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This paper focuses on modeling and analyzing a nonlinear fractional Maxwell fluid across a vertical plate using a novel definition of the Caputo fractional derivative. Specifically, the newly formulated fractional definition is tailored for the finite difference method. Buoyancy effects are considered to bring the model closer to the natural occurrences of the magnetic field. The novel fractional derivative definition of the designed model has been implemented, and the results have been further analyzed using theoretical and numerical methods. The range of fractional parameters α and β is 0 < α ≤ 1 and 0 < β ≤ 1 , respectively. Significant theorems are also included in the study. A simulation was performed to determine which fractional derivative operator is more accurate than the others. The designed definition demonstrates a high convergence rate and decreased computational cost, both of which are primary objectives. Graphs have also been plotted to illustrate the velocity and temperature distributions as the physical parameters vary. [ABSTRACT FROM AUTHOR]

This study focuses on the Fokas system equation, which explains the nonlinear pulse propagation in single-mode fibres. The proposed new sub-equation approach and the modified Khater's method extract a more efficient and broad range of soliton solutions. These include multiple anti-kink, anti-kink, flat anti-kink, combined bright-periodic, multiple bright, v-shaped, bell-shaped, and flat bell-shaped singular soliton. The primary innovation of this paper lies in its discovery of a set of novel soliton solutions that provide insight into the interaction mechanisms of nonlinear waves within the realm of nonlinear physics. The travelling wave patterns of the model are graphically presented with suitable parameter values using the modern software Mathematica. The visual representation of the solutions in 3D, 2D, and contour surfaces enhances understanding of parameter impact. We conducted a sensitivity analysis on a newly developed dynamic framed structure that controls pulses using the velocity factor of the soliton wave. The proposed model's dynamics were observed and presented as bifurcation analysis, quasi-periodic chaotic, periodic systems and quasi-periodic. We anticipate that our research findings will catalyze advancing the study of nonlinear theory and its applications in optical fibres. This analysis confirms the effectiveness and reliability of the method employed, demonstrating its applicability in discovering travelling wave solitons for a wide range of nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

In this study, the generalized perturbed-KdV partial differential equation is examined. Furthermore, symmetry generators address the Lie invariance criteria. The suggested approach produces the Lie algebra, where translation symmetries in space and time are associated with mass conservation and conservation of energy respectively, the other symmetries are scaling or dilation. The optimal system of the obtained system developed. By using Lie Group methods, the generalized perturbed-KdV partial differential equation is changed using suitable similarity transformations through a system of highly nonlinear ordinary differential equations. The new extended direct algebraic approach is applied to get the soliton solutions. As a result, a plane solution, periodic stumps, compacton, smooth soliton, mixed hyperbolic solution, periodic and mixed periodic solutions, mixed trigonometric solution, trigonometric solution, peakon soliton, anti-peaked with decay, shock solution, mixed shock singular solution, mixed singular solution, complex solitary shock solution, singular solution and shock wave solutions are developed. The behavior of certain solutions is shown in 3-D and 2-D for specific values of the physical components in the studied equation. The outcomes hold significance for elevating research to a more impactful and effective level. The whole set of local conservation laws for the generalized perturbed-KdV equation for any arbitrary constant coefficients is found by applying the conservation laws multiplier. These findings are pivotal for advancing the current understanding and pushing the boundaries of knowledge to new heights. [ABSTRACT FROM AUTHOR]

In this article, we present an idea to attain the variety of novel optical solitons solutions for nonlinear time-fractional Clannish Random Walker's parabolic (CRWP) equation in the manner of β -derivative using a well-known analytical technique namely, extended direct algebraic method. A diversity of bright, singular, dark, periodic-singular and combo dark-bright solitons solutions are assembled in the shape of rational, hyperbolic and exponential functions. Some of the extracted results are sketched in the pattern of 3D, 2D and contour plots, for the purpose to demonstrate the physical behavior of the attained solutions. We also investigate the influence of the fractional order parameter on the obtained solutions which demonstrates how changes in this parameter affect the properties and behavior of the soliton solutions in the context of the time-fractional CRWP model. The novelty of the extracted solutions is determined by comparing with some other solutions which are already listed in the works for the CRWP equation, which shows the effectiveness and authenticity of the proposed method. The under consideration method is also utilized to any other nonlinear integer and fractional model appears in physics and engineering. [ABSTRACT FROM AUTHOR]

The prime motive to conduct this communication is to explicate hydrothermal attributes of water by inducing new composition of nanoparticles termed as ternary particles. For this purpose, two differently natured groups one with lesser densities (Carbon nanotubes, Graphene and Aluminium oxide) and with higher densities (Copper oxide, Copper and Silver) are accounted. A 3D permeable surface is considered as a physical configuration of problem by providing dual stretching. Initially, mathematical structuring in dimensional representation expressing the constitutive relations for mass, momentum and energy conservation is manifested. Later on, a set of similar variables are executed to express attained coupled system into ordinary form. Numerical simulations are performed to find solution by employing shooting and RK-4 methods in conjunction. Description about change is displayed through graphical visualization. Subsequently, temperature distribution and heat flux coefficient against sundry variables are also measured and comprehensively discussed in pictorial and tabular format. Wall drag coefficients along (x, y) directions are also computed. It is inferred from the outcomes that velocity, temperature and concentration of base fluid is higher for ternary group 1 containing particles of low densities than for group 2 with more denser particles. It is also deduced that elevation in temperature of fluid is revealed against Soret number whereas contrary aspects is observed in view of concentration distribution. Dufour number has declining impact on temperature profile whereas it upsurges the mass distribution. It is depicted that skin friction in case of group containing particles with less densities are more than other group. [ABSTRACT FROM AUTHOR]

The present study examines the nonlinear time-fractional model in the sense of a solitonic structure. A non-linear Schrödinger equation has applications in light scattering, indirect optical pulses as well as planer waves and to Bose-Einstein condensates enclosed in an anisotropic-shaped cigar, in a mean-field state, etc. A new extended direct algebraic method is utilized to get the soliton solutions with modified M-truncated and Atangana–Baleanu fractional operators which have Mittag-Leffler kernel. The obtained solutions contain new families of functions such as trigonometric, hyperbolic, rational, and exponential functions. The graphical 2D, 3D, contour, and also 3D spherical presentation pictorial the analysis with the feasible parametric values. On the evidence of the acquired solutions, it can be presumed that this technique is more effective and generalized to obtain solutions of many other non-linear partial differential equations that appear in different scientific disciplines. [ABSTRACT FROM AUTHOR]

This research examines the fractional Peyrard–Bishop DNA dynamical governing system, which displays the proliferation of optical pulses in field of plasma and the optical fibre. The analytical method is utilized to find travelling wave solutions because the inverse scattering transform cannot solve the Cauchy problem for this equation. The Φ 6 -model expansion method is an efficient and dependable technique for generating the generalised solitonic wave profiles with a wide range of soliton families. The main advantage of the offered analytical strategy is that it specifies a constraint for each solution to guarantee its existence. As a result, solitonic wave structures get attributes such as the Jacobi elliptic function, periodicity, brightness, dark-brightness, singularity, exponential, trigonometry, and rational solitonic structures, among others, under existence conditions that have not been explored previously. The results are represented graphically in 2-D, 3-D, and contour glimpses to illustrate the behavioural responses to pulse propagation by inferring the fitting values of system parameters. The stabilty of the considered model is discussed and develope the stability condition. The fractional parameter is responsible for reducing singularity and continuing to increase the smoothness in wave patterns. It is easy to employ the Φ 6 -model expansion method to other complicated non-linear systems and acquire solitary waves pattern. [ABSTRACT FROM AUTHOR]

The weakly nonlinear wave propagation that occurs in the presence of magnetic fields, in which energy is concentrated in a narrow band of wave-numbers in a dispersive and dissipative fluid. The main objective of this paper is to analyze the (2 + 1) - dimensional elliptic nonlinear Schrodinger equation under the influence of three different fractional operators. The generalized fractional soliton solutions and propagation of magnetohydrodynamics fluid in sort of solition will be visualized. The Conformable, β and M-truncated fractional operator applied to classical evolution Schrodinger equation. In order to get the analytical closed form solution, one of the generalized approach new extended direct algebraic method is utilized. The fractional nonlinear elliptic Schrodinger equation is developed in three different fractional sense. The similarity transformation technique converted the controlling fractional system to ordinary differential equations. The fractional analytical solutions such as, plane solution, mixed hyperbolic solution, periodic and mixed periodic solutions, mixed trigonometric solution, trigonometric solution, shock solution, mixed shock singular solution, mixed singular solution, complex solitary shock solution, singular solution and shock wave solutions are obtained. The graphical 2-D and 3-D representation of the results is shown to express the propagation of fluid with the magnetic field by assuming the appropriate values of the involved parameters. The graphical performance of the obtained solution at various settings of parametric values and fractional order reveals new perspectives and fascinating model phenomena. The attained outcomes have significant applications and have opened up innovative development areas for research across numerous scientific fields. [ABSTRACT FROM AUTHOR]

In this paper, the fractional model of the bioconvection flow of a MHD viscous fluid for vertical surface has been investigated. Introducing dimensionless variables, the governing equations are solved by Laplace transform technique. Classical governing model is drawn out to fractional order technique with non-singular kernel which can be used to explain the memory for natural phenomena. Since this operator describes the rate of change at each point of the measured interval, so we use this operator. To see the behavior of related parameters physically, some graphs have been plotted in conclusion section. In the end, some useful conclusions have been attained. It is resulted that constant proportional Caputo fractional derivative measures the memory strong in comparison with Caputo and Caputo–Fabrizio fractional approaches. Further, on comparison between different kinds of viscous fluid (Water, Air, Kerosene) and found that temperature and velocity of air are higher than water and kerosene, respectively. The impacts of dimensionless numbers on velocity field have been discussed graphically and concluded that velocity increased by increasing Gr and showed opposite behavior for M and Ra . [ABSTRACT FROM AUTHOR]

By applying the Sardar subequation method, we study new coupled Konno–Oono (CKO) equation and obtained soliton type solutions in the form of bright, dark, singular, periodic singular and combined solutions. In nonlinear sciences, CKO equation is very important and has applications in magnetic field. In order to better explanation of the developed solutions, different sketches of some solutions are drawn. The proposed approach is simple, direct and effective to obtain new results for nonlinear equations in mathematical sciences. [ABSTRACT FROM AUTHOR]

The wave propagation has the significant role in the field of coastal engineering and ocean. In the geographical fields, waves are primary source of environmental process owed to energy conveyance on floating structure. This study aims to investigate the system of cold bosonic atoms in zig-zag optics lattices. The solitonic patterns of the considered model successfully surveyed by using two integrated analytical techniques new extended direct algebraic and G ′ G 2 expansion method. The exact solutions are presented by rational, trigonometric, hyperbolic and exponential functions. On the basis of solitons, we need to show that which one is more integrated and robust scheme. These solutions will help to understood the dynamics of cold bosonic atoms in zig-zag optical lattices and the several other systems. Three dimensional as well as two dimensional comparison presented for a cold bosonic atoms model solutions which are revealed diagrammatically for appropriate parameters by using Mathematica. This study will help physicists to predict some new hypothesis and theories in the field of mathematical physics. [ABSTRACT FROM AUTHOR]

This work deals with the applications of thermodynamical geometries on conformal gravity black holes (CGBH) consisting of conformal parameters a and k. The stability of black hole (BH) addressed with the aid of small, middle, large and divergency roots, respectively. For this purpose, graphical behavior of heat capacity and temperature versus horizon radius is presented which help us to show the stability conditions. Further, studied the different geometries like Weinhold, Ruppeiner, Geometrothermodynamics (GTD) and Hendi-Panahiyah-Eslam-Momennia (HPEM), and found relationship between divergency of scalar curvature and zeros of heat capacity. As a result, it is noticed that Ruppeiner, HPEM and GTD metric exhibit more important information as compared to Weinhold. [ABSTRACT FROM AUTHOR]

This study deals with the perturbed Chen-Lee-Liu governing mode which portrays the propagating phenomena of the optical pulses in the discipline of optical fiber and plasma. The Cauchy problem for this equation cannot be solved by the inverse scattering transform and we use an analytical approach to find traveling wave solutions. One of the generalized techniques Φ 6 - model expansion method is exerted to find new solitary wave profiles. It is an effective, and reliable technique that provides generalized solitonic wave profiles including numerous types of soliton families. As a result, solitonic wave patterns attain, like Jacobi elliptic function, periodic, dark, bright, singular, dark-bright, exponential, trigonometric, and rational solitonic structures, etc. The constraint corresponding to each obtained solution provides the guarantee of the existence of the solitary wave solutions. The graphical 2-D, 3-D, and contour visualization of the obtained results is presented to express the pulse propagation behaviors by assuming the appropriate values of the involved parameters. The Φ 6 - model expansion method is simple which can be easily applied to other complex non-linear models and get solitary wave structures. [ABSTRACT FROM AUTHOR]

Two efficient integration schemes, new extended hyperbolic function and generalized tanh are employed to discover optical soliton solutions to magneto-optic waveguides that retains anti-cubic form of nonlinear refractive index. Bright, dark, periodic singular, singular, and combo soliton solutions have created. These solutions expose the comprehensive variety of soliton solutions. [ABSTRACT FROM AUTHOR]

In the present paper, unsteady free convection flow of Maxwell fluid containing clay-nanoparticles is investigated. These particles are hanging in water, engine oil and kerosene. The values for nanofluids based on the Maxwell-Garnett and Brinkman models for effective thermal conductivity and viscosity are calculated numerically. The integer order governing equations are being extended to the novel non-integer order fractional derivative. Analytical solutions of temperature and velocity for Maxwell fluid are build using Laplace transform technique and expressed in such a way that they clearly satisfied the boundary conditions. To see the impact of different flow parameters on the velocity, we have drawn some graphs. As a result, we have seen that the fractional model is superior in narrate the decay property of field variables. Some limiting solutions are obtained and compared with the latest existing literature. Moreover, significant results can be observed for clay nanoparticles with different base fluids. [ABSTRACT FROM AUTHOR]

The unsteady flow of micropolar hybrid nanofluids through an oscillating vertical plate having infinite length has been analyzed in this study. This flow goes through Newtonian heating and thermal radiation. To enrich the thermal properties, we add copper and alumina nanoparticles to micropolar fluid. For the enhancement of heat transfer, water is taken as base liquid. The dimensionless reduced form of the governing equations for velocity, temperature and microrotation is formed by using dimensional analysis. We used Laplace transform method to have the exact solutions of these governing equations. MathCad's software has been used to visualize the effects on the flow of fluid graphically. We found that microrotation, velocity and temperature are inversely proportional to Prandtl number but directly proportional to Grashof number. Further, temperature can be enhanced for increasing value of Newtonian heating, radiation and volume fraction of nanoparticles copper and alumina. [ABSTRACT FROM AUTHOR]

In this paper, it has been discussed a nonlocal fractional model of viscous nanofluid holding a hybrid nanostructure. Hybridized copper (Cu) and aluminium oxide (Al 2 O 3) nanoparticles were liquefied in base fluid water (H 2 O) to form a hybrid nanofluid. The MHD free convection flow of the nanofluid (Cu - Al 2 O 3 - H 2 O) was considered in a microchannel. The viscous nanofluid model was comprehended by a nonlocal constant proportional Caputo (CPC) fractional operator with actual thermophysical properties. The governing equations of the model were exposed to physical initial and boundary conditions. The analytical solutions were gained by the fractional Laplace transform technique. To see the physical significance of parameters, graphs were made by Mathcad software. It was invented that the consequences were universal, consistent and realistic. An outstanding covenant between the theoretic and experiments results could be found by arranging the fractional parameter. In the end, a comparison of CPC with the recent literature and significant findings are obtained. Further, temperature can be enhanced for larger values of heat generation and volume fraction parameters, respectively. [ABSTRACT FROM AUTHOR]

Temperature transfer by virtue of natural convection for visualizing heat transport characteristics through heatline method within a prismatic cavity filled with Cu-H2O nanofluid considering two different temperature boundary conditions is performed numerically. Two top inclined walls are warmed-up at low temperature whilst the bottom wall is heated two different heated conditions such as uniform temperature condition and linear temperature condition. Two vertical walls are insulated. Finite element technique of Galerkin weighted residual form is employed for solving nonlinear partial differential equations for numerical calculation. Heatlines, isotherm contours, streamline contours, and Nusselt number are employed for displaying numerical simulated results for the model parameters entitled nanoparticles volume fraction, Hartmann number and Rayleigh number. The outcomes indicate that heat transfer rate has a significant impact on thermal boundary condition and shape of the nanoparticles. The temperature transfer value enhances significantly for higher Rayleigh number as well as nanoparticles volume fraction. Hartmann number has a positive impact on fluid flow and temperature transport. The characteristics of heat transport using heatlines method are also performed for predicting the better energy transform compared to isotherm contours. In addition, different types of nanofluids are also employed to examine the best heat transport performance. [ABSTRACT FROM AUTHOR]

In this work, analytical solution of hybrid Maxwell nanofluid of the vertical channel due to pressure gradient is discussed. By introducing dimensionless variables the governing equations with all levied initial and boundary conditions is converted into dimensionless form. Fractional model for Maxwell fluid is developed by Caputo time fractional differential operator by using the constitutive relation. The dimensionless expression for temperature and velocity are found using Laplace transform. Draw graphs of temperature and velocity by Mathcad software and discuss the behavior of flow parameters and the effect of fractional parameters. As a result, we have found by increasing the volumetric fraction of copper and alumina temperature increases and velocity decreases. Also, fluid flow properties showed dual behavior for small and large time, respectively, by increasing fractional parameters values. [ABSTRACT FROM AUTHOR]

Free convective, unsteady flow of Jeffrey liquid under the influence of magnetic field between two hot upright parallel plates fixed in porous medium is investigated in this paper. First plate is moving with time-dependent velocity U o f (t) in its own plane while other is fixed. Mathematical model is developed using law of conservation of momentum, Fourier's law of heat transfer. Equations for temperature and velocity fields are reduced to dimensionless form by applying suitable dimensionless variables. The Laplace transform method is used to find exact solutions of temperature and velocity. Finally, we have presented the effects of material and flow parameters and illustrated graphically. As a result, through this study, we found that coefficient of heat transfer shows dual behavior for small and large time. Also, the obtained results are reduced to the recently published work. [ABSTRACT FROM AUTHOR]

In this work, influence of hybrid nanofluids on heat transfer flow of a viscous fluid due to pressure gradient is discussed with innovative constant proportional Caputo fractional derivative. For this purpose, we consider an infinite vertical wall which is exponentially moving in the x-direction with variable temperature. Nanosized particles of Cu and Al 2 O 3 are suspended in water, the base fluid. The governing equations of the problem are converted into dimensionless form. Further, we develop the constant proportional Caputo fractional model with a new operator with power law kernel which can be used to study the fluid behaviour for different values of fractional parameter at the present time. We applied the Laplace transform method to obtain the solutions and to see the impact of hybrid nanofluids and fractional parameter α respectively. We compared the present results with the recently published work (Nehad et al, Adv. Mech. Eng.11(7): 1 (2019)) with Caputo fractional derivative. As a result, we have found that the present solutions are best to describe the memory concept of temperature and velocity. For small values of fractional parameter, temperature and velocity have maximum values and for larger values of fractional parameter, temperature and velocity have minimum values. Further, rate of heat transfer and skin friction are also computed in tabular forms and it is found that Nusselt number with CPC is much less than that is computed with Caputo fractional derivative for greater values of fractional parameter α . [ABSTRACT FROM AUTHOR]

A three dimensional (3D) numerical solution of unsteady, Ag-MgO hybrid nanoliquid flow with heat and mass transmission caused by upward/downward moving of wavy spinning disk has been scrutinized. The magnetic field has been also considered. The hybrid nanoliquid has been synthesized in the presence of Ag-MgO nanoparticles. The purpose of the study is to improve the rate of thermal energy transmission for several industrial purposes. The wavy rotating surface increases the heat transmission rate up to 15%, comparatively to the flat surface. The subsequent arrangement of modeled equations is diminished into dimensionless differential equation. The obtained system of equations is further analytically expounded via Homotopy analysis method HAM and the numerical Parametric continuation method (PCM) method has been used for the comparison of the outcomes. The results are graphically presented and discussed. It has been presumed that the geometry of spinning disk positively affects the velocity and thermal energy transmission. The addition of hybrid nanoparticles (silver and magnesium-oxide) significantly improved thermal property of carrier fluid. It uses is more efficacious to overcome low energy transmission. Such as, it provides improvement in thermal performance of carrier fluid, which play important role in power generation, hyperthermia, micro fabrication, air conditioning and metallurgical field. [ABSTRACT FROM AUTHOR]

The main feature of the present numerical model is to explore the behavior of Maxwell nanoliquid moving within two horizontal rotating disks. The disks are stretchable and subjected to a magnetic field in axial direction. The time dependent characteristics of thermal conductivity have been considered to scrutinize the heat transfer phenomena. The thermophoresis and Brownian motion features of nanoliquid are studied with Buongiorno model. The lower and upper disk's rotation for both the cases, same direction as well as opposite direction of rotation is investigated. The subsequent arrangement of the three dimensional Navier Stoke's equations along with energy, mass and Maxwell equations are diminished to a dimensionless system of equations through the Von Karman's similarity framework. The comparative numerical arrangement of modeled equations is further set up by built-in numerical scheme "boundary value solver" (Bvp4c) and Runge Kutta fourth order method (RK4). The various physical constraints, such as Prandtl number, thermal conductivity, magnetic field, thermal radiation, time relaxation, Brownian motion and thermophoresis parameters and their impact are presented and discussed briefly for velocity, temperature, concentration and magnetic strength profiles. In the present analysis, some vital characteristics such as Nusselt and Sherwood numbers are considered for physical and numerical investigation. The outcomes concluded that the disk stretching action opposing the flow behavior. With the increases of magnetic field parameter M the fluid velocity decreases, while improving its temperature. We show a good agreement of the present work by comparing with those published in literature. [ABSTRACT FROM AUTHOR]

In this work, influence of hybrid nanofluids (Cu and Al 2 O 3 ) on MHD Maxwell fluid due to pressure gradient are discussed. By introducing dimensionless variables the governing equations with all levied initial and boundary conditions are converted into dimensionless form. Fractional model for Maxwell fluid is established by Caputo time fractional differential operator. The dimensionless expression for concentration, temperature and velocity are found using Laplace transform. As a result, it is found that fluid properties show dual behavior for small and large time and by increasing volumetric fraction temperature increases and velocity decreases respectively. Further, we compared the Maxwell, Casson and Newtonian fluids and found that Newtonian fluid has greater velocity due to less viscosity. Draw the graphs of temperature and velocity by Mathcad software and discuss the behavior of flow parameters and the effect of fractional parameters. [ABSTRACT FROM AUTHOR]

The present paper is about the application of a non-local and non-singular fractional derivative to the unsteady flow of incompressible MHD viscous fluid with uniform heat flux through a porous medium. In addition the Soret, radiation, heat sink and chemical reaction effects were also applied to this physical model. The transport model of an ordinary derivative can be extended to the fractional model of a non-integer order derivative with non-local and non-singular kernel. The Laplace transform was used to obtain the expressions for temperature, concentration and velocity fields in non-dimensional form and numerical techniques were used to obtain the inverse Laplace transform for these expressions. At the end, influence of flow parameters on temperature, concentration and velocity fields have been discussed and explained graphically. As a result, the flow behavior can be enhanced with the non-integer order fractional parameter of non-local and non-singular kernel. [ABSTRACT FROM AUTHOR]