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The human body regularly produces mucus, and its presence does not mean that something bad is going on. The boundary tissues of mammals, including the nose, throat and lungs, serve as a barrier against pollution and are produced by our respiratory system. Our work describes the flow comprehension of the components moulding the fluid elements of respiratory irresistible infections. A mathematical model is introduced to concentrate on the mucus fluid flow driven by natural convection through asymmetric channel. This study also summarized biological disorder of human lung build-up fluid. The nonlinear governing equations contain importance of heat transfer and fluid motion of mucus fluid. The analytic solutions of unsteady mucus flow through an isothermal channel with thermal conduction are accomplished adopting Laplace transform method. Significance of mucus fluid heat source thermal conductivity on momentum distribution and energy distribution is enumerated. The comparison of various dimensionless parameters is shown graphically with the help of MATLAB software. In order to provide some understanding on the behavior of mucus fluids and heat transfer mechanisms, extension of this study explores at how temperature influences mucus flow properties.

The time-fractional equal-width equation plays a vital role in plasma physics, serving as an essential tool for elucidating the dynamics of hydromagnetic waves in cold plasma, a critical component for the comprehensive understanding of plasma-related phenomena. Furthermore, this equation finds application in fluid mechanics, which models the transmission of nonlinear waves across shallow waters. In this study, we employ an innovative technique to solve time-fractional equal-width equation, demonstrating the q-homotopy analysis transform technique. This considered technique was implemented to find the effective approximated solution of the considered equations. The obtained results are discussed through the 3D plots and graphs that express the physical representation. The results derived from the q-homotopy analysis transform method are presented in a series form, exhibiting swift convergence and capturing the behavior of the equal-width equation’s solution with minimal error, and the convergence analysis and uniqueness of the solution using the projected method have been secured using the fixed-point theory. The projected method we used does not require linearization, perturbations, or discretization, which significantly reduces computations. The results disclose that the proposed technique is highly effective, methodical, and easy to apply for complex and nonlinear systems, helping us to capture the associated behavior of diverse classes of phenomena. The numerical simulations presented ensure higher accuracy and are compared with other techniques to evaluate approximate errors. In comparison with other techniques, the considered method is a competent tool to get an analytical solution of an considered equation and the methodology employed herein for the considered equation is demonstrated to be both efficient and dependable, marking a significant advancement in the field.

In recent times, researchers have increasingly directed their focus toward Reaction-Diffusion models, attracted by their versatile applications across various scientific domains. Within these models, the Schnakenberg Reaction-Diffusion System (SRDS) has gained significant attention for its ability to explain intricate phenomena such as oscillatory behavior, limit cycles, pattern formations and diffusion in biochemistry. This paper specifically delves into the Fractional Schnakenberg Reaction Diffusion System (FSRDS), an extension of SRDS that incorporates principles of fractional calculus. This extension provides a more comprehensive framework for understanding complex dynamics. The unique aspect of this work lies in the innovative approach used to derive an analytical solution for FSRDS – the Residual Power Series Method with Laplace Transform (L.T.)/Laplace Residual Power series Method (LRPSM). By employing LRPSM and considering the provided initial conditions, our objective is to unveil an analytical solution for FSRDS.

This research paper introduces a novel formulation of the modified Atangana–Baleanu (AB) Fractional Operators (FrOs). The paper begins by discussing the boundedness of the novel fractional derivative operator. Some fractional differential equations corresponding to different choices of functions as well as comparative graphical representations of a function and its derivative are provided. Furthermore, the paper investigates the generalized Laplace transform for this newly introduced operator. By employing the generalized Laplace transform, a wide range of fractional differential equations can be effectively solved. Additionally, the paper establishes the corresponding form of the AB Caputo fractional integral operator, examines its boundedness and obtains its Laplace transform. It is worth noting that the FrOs previously documented in the existing literature can be derived as special cases of these recently explored FrOs.

The development of thermal therapy always requires more reasonable temperature distribution predictions. Controlling the amount of heating is a common practice within general thermal therapy operations. This paper used a modified three-phase lag (TPL) bioheat transfer equation to describe the behavior of heat conduction in tissue with thermoelastic effect. To explore the effect of thermal load, the tissue was subjected to a constant surface temperature and a pulsed surface heat flux, respectively. The modified TPL bioheat transfer equation involves mixed derivative terms and higher-order time derivatives of temperature. In analyzing such problems, there are mathematical difficulties. Therefore, the hybrid numerical scheme based on the Laplace transform and an improved discrete method was proposed to solve the present problem. The influence of thermoelastic parameters on the behavior of heat transfer in tissue has been investigated. The results depict the effect of thermal load on thermal response within heat transfer medium is obvious. The thermoelastic effect excites the thermal response oscillation, which is intensified for the reduction of material constant characteristic $k_v$ and phase lag ${\tau }_v$, in the heat conduction medium.

To model several engineering and physical models, the approach of the fractional derivative is highly anticipated. As compared to the ordinary derivatives, the fractional derivatives with more flexibility can estimate the data due to the involvement of the fractional-order derivatives. Due to these advantages of the fractional approach, this study communicates with the determination of the fractional-based exact outcomes of an oscillatory rectangular duct problem of a generalized second-grade fluid. The approach of the fractional operator is involved in the relationship of the constitutive equations. For cosine oscillation of the rectangular duct, exact results of the magnetized unsteady flow problem are evaluated through the technique of Laplace transform with double finite Fourier sine transform. This study concludes that the velocity field exhibits escalating behavior relative to the improved fractional parameter. Moreover, the magnetic parameter with increasing values declines the flow field while the accelerating values of the fluid parameter enhance the velocity field.

This paper aims to contribute to and broaden the current understanding of fractional calculus and its uses in science. In general, determining the fractional derivative of a product of two functions is an obstacle. In this paper, we overcome this issue for Caputo and Riemann–Liouville fractional derivatives. Specifically, we provide explicit formulas for the derivatives of a product of two functions of order $\alpha \in (0,1)$ for these fractional derivatives. We also provided a few examples to help clarify and bolster our calculations.

A general solution for propagating waves in a generalized piezo-photo-thermoelastic medium for the one-dimensional (1D) problem under the hyperbolic two-temperature theory is investigated. The governing equations of the elastic waves, carrier density (plasma wave), quasi-static electric field, heat conduction equation, hyperbolic two temperature coefficient and constitutive relationships for the peizo-thermoelastic medium are obtained using Laplace transformation method in 1D. On the interface adjacent to the vacuum, mechanical stress loads, thermal and plasma boundary conditions are applied to obtain the main basic physical quantities in the Laplace domain. The inversion of Laplace transform by a numerical method is applied to obtain the complete solutions in the Laplace time domain for the main physical fields in this phenomenon. The effects on the force stress, displacement component, temperature distribution and carrier density of the thermoelastic, thermoelectric and hyperbolic two-temperature parameters by the applied force were graphically discussed.

For modeling of complex systems, in signal processing, and the explorations of nonlinear dynamics and memory effects, the utilization of noninteger-order dynamics provides adaptable control over oscillation patterns and frequencies. Moreover, various waveforms result in unique sonic qualities. By manipulation of these waveforms, musicians and sound designers have the capacity to craft a wide spectrum of auditory experiences, ranging from basic tones to intricate sound scape. This paper is about study of such vibrating systems using the noninteger-order derivative operator approach. In particular, we will discuss fractional relaxation oscillator and Scott-Blair oscillator employing constant proportional Caputo fractional derivative operator. Laplace transform method and Tzou’s numerical inversion algorithm are utilized to solve these vibrating models with respective initial conditions. A thorough graphical analysis is done to discuss the control of noninteger-order parameters and the parameters involved in the definition of fractional derivative operator. Finally, useful conclusions are recorded to highlight the behavior of oscillators and influence of noninteger-order parameters, the parameters involved in the definition of fractional derivative operator and system parameters that are helpful in controlling over oscillation pattern, frequencies and shape of vibrations produced by the vibrating systems.

Exact bound state solutions of the Dirac equation for the Kratzer potential in the presence of a tensor potential are studied by using the Laplace transform approach for the cases of spin- and pseudo-spin symmetry. The energy spectrum is obtained in the closed form for the relativistic as well as non-relativistic cases including the Coulomb potential. It is seen that our analytical results are in agreement with the ones given in the literature. The numerical results are also given in a table for different parameter values.

With an analytical solutions of DGLAP evolution equations based on the Laplace transform method, we find the fragmentation functions (FFs) of neutral mesons, ${\pi }^0$ and $k^0$ at NLO approximation. We also calculated the total fragmentation functions of these mesons and compared them with experimental data and those from global fits. The results show a good agreement between our solutions and other models and they are compatible with experimental data.

The convection–dispersion equation has always been a classic equation for studying pollutant migration models. There are certain deviations in scientific research because of the existence of the impurity of the medium and the nonsmooth boundary. In this paper, we introduced the one-dimensional convection–dispersion equation with fractal derivatives in fractal space, and established the fractal variational formula of the equation through the semi-inverse method. The fractal variational formula we have obtained can provide the conservation laws in an energy form in the fractal space and possible solution structures of the given equation. An analytical solution is obtained through the two-scale transform method and Laplace transform.

In this paper, we study a nonlinear fractional Damped Burger and Sharma–Tasso–Olver equation using a new novel technique, called homotopy perturbation transform method (FHPTM). There are three examples used to demonstrate and validate the proposed algorithm’s efficiency. This nonlinear model depicts nonlinear wave processes in fluid dynamics, ecology, solid-state physics, shallow-water wave propagation, optical fibers, fluid mechanics, plasma physics, and other applied science, engineering, and mathematical physics disciplines, as well as other phenomena. Numerous algebraic properties of the fractional derivative Caputo–Fabrizio operator are illustrated concerning the Laplace transformation to demonstrate their utility. Different graphs and tables compare the results obtained by R. Nawaz et al. [Alex. Eng. J. 60, 3205 (2021)] and M. S. Rawashdeh [Appl. Math. Inform. Sci. 9, 1239 (2015)]. The proposed scheme accelerates the convergence of the series solution and guarantees the convergence associated with the homotopy parameter. Furthermore, the physical nature of various fractional orders has been captured in plots. The obtained results demonstrate that the employed solution procedure is dependable and methodical in investigating the behaviors of nonlinear models of both integer and fractional orders.

In the field of molecular nonequilibrium transports, physical mechanisms of multiple reaction dynamics of these systems are the core of deep understanding complex reactions and transport mechanisms. In order to explore related mechanisms, establishing multiple systems coupled with tremendous exit dynamics and studying their exit dynamics properties are quite vital. Beyond previous researches, new stochastic transport processes are emphasized here. Multiple new exit dynamic systems are established, which are motivated by the multiplicity of paths and products of real biochemical processes in organisms. In order to ensure research universality, core system modeling factors are fully considered. Countable parallel orbits, uniform connection with external sources, countable parallel orbits as subsystems in middle lattices and influences of all lattices on transport trajectories on dynamic properties are analyzed. Dynamic properties of different particles located in orbits are explored by deeply studying average exit time and time scale. Quantitative spatiotemporal impacts are extensively studied. The rationality of average exit time as a time scale in the universal exit dynamic system is proved. Main findings and fruitful results can not only serve as theoretical bases for broadening reaction path modeling, but also be helpful to support understanding nonequilibrium transport mechanisms, especially stochastic biochemical processes.

Current-carrying nanowires are expected to be building blocks of the upcoming micro-nano-electromechanical devices, however, little is known on their dynamic interactions in a bundle. As a pivotal step towards realizing such a crucial mechanism, this work is devoted to vibrations and instability of a double-nanowire-system as an electric current carrier. Using Biot–Savart law, the Lorentz interactional forces between doubly parallel current-carrying nanowires are evaluated. Accounting for the surface elastic energy, equations of motion pertinent to the in-plane and out-of-plane vibrations are established. Using analytical techniques, the explicit expressions of both static and purely dynamic parts of the nanowires’ displacements are obtained. For each component of the transverse displacement field, two major vibration modes are observed: in-phase and out-of-phase modes. The frequencies associated with these vibration modes are analytically calculated. Further, the condition corresponds to the dynamic instability of the system is discovered, and the roles of initial tensile force, electric current, and interwire distance on frequencies and stability of the system are addressed.

In view of the generalization of Miller–Ross kernel in the sense of Riemann–Liouville type, we propose the new definitions of the general fractional integral (GFI) and general fractional derivative (GFD) to discuss the anomalous diffusion equation, which is distinct from those classic calculus operators. The obtained analytical solution of the application described in the graph is effective and accurate making the use of Laplace transform.

In this paper, the Homotopy perturbation Laplace method is implemented to investigate the solution of fractional-order Whitham–Broer–Kaup equations. The derivative of fractional-order is described in Caputo’s sense. To show the reliability of the suggested method, the solution of certain illustrative examples are presented. The results of the suggested method are shown and explained with the help of its graphical representation. The solutions of fractional-order problems as well as integer-order problems are determined by using the present technique. It has been observed that the obtained solutions are in significant agreement with the actual solutions to the targeted problems. Computationally, it has been analyzed that the solutions at different fractional-orders have a higher rate of convergence to the solution at integer-order of the derivative. Due to the analytical analysis of the problems, this study can further modify the solution of other fractional-order problems.

The primary goal of this paper is to seek solutions to the coupled nonlinear partial differential equations (CNPDEs) by the use of q-homotopy analysis transform method (q-HATM). The CNPDEs considered are the coupled nonlinear Schrödinger–Korteweg–de Vries (CNLS-KdV) and the coupled nonlinear Maccari (CNLM) systems. As a basis for explaining the interactive wave propagation of electromagnetic waves in plasma physics, Langmuir waves and dust-acoustic waves, the CNLS-KdV model has emerged as a model for defining various types of wave phenomena in mathematical physics, and so forth. The CNLM model is a nonlinear system that explains the dynamics of isolated waves, restricted in a small part of space, in several fields like nonlinear optics, hydrodynamic and plasma physics. We construct the solutions (bright soliton) of these models through q-HATM and present the numerical simulation in form of plots and tables. The solutions obtained by the suggested approach are provided in a refined converging series. The outcomes confirm that the proposed solutions procedure is highly methodological, accurate and easy to study CNPDEs.

In this paper, the natural convective flow of hybrid nanofluid over the vertical plate has been examined. Cattaneo law of thermal flux is used to characterize thermal transport. The novel model for fractional constitutive equation is expressed by the time fractional Caputo–Fabrizio integral. The analytical solution of the generalized convective flow of viscous fluid over the vertical plate along generalized conditions is obtained by using the Laplace transform. Mathcad is used to determine numerically the influence of the physical and fractional parameters on the temperature and velocity field to depict the results graphically.

A fractional technique is used to evaluate the temperature, mass, and velocity flow of single and double wall CNTs over a vertical plate. Slip boundary conditions and applied magnetic force are addressed. Human blood is used to examine how base fluid behaves. Applying the proper dimensionless variables results in the dimensionless formulation of initial and boundary conditions related to the governed dimensional concentration, momentum, and energy equations. The Laplace transform technique is used to resolve the dimensionless governing partial differential equations and get the solutions. The constant proportional Caputo (CPC) time-fractional derivative is a unique class of fractional model used in the simulation technique. The fundamental definitions are used to support the said model first. Using MWCNTs and SWCNTs in comparison to the flow characteristics, a thermal and mass study is given. The heat and mass transfer processes for single-walled carbon nanotubes (SWCNTs) have been shown to typically be progressive. The momentum profile decreases as the fractional variables rise. Multi-walled carbon nanotubes (MWCNTs) show more progressive velocity control as a result of the magnetic parameter. Graphs demonstrate the influence of embedding factors on the velocity, energy, and concentration profiles.