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Understanding the intricate Korteweg–de Vries ( c K d V ) equation is paramount for comprehending various nonlinear wave phenomena, owing to its capacity to depict the propagation of nonlinear waves in media characterized by complex-valued dispersive and nonlinear coefficients. The practical ramifications of the c K d V equation are extensive, spanning disciplines such as optics, plasma physics, and other realms dealing with intricate media. This investigation employs the Khater II method and a generalized rational approach as analytical methodologies to generate novel exact solutions for the c K d V equation. The Khater II method is employed to transform the nonlinear partial differential equation into a nonlinear ordinary differential equation, thus facilitating a systematic resolution. Moreover, the generalized rational technique utilizes a rational ansatz function to derive diverse forms of solutions. Application of these methodologies leads to the discovery of fresh exact solutions, encompassing solitary and periodic wave solutions for the c K d V equation. These solutions are expressed in rational forms featuring arbitrary functions, thereby expanding the repertoire of known solutions for the model. The efficacy of the analytical methodologies employed becomes evident through the discovery of these novel exact solutions, thereby enriching our understanding of the physical interpretations and wave characteristics associated with the c K d V equation. The derived solutions augment the existing body of knowledge pertaining to the model. This research enhances our comprehension of the c K d V equation, thereby advancing nonlinear wave analysis with potential applications in physics, optics, plasma science, and allied engineering domains. Future investigations may explore the extension of these methodologies to address other nonlinear wave equations. [ABSTRACT FROM AUTHOR]

Hybrid nanofluids (HNF) are the advanced form of nanofluids used for improved heat transfer purposes. Taking this point in mind, the objective of the presented endeavor is to examine the Yamada–Ota HNF flow model comprising (gold–silver/engine oil) over a stretched cylindrical surface and a sheet (as a limiting case) in a permeable medium. The novelty of this research is the consideration of the surface-catalyzed reaction along with the homogeneous–heterogeneous reactions to accelerate the chemical reactions in the shortest possible time. The heat transport phenomenon is strengthened with the support of Joule heating, heat absorption/generation, and the convective heat boundary condition at the surface of the cylinder. The obtained ordinary differential equations are reduced from the partial differential equations using boundary layer theory and are numerically computed using the Keller box method. It is witnessed that for varied estimates of the magnetic parameter, the thermal profile enhances while the velocity field reduces. It is also noted that the fluid concentration is reduced when the surface-catalyzed parameter is enhanced. The validation of the envisioned model in a limiting case is also added to this investigation. [ABSTRACT FROM AUTHOR]

The current flow model is dedicated to capture the role of Dufour and Soret on heat and mass transmission of rotating flow of γ Al 2 O 3 - C 2 H 6 O 2 and γ Al 2 O 3 - water nanoparticles due to exponential stretching under the action of thermal radiation, magnetic, and Eckert numbers. The problem is modelled in terms if partial differential equations (PDEs) with associated physical conditions. The ordinary differential equations (ODEs) are obtained via suitable transformations. The reduced nonlinear ODEs set is tackled via a new scheme. We suggested significant improvements in the traditional technique and further formulated an extended version of wavelets scheme-based Chebyshev polynomials thoughts. The detailed procedure of the wavelet scheme and flow chart are provided. To validate the numerical outcomes; a comparative study with numerical technique RK (order-4) is also provided. Furthermore; numerical consequences of velocity, concentration and temperature profiles are further examined using several plots. The graphical plots, compared and convergence analyses are endorsing that our proposed modifications are worthy. Velocities profiles in view of γ Al 2 O 3 - C 2 H 6 O 2 nanofluid are lower than the γ Al 2 O 3 - H 2 O nanofluid. Temperature and concentration profiles are dominant when γ Al 2 O 3 - H 2 O nanofluid is considered. [ABSTRACT FROM AUTHOR]

This study discusses entropy generation analysis for a peristaltic flow in a rotating medium with generalized complaint walls. The goal of the current analysis is to understand the fluid flow phenomena particular to micro devices. Nano materials with a size less than 100 nm have applications in micro heat exchangers to cool electronic circuits, blood analyzers, biological cell separations, etc. For this study, we considered the effects of radiation, viscous dissipation and heat flux on the flow of nanomaterial inside a cylindrical micro-channel. To investigate the slip effects on the flow, the second order slip condition for axial velocity, the first order slip condition for secondary velocity and the thermal slip conditions were used. The flow was governed by partial differential equations (PDE's), which were turned into a system of coupled ordinary differential equations (ODE's) that were highly non-linear and numerically solved using the NDSolve command in Mathematica. The impacts of different involved parameters on the flow field were investigated with the aid of graphical illustrations. Entropy generation and the Bejan number were given special attention, and it was found that they decreased as the Hartman number, rotation, and radiation parameters increased. [ABSTRACT FROM AUTHOR]

The nanofluid flows play a vital role in many engineering processes owing to their notable industrial usage and excessive heat transfer abilities. Lately, an advanced form of nanofluids namely "hybrid nanofluids" has swapped the usual nanofluid flows to further augment the heat transfer capabilities. The objective of this envisaged model is to compare the performance of two renowned hybrid nanofluid models namely Hamilton–Crosser and Yamada–Ota. The hybrid nanoliquid (TiO2-SiC/DO) flow model is comprised of Titanium oxide (TiO2) and Silicon carbide (SiC) nanoparticles submerged into Diathermic oil (DO). The subject flow is considered over a stretched surface and is influenced by the magnetic dipole. The uniqueness of the fluid model is augmented by considering the modified Fourier law instead of the traditional Fourier law and slip conditions at the boundary. By applying the suitable similarity transformations, the system of ordinary differential equations obtained from the leading partial differential equations is handled by the MATLAB solver bvp4c package to determine the numerical solution. It is divulged that the Yamada–Ota model performs considerably better than the Hamilton–Crosser flow model as far as heat transfer capabilities are concerned. Further, the velocity reduces on increasing hydrodynamic interaction and slip parameters. It is also noted that both temperature profiles increase for higher hydrodynamic interaction and viscous dissipation parameters. The envisioned model is authenticated when compared with an already published result in a limiting case. [ABSTRACT FROM AUTHOR]

Prandtl–Eyring hybrid nanofluid (P-EHNF) heat transfer and entropy generation were studied in this article. A slippery heated surface is used to test the flow and thermal transport properties of P-EHNF nanofluid. This investigation will also examine the effects of nano solid tubes morphologies, porosity materials, Cattaneo–Christov heat flow, and radiative flux. Predominant flow equations are written as partial differential equations (PDE). To find the solution, the PDEs were transformed into ordinary differential equations (ODEs), then the Keller box numerical approach was used to solve the ODEs. Single-walled carbon nanotubes (SWCNT) and multi-walled carbon nanotubes (MWCNT) using Engine Oil (EO) as a base fluid are studied in this work. The flow, temperature, drag force, Nusselt amount, and entropy measurement visually show significant findings for various variables. Notably, the comparison of P-EHNF's (MWCNT-SWCNT/EO) heat transfer rate with conventional nanofluid (SWCNT-EO) results in ever more significant upsurges. Spherical-shaped nano solid particles have the highest heat transport, whereas lamina-shaped nano solid particles exhibit the lowest heat transport. The model's entropy increases as the size of the nanoparticles get larger. A similar effect is seen when the radiative flow and the Prandtl–Eyring variable-II are improved. [ABSTRACT FROM AUTHOR]

In this paper, we study a fourth-order nonlinear symmetric regularized long-wave equation which arises in several physical applications including ion sound waves in plasma. This model also describes nonlinear ion-acoustic and space-charge waves. Lie group analysis and the tanh-coth method is utilized for the construction of solitary wave solutions to the equation. Based on each symmetry, we reduce the considered partial differential equation into different ordinary differential equations. The obtained equations are solved and new solitary wave patterns are reported. To understand the physical interpretation, some solitary wave structures are exhibited graphically. Also, by using the multiplier approach, we constructed the local conservation laws of the considered equation. [ABSTRACT FROM AUTHOR]

Summary: A solar thermal collector collects heat by absorbing sunlight and converting this incoming radiation into useful thermal energy. The parabolic trough solar sollector (PTSC) is widely installed in concentrated solar power technology with a temperature range of 325‐700 K. The thermodynamic efficiency of the PTSC can be intensified by using nanofluids owing to their enhanced thermophysical properties. In this paper, the two‐dimensional flow over a stretched sheet in PTSC under Williamson nanofluid effects is explored. With the help of useful transformations, the partial differential equations arising from boundary layer equations of the Williamson nanofluid model are changed into nonlinear ordinary differential equations. The Keller box numerical scheme is then used to find the solution of the resulting nonlinear ordinary differential equations. Two nanofluids, copper‐engine oil (Cu‐EO) and alumina‐engine oil (Al2O3‐EO) are considered to address the performance analysis of PTSC in this study. It is observed that the parameters of the porous media reduced the heat transfer rate but increased the velocity gradient. In order to examine the effect on various performance parameters of the device, the nanoparticle concentration is also varied. With the increase in the Reynolds and Brinkman numbers, the entropy is found to increase. The thermodynamic performance of the Cu‐EO nanofluid is remarkably better than that of the Al2O3‐EO nanofluid under the same conditions. The thermal efficiency of Cu‐EO over Al2O3‐EO is seen with a minimum of 1.5% and a maximum of 9.1%. The theoretical simulations documented here may be effective in improving solar thermal energy systems. [ABSTRACT FROM AUTHOR]

In this paper, a generalized (3 + 1)-dimensional nonlinear evolution equation (NLEE) is proposed and examined. This investigated NLEE is undoubtedly a robust mathematical model in real-world applications. Because several essential and commonly used relative differential equations can be obtained by considering some specific choices in the given general form. In this way, many new solitary wave solutions are formally obtained via the generalized exponential rational function method (GERFM). The general idea of solving the equation in this method is to use an appropriate variable change that can transform the given partial differential equation into an ordinary differential equation. We then provide a general symbolic form for the solutions and determine the solution of the equation. To understand the dynamic concepts of numerical solutions better, several numerical simulations are included. The dependence of the properties of the solutions on the selection of the parameters in the equation can be examined in the results. Finally, the results verify the efficiency and reliability of the considered method. [ABSTRACT FROM AUTHOR]

This paper provides a substantial amount of study related to coupled fluid flow and heat conduction of an upper-convected-Maxwell viscoelastic liquid over a stretching plane with slip velocity. A new model, presented by Christov, for thermal convection is employed. The partial differential equations are converted to ordinary differential equations by using appropriate transformation variables. The transformed equations are solved analytically by using the Galerkin method. For the sake of soundness, a comparison is done with a numerical method, and good agreement is found. The impacts of various parameters like slip coefficient, elasticity number, the thermal relaxation time of heat flow, and the Prandtl number over the temperature and velocity fields are studied. Furthermore, the Cattaneo–Christov heat flux model is compared with Fourier's law. Additionally, the present results are also verified by associating with the published work as a limiting case. [ABSTRACT FROM AUTHOR]

The major purpose of this article is to exploit the strength of the Intelligent Back-propagated Neural Networks of Levenberg Marquardt Technique (IBNN-LMT) to study three-dimensional (3D) squeezed flow model of carbon nanotubes (SFM-CNTs) relying on water in a revolving network with a rigid ground wall. The impact of magnetohydrodynamic (MHD) and thermal radiation has been investigated. Similarity transformation is utilized to convert partial differential equations (PDEs) into ordinary differential equations (ODEs). For numerous nanofluids constructed of SWCNT and MWCNT, the influence of each parameter on velocity, temperature, Bejan number, and entropy production is estimated. The overall entropy optimization was also computed. The major outcome of this research is to examine the impacts of flowing parameters on CNTs when squeezing parameters are changed. Moreover, the estimated solution is evaluated for designed SFM-CNTs by using a testing method and comparing it to a standard solution that has been backed up by performance studies based on MSE convergence, error histogram, and regression studies. The velocity profile f (χ) along y-axis intensifies with the intensification in S , φ ,and A. Where Ω gives a dual performance with the increment in the function, as with the increase in M , f (χ) provides decreasing performance. The temperature function θ (χ) increase as the increment in S , Ec ,and Ec m and decrease with the increase in φ and R. The solution is accurate up to 4 to 10 decimal places which proves the worth and effectiveness of the proposed IBNN-LMT solver. [ABSTRACT FROM AUTHOR]

In this analysis, we aim to examine the heat transfer and flow characteristics of a copper-aluminum/water hybrid nanofluid in the presence of viscous dissipation, magnetohydrodynamic (MHD), and porous medium effect over the shrinking sheet. The governing equations of the fluid model have been acquired by employment of the model of Tiwari and Das, with additional properties of the hybrid nanofluid. The system of partial differential equations (PDEs) has been converted into ordinary differential equations (ODEs) by adopting the exponential similarity transformation. Similarity transformation is an essential class of phenomenon where the symmetry of the scale helps to reduce the number of independent variables. Note that ODE solutions demonstrate the PDEs symmetrical behavior for the velocity and temperature profiles. With BVP4C solver in the MATLAB program, the system of resulting equations has been solved. We have compared the present results with the published results and found in excellent agreements. The findings of the analysis are also displayed and discussed in depth graphically and numerically. It is discovered that two solutions occur in definite ranges of suction and magnetic parameters. Dual (no) similarity solutions can be found in the range of S c ≤ S   and   M c ≤ M ( S c > S   and   M c > M ). By performing stability analysis, the smallest values of eigenvalue are obtained, suggesting that a stable solution is the first one. Furthermore, the graph of the smallest eigenvalue shows symmetrical behavior. By enhancing the Eckert number values the temperature of the fluid is raised. [ABSTRACT FROM AUTHOR]

This exploration examines the nonlinear effect of radiation on magnet flow consisting of hybrid alloy nanoparticles in the way of stream-wise and cross flow. Many experimental, as well as theoretical explorations, demonstrated that the thermal conductivity of the regular liquid increases by up to 15 to 40% when nanomaterials are mixed with the regular liquid. This change of the thermal conductivity of the nanoliquid depends on the various characteristics of the mixed nanomaterials like the size of the nanoparticles, the agglomeration of the particles, the volume fraction, etc. Researchers have used numerous nanoparticles. However, we selected water-based aluminum alloy (AA7075) and titanium alloy (Ti6Al4V) hybrid nanomaterials. This condition was mathematically modeled by capturing the Soret and Dufour impacts. The similarity method was exercised to change the partial differential equations (PDEs) into nonlinear ordinary differential equations (ODEs). Such nonlinear ODEs were worked out numerically via the bvp4c solver. The influences of varying the parameters on the concentration, temperature, and velocity area and the accompanying engineering quantities such as friction factor, mass, and heat transport rate were obtained and discussed using graphs. The velocity declines owing to nanoparticle volume fraction in the stream-wise and cross flow directions in the first result and augment in the second result, while the temperature and concentration upsurge in the first and second results. In addition, the Nusselt number augments due to the Soret number and declines due to the Dufour number in both results, whereas the Sherwood number uplifts due to the Dufour number and shrinks due to the Soret number in both results. [ABSTRACT FROM AUTHOR]

This paper investigates the steady, two dimensional, and magnetohydrodynamic flow of copper and alumina/water hybrid nanofluid on a permeable exponentially shrinking surface in the presence of Joule heating, velocity slip, and thermal slip parameters. Adopting the model of Tiwari and Das, the mathematical formulation of governing partial differential equations was constructed, which was then transformed into the equivalent system of non-linear ordinary differential equations by employing exponential similarity transformation variables. The resultant system was solved numerically using the BVP4C solver in the MATLAB software. For validation purposes, the obtained numerical results were compared graphically with those in previous studies, and found to be in good agreement, as the critical points are the same up to three decimal points. Based on the numerical results, it was revealed that dual solutions exist within specific ranges of the suction and magnetic parameters. Stability analysis was performed on both solutions in order to determine which solution(s) is/are stable. The analysis indicated that only the first solution is stable. Furthermore, it was also found that the temperature increases in both solutions when the magnetic parameter and Eckert number are increased, while it reduces as the thermal slip parameter rises. Furthermore, the coefficient of skin friction and the heat transfer rate increase for the first solution when the magnetic and the suction parameters are increased. Meanwhile, no change is noticed in the boundary layer separation for the various values of the Eckert number in the heat transfer rate. [ABSTRACT FROM AUTHOR]

Symmetry and fluid dynamics either advances the state-of-the-art of mathematical methods and extends the limitations of existing methodologies to new contributions in fluid. Physical scenario is modelled in terms of differential equations as mathematical models in fluid mechanics to address current challenges. In this work a physical problem to examine the unsteady flow of a third-grade non-Newtonian liquid induced through a permeable shrinking surface containing nanoliquid is considered. The model of Buongiorno is utilized comprising the thermophoresis and Brownian effects through nonlinear thermal radiation and convective condition. Based on the flow symmetry, suitable similarity transformations are employed to alter the partial differential equations into nonlinear ordinary differential equations and then these ordinary differential equations are numerically executed via three-stage Lobatto IIIa formula. The flow symmetry is discussed for interesting physical parameters and thus this work is concluded. More exactly, the impacts of pertinent constraints on the concentration, temperature and velocity profiles along together drag force, Sherwood and Nusselt numbers are explained through the aid of the tables and plots. The outcomes reveal that the dual nature of solutions is gained for a specific amount of suction and flow in the decelerating form A < 0 . However, the unique result is obtained for flow in accelerating form A ≥ 0 . In addition, the non-linear parameter declines the liquid velocity and augments the concentration and temperature fields in the first result, whereas the contrary behavior is scrutinized in the second result. [ABSTRACT FROM AUTHOR]

This paper reflects the effects of velocity and thermal slip conditions on the stagnation-point mixed convective flow of Cross liquid moving over a vertical plate entrenched in a Darcy–Forchheimer porous medium. A Cross liquid is a type of non-Newtonian liquid whose viscosity depends on the shear rate. The leading partial differential equations (PDEs) are altered to nonlinear ordinary differential equations (ODEs) via feasible similarity transformations. These transmuted equations are computed numerically through the bvp4c solver. The authority of sundry parameters on the temperature and velocity distributions is examined graphically. In addition, the characteristics of heat transfer are analyzed in the presence of the impact of drag forces. The outcomes reveal that the permeability parameter decelerates the drag forces and declines the rate of heat transfer in both forms of solutions. Moreover, it is found that the drag forces decline with the growing value of the Weissenberg parameter in the upper branch solutions, while a reverse trend is revealed in the lower branch solutions. However, the rate of heat transfer shows a diminishing behavior with an increasing value of the Weissenberg parameter. [ABSTRACT FROM AUTHOR]

The present study accentuates the heat transfer characteristics of a convective condition of micropolar nanofluid on a permeable shrinking/stretching inclined surface. Brownian and thermophoresis effects are also involved to incorporate energy and concentration equations. Moreover, linear similarity transformation has been used to transform the system of governing partial differential equations (PDEs) into a set of nonlinear ordinary differential equations (ODEs). The numerical comparison has been done with the previously published results and found in good agreement graphically and tabular form by using the shooting method in MAPLE software. Dual solutions have been found in the specific range of shrinking/stretching surface parameters and the mass suction parameter for the opposing flow case. Moreover, the skin friction coefficient, the heat transfer coefficient, the couple stress coefficient, and the concentration transfer rate decelerate in both solutions against the mass suction parameter for the augmentation of the micropolar parameter respectively. The first (second) solution is the stable (unstable) solution and can (not) be considered as a real solution as the values of the smallest eigenvalues are positive (negative). [ABSTRACT FROM AUTHOR]

The principal objective of the current study is to analyze the Brownian motion and thermophoretic impacts on micropolar nanofluid flow over a nonlinear inclined stretching sheet taking into account the Soret and Dufour effects. The compatible similarity transformations are applied to obtain the nonlinear ordinary differential equations from the partial differential equations. The numerical solution of the present study obtained via the Keller-Box technique. The physical quantities of interest are skin friction, Sherwood number, and heat exchange, along with several influences of material parameters on the momentum, temperature, and concentration are elucidated and clarified with diagrams. A decent settlement can be established in the current results with previously published work in the deficiency of incorporating effects. It is found that the growth of the inclination and nonlinear stretching factor decreases the velocity profile. Moreover, the growth of the Soret effect reduces the heat flux rate and wall shear stress. [ABSTRACT FROM AUTHOR]

The key objective of the study under concern is to probe the impacts of Brownian motion and thermophoresis diffusion on Casson nanofluid boundary layer flow over a nonlinear inclined stretching sheet, with the effect of convective boundaries and thermal radiations. Nonlinear ordinary differential equations are obtained from governing nonlinear partial differential equations by using compatible similarity transformations. The quantities associated with engineering aspects, such as skin friction, Sherwood number, and heat exchange along with various impacts of material factors on the momentum, temperature, and concentration, are elucidated and clarified with diagrams. The numerical solution of the present study is obtained via the Keller-box technique and in limiting sense are reduced to the published results for accuracy purpose. [ABSTRACT FROM AUTHOR]

In present study, the nano-material flow of Ree-Eyring fluid model (NF-REFM) is examined by utilizing the technique of Levenberg Marquardt with backpropagated neural networks (TLM-BNNs). The flow is examined between two disks and the impacts of porosity and velocity slip are also analyzed. The partial differential equations (PDEs) representing the NF-REFM are transformed into system of ordinary differential equations (ODEs). Homotopy analysis method (HAM) is used to solve the ODEs and interpret the reference dataset for TLM-BNN. This dataset helps to compute the approximated solution of NF-REFM in MATLAB software. Regression analysis, Error histogram and MSE results, validates the performance of TLM-BNN. The flow effects on the velocity profile, temperature distribution and concentration profile are examined for different parameters. The results for entropy generation, Bejan number, Nusselt number, Sherwood number and skin friction coefficient are also discussed in this article. [ABSTRACT FROM AUTHOR]