanics and to facilitate the design of more efficient and effective microfluidic devices.

Nanofluid bioconvective channel flow is an essential aspect of the recent healthcare industry applications, such as biomedical processing systems. Thus, the present work examined the influence of nth order chemical reaction in an unsteady nanofluid bioconvective channel flow in a horizontal microchannel with expanding/contracting walls. The suitable form of the similarity transformation is exercised to transform the governing boundary layer equations into a more straightforward form of system to ease the computation process. The Runge–Kutta method of fifth-order integration technique solved the reduced boundary layer system and generated the numerical results as the governing parameters vary. It is found that the destructive second-order chemical reaction enhances the mass transfer rate at the lower wall but deteriorates the mass transfer rate at the upper wall. The upper channel wall has a better heat transfer rate than the lower wall when the Reynolds number increases.

In the present paper, we establish some compositions formulas for Marichev-Saigo-Maeda (MSM) fractional calculus operators with k-Struve function Sν,ckS_{\nu ,c}^k as of the kernel. The results are presented in terms of generalized k-Wright function pΨqk_p\Psi _q^k .

Tick-borne encephalitis (TBE) is a flaviviral vector-borne disease, which is spread by a tick named Ixodes persulcatus in domestic animals as well as in humans. In this article, susceptible, exposed, infected, recovered model; with no immunity after getting recovered is taken. The only possible immunity is before getting the disease (in our model). The vaccination details are also discussed in the article. Hence, SEIS (susceptible, exposed, infected and again susceptible with zero removal from the specie compartment) is used to construct a mathematical model of TBE. TBE is acute inflammation of the brain parenchyma. After becoming viral in European states and some Asian countries, especially in China, this is an emerging viral disease in Pakistan. After constructing a model, formula for the basic reproduction number R 0-like threshold has been derived by using the next-generation matrix method. The formula for R 0-like threshold is used to evaluate whether the disease is going to be outbroken in the respective area from which the specific data are taken into consideration. The main motivation behind selection of this topic is to address the unawareness of this disease specifically in Pakistan and in its neighboring countries when there persists probability for the outbreak of this disease. Some equilibrium points and their local stability is also discussed. Numerical computations and graphs are also presented to validate the results.

This article concerns with the development of the number of focal values. We analyzed periodic solutions for first-order cubic non-autonomous ordinary differential equations. Bifurcation analysis for periodic solutions from a fine focus z=0{\mathfrak{z}}=0 is also examined. In particular, we are interested to detect the maximum number of periodic solutions for various classes of higher order in which a given solution can bifurcate under perturbation of the coefficients. We calculate the maximum number of periodic solutions for different classes, namely, C10,5{C}_{10,5} and C12,6{C}_{12,6} with trigonometric coefficients, and they are found with nine and eight multiplicities at most. The classes C8,3{C}_{8,3} and C8,4{C}_{8,4} with algebraic coefficients have at most eight limit cycles. The new formula ϰ10{\varkappa }_{10} is developed by which we succeeded to find highest known multiplicity ten for class C9,3{C}_{\mathrm{9,3}} with polynomial coefficient. Periodicity is calculated for both trigonometric and algebraic coefficients. Few examples are also considered to explain the applicability and stability of the methods presented.

In this study, the behavior of a microchannel flow is examined. The fluid is considered to be a nanofluid, which moves between two parallel flat plates in the presence of an electrical double layer. The Buongiorno nanofluid is considered with body force. In this study, the unphysical supposition presented in the preceding work to the discontinuity of the flow fled where the electrostatic potential in the central of the canal must be equal to zero is removed. The incorrect supposition that the pressure constant is preserved, which is considered a known form, is corrected. The current fresh model equation is modified by using dimensionless parameters to convert partial differential equations into ordinary differential equations. The transformed nonlinear equations are solved by the homotopy analysis method. The physical parameters, magnetic parameters, Eckert number, Lewis number, Brownian motion parameters, thermophoresis parameters, and Prandtl number are analyzed. The influence of both the viscous and Joule dissipation in the presence of magnetohydrodynamic effect is examined.

The main purpose of this paper is to introduce a class of new integrals involving generalized Bessel functions and generalized Struve functions by using operational method and umbral formalization of Ramanujan master theorem. Their connections with trigonometric functions with several distinct complex arguments are also presented.