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Ternary hybrid nanofluids possess improved thermal characteristics, enhanced stability, better physical strength, and multi-functionality as compared to hybrid or usual nanofluids. The aim of the ongoing study is to explore the novel thermal attributes of hybrid and trihybrid nanofluids through a porous medium. Whereas the nano-composition of cobalt (Co), gold (Au), and zirconium oxide (ZrO2) make amalgamation in the paraffin (Pfin) which is a base fluid. This nano-composition of the proposed nanoparticles, specifically, subject to the base fluid Pfin has not been interpreted before. The analysis not only covers the features of trihybrid nanofluids (Co–Au–ZrO2–Pfin) but it also describes the characteristics of hybrid (Co–Au–Pfin) as well as pure nanofluids (Co–Pfin). An efficient numerical algorithm is developed for which the numerical simulations are carried out. The approximations are performed in MATLAB software using “Successive under Relaxation (SUR)” technique. A comparison, under certain limiting conditions, with the established results appraises the efficiency of the numerical code. The outcomes evidently designate that temperature raises with the change in thermal radiation and volume fraction of gold and zirconium oxide in either case of pure, hybrid, or ternary nanofluids. The concentration ϕ3{\phi }_{3} of ZrO2 has a significant impact on Nusselt number rather than the concentration ϕ1{\phi }_{1} of cobalt and ϕ2{\phi }_{2} of gold. It has been comparatively noticed that the ternary nanofluids (Co–Au–ZrO2–Pfin) portray embellished and improvised thermal characteristics as compared to the other two cases.

In this article, the solution to the time-fractional Fisher equation is determined using two well-known analytical techniques. The suggested approaches are the new iterative method and the optimal auxiliary function method, with the fractional derivative handled in the Caputo sense. The obtained results demonstrate that the suggested approaches are efficient and simple to use for solving fractional-order differential equations. The approximate and exact solutions of the partial fractional differential equations for integer order were compared. Additionally, the fractional-order and integer-order results are contrasted using simple tables. It has been confirmed that the solution produced using the provided methods converges to the exact solution at the appropriate rate. The primary advantage of the suggested method is the small number of computations needed. Moreover, it may be used to address fractional-order physical problems in a number of fields.

In the current study, we delve into examining the movement of a nanofluid within a Williamson boundary layer, focusing on the analysis of heat and mass transfer (HMT) processes. This particular flow occurs over a sheet that undergoes nonlinear stretching. A significant facet of this investigation involves the incorporation of both the magnetic field and the influence of viscous dissipation within the model. The sheet is situated within a porous medium, and this medium conforms to the Darcy model. Since more precise outcomes are still required, the model assumes that both fluid conductivity and viscosity change with temperature. In this research, we encounter a system of extremely nonlinear ordinary differential equations that are treated through a numerical technique, specifically by employing the spectral collocation method. Graphical representations are used to illustrate how the relevant parameters impact the nanoparticle volume fraction, velocity, and temperature profiles. The study involves the computation and analysis of the effect of physical parameters on the local Sherwood number, skin friction coefficient, and local Nusselt number. Specific significant findings emerging from the present study highlight that the rate of mass transfer is particularly influenced by the thermophoresis factor, porous parameter, and Williamson parameter, showing heightened effects, while conversely, the Brownian motion parameter demonstrates an opposing pattern. The results were computed and subjected to a comparison with earlier research, indicating a notable degree of conformity and accord.

This work examines the behaviour of flow and heat transmission in the presence of hybrid nanofluid in thermal radiation, heat generation, and magnetohydrodynamics. The hybrid state in this model is represented by two different fluids, TiO2 (titanium dioxide) and Ag (silver). The enclosure is wavy and slanted, with curving walls on the left and right. The finite difference approximation method was utilized to resolve the fundamental equations after they were non-dimensionalized, which are further reduced to a fourth-order bi-harmonic equation and are numerically solved based on the biconjugate gradient-stabilized approach method. The simulations are performed with various Rayleigh numbers, Hartmann numbers, an inclination angle of the enclosure, radiation parameters, heat generation parameters, inclination angle of the magnetic field, and volume fraction of hybrid nanoparticles. The streamlines, isotherms, and average Nusselt number contours are used to depict the thermo-fluid patterns. The findings show that the average Nusselt number relies on ϕ and increases as ϕ rises. The investigation’s findings demonstrated that the transfer of heat on the heated bottom wall significantly increases with the Rayleigh number (Ra = 105 and 106). At a cavity inclination of 45°, interesting multi-vortex structures are observed. The results of this study may enhance the effectiveness of solar collectors, heat exchangers, and other similar systems that depend on convective heat transfer in nature.

Thermophoretic particle deposition (TPD) and thermal radiation have significant uses in engineering and research, such as projectiles, electrical fuel, and production of coating sheets, thermal transference, nuclear plants, renewable energy, aerospace engineering, and gas turbines. In light of the above applications, the present analysis examines the stagnation point flow of hybrid nanofluid (hnf) around a revolving sphere. The hnf is prepared with the addition of Cu and Al2O3 nanoparticles in the water. The flow is examined under the impact of chemical reaction, thermal radiation, TPD, and activation energy. The flow equations are reformed into a dimensionless set of ordinary differential equations and then solved through the numerical approach parametric continuation method. The graphical and numerical results are demonstrated through graphics and tables. It has been noted that the effects of acceleration and rotational parameters boost the hnf (Cu and Al2O3/water) velocity. Furthermore, the energy outline reduces with the effect of acceleration parameter and nanoparticle volume friction. The influence of the rotation factor and acceleration parameters boosts the rate of skin friction. The influence of thermal radiation enriches the energy transmission rate.

Thermal energy from the Sun comes mostly from sunlight. These energies might be used in photovoltaic cells, sustainable power systems, solar light poles, and water-collecting solar pumps. This age studies solar energy and how direct sunshine might improve solar panel efficiency. Solar energy, especially solar tiles, is widely used in manufacturing today. The literature includes a modified Buongiorno hybrid nanofluid prototype. There are no studies that have examined the impact of tri-hybrid and unique tetra hybridity nanomolecules integrated with the Buongiorno nanofluid prototype on liquid moving on a flexible surface. This study examines the effects of an improved Buongiorno tetra hybrid nanoliquid prototypical with Buongiorno and Tiwari–Das nanofluid on magnetized double-diffusive binary nanofluid with cross fluid and Maxwell liquid flowing with variant thermal conductance over a porous medium. Different profiles include diffusion thermo and thermo diffusion. The LobattoIIIA scheme’s convergence and stability are examined in terms of residual error, mesh points for ordinary differential equations (ODEs), and boundary conditions. Leading equations about liquid flow continuity, impetus, temperature, and concentricity are obtained using continuity, conservation of momentum, the second law of thermodynamics, Fick’s second law of diffusion, and boundary layer expectations. The system of partial differential equations obtained from the given assumption becomes a system of ODEs and well-established LobattoIII. Their numerical solution is obtained using a numerical technique. Statistical charts and tables provide numerical solutions. The heat transport rate of tetra-hybrid nanomolecules increases dramatically, unlike tri- and di-hybrid nanomolecules. The improved Buongiorno tetra hybrid nanofluid (BTHNF) model produces more heat when radiation Rd{\rm{Rd}}, Brownian diffusion Nb{\rm{Nb}}, and thermal conductivity are increased. The data show that the diffusion factor LL, Brinkman number Br{\rm{Br}}, and Reynolds number Re increase entropy production, but Bejan number reduces it owing to an increase in Be{\rm{Be}} and Re\mathrm{Re}. A statistical regression study shows that retaining the Maxwell fluid parameter constant and increasing the Weissenberg number We{\rm{We}} decrease the drag coefficient error. A BTHNF model containing tetra hybrid nanoparticles has not been utilized to examine heat and mass transferences in non-Newtonian fluids, considering diffusion, thermo, and thermo diffusion. Entropy generation in a binary fluid with tetra hybrid nanoparticles and BTHNF has not been studied. Tetra hybrid nanofluid is not mentioned in the literature. This effort aims to create a new tetra-hybrid nanofluid model. This article is novel because it investigates the effects of thermal radiation, thermal conductivity, porosity, Darcy–Forchheimer, and Buongiorno models on a tetra-hybrid nanofluid flow under an extensible sheet.

In this article, we investigated the Landau–Ginzburg–Higgs (LGH) equation, focusing on the analysis of isolated soliton solutions and their stability. To compute the isolated soliton solutions, we used the advanced auxiliary equation (AAE) approach, which has proven to be a powerful and efficient method for extracting soliton solutions in various nonlinear partial differential equations (NLPDEs). We provided a detailed explanation, both graphically and physically, of the obtained soliton solutions in this article. Furthermore, we used the linear stability technique to conduct a stability analysis of the LGH equation. Additionally, we studied the bifurcation and stability of the equilibria and performed phase plane analysis of the model. We also provided a discussion on the comparisons between the AAE method and two other well-known approaches: the generalized Kudryashov method and the improved Bernoulli sub-equation function method. The application of the AAE approach in this study demonstrates its effectiveness and capability in analysing and extracting soliton solutions in NLPDEs.

In this article, we used a novel semi-analytical approach, named the optimal auxiliary function method (OAFM), to solve integro-differential equations (IDEs). The OAFM includes an auxiliary function and convergence control parameters, which expedite the convergence of the method. To apply the proposed method, some assumptions regarding small or large parameters in the problem are necessary. We present numerical outcomes acquired via the OAFM alongside those obtained from other numerical techniques in tables. Furthermore, we demonstrate the efficacy and ease of implementing the proposed method for various IDEs using 2D graphs.

The fundamental objective of this work is focused to achieve a class of advanced and impressive exact estimations to the Zoomeron equation and the time-fraction biological population model through contrivance by a couple of important and magnificent techniques, namely, the modified extended tanh-function method which depend on the balance theory and the Ricatti–Bernoulli sub-ODE method which is independent of the balancing principle. The suggested model is one of the major concerns for studying population distribution dynamics as well as the quantum field theory which is an important discipline for the description of interactions between light and electrons. The two suggested reliable, effective techniques are considered famous among ths ansatz methods that have various visions to realize the exact solutions to the non linear partial differential equation that reduce the volume of calculations examined before and usually give good results. It is solicited for this contrivance finding new exact solutions for two models in terms of some variable. The models are significant in quantum field theory, description of interactions between light and electron, quantum electrodynamics, demographic model, important to bring it into line with the reasonable distribution of wealth, resources, income, etc. The achieved results predict many types of solutions as trigonometric functions, hyperbolic functions, perfect periodic soliton solutions, singular periodic soliton solutions, and other rational solitons solutions. The efficiency of the techniques is demonstrated by the satisfactory results obtained through the derivation of closed-form soliton solutions from the exact solution by assigning definite values to the variables present in it.

Enhancement in thermal distribution of Williamson hybrid nanofluid flow is articulated in this research. Nichrome and TC4 nanoparticles are homogenously diffused in the water, which is the base fluid. An elongating surface holds the flow and thermal transition phenomenon in the existence of uniform sources of magnetic field and heat radiation. The boundary of wall obeys a suction and slip condition. The formulation for physical conservation laws is made as a system of partial differential equations. For the solution purpose, their boundary-value problem is transmuted into the ordinary differential form. Then, Matlab code involving Runge–Kutta procedure is run to compute the variation in velocity as well as temperature profiles under impacts of the controlling factors. The comparative computations are made for two cases: nanofluids (TC4+water)\left({\rm{TC}}4+{\rm{water}}) and hybrid nanofluids (TC4,Nichrome+water)\left({\rm{TC}}4,{\rm{Nichrome}}+{\rm{water}}). The heat for that hybrid nanofluid case is larger than that for the nanofluids. The velocity curve is decreased against increasing magnetic field strength and Williamson parameter. Enhancement in thermal distribution is observed with increasing concentration ϕ2{\phi }_{2} of Nichrome.

We use a general transformation, to find exact solutions for the Manakov system with variable coefficients (depending on the time ε\varepsilon ) using an improved tanh–coth method. The solutions obtained in this work are more general compared to those in other works because they involve variable coefficients. The implemented computational method is applied in a direct way on the reduced system, avoid in this way the reduction to only one equation, as occurs in the works respect to exact solutions, made by other authors. Clearly, from the solutions obtained here, new solutions are derived for the standard model (constant coefficients), complementing in this way the results obtained by other authors mentioned here. Finally, we give some discussion on the results and give the respective conclusions.

The compounded Bell generalized class of distributions is proposed in this article as an alternative to the compounded Poisson generalized family of distributions. Some properties and actuarial measures are presented. The properties of a special model named Bell Weibull (BellW) are obtained such as the linear representation of density, rth moment, incomplete moment, moment generating function using Wright generalized hypergeometric function and Meijer’s G function, the pth moment of order statistics, reliability, stochastic ordering, and residual and reversed residual life. Moreover, some commonly used entropy measures, namely, Rényi, Havrda and Charvat, and Arimoto and Tsallis entropy are obtained for the special model. From the inferential side, parameters are estimated using maximum likelihood estimation. The simulation study is performed to highlight the behavior of estimates. Some actuarial measures including expected shortfall, value at risk, tail value at risk, tail variance, and tail variance premium for the BellW model are presented with the numerical illustration. The usefulness of the proposed family is evaluated using insurance claims and COVID-19 datasets. Convincing results are obtained.

Differential equations (DEs) performed a vital role in the implementation of almost all the mechanical, physical, or biological processes. Higher order DEs had always been challenging to solve for the researchers so numerous numerical techniques were developed to attain the vital numerical approximations of such types of problems. In this work, highly advanced numerical techniques are established for the approximation of the fourteenth (14th)-order boundary value problems using Adomian decomposition method. The mathematical outcomes of the equations are attained in the form of convergent series that have effortlessly assessable components having step size h = 10. Some numerical examples are also deliberated to demonstrate the capability and application of the established procedure.

In this article, first integral method (FIM) is used to acquire the analytical solutions of (3+1)-D Wazwaz–Benjamin–Bona–Mahony and (2+1)-D cubic Klein–Gordon equation. New soliton solutions are obtained, such as solitons, cuspon, and periodic solutions. FIM is a direct method to acquire soliton solutions of nonlinear partial differential equations (PDEs). The proposed technique can be used for solving higher dimensional PDEs. FIM can be implemented to solve integrable and ion-integrable equations.

The natural streamflow of the River is encouraged to forecast through multiple methods. The impartiality of this study is the comparison of the forecast accuracy rates of the time-series (TS) hybrid model with the conventional model. The behavior of the natural monthly statistical chaotic streamflow to use in the forecasting models has been compiled by projecting two distinguished rivers, the Indus and Chenab of Pakistan. Therefore, this article is based on the monthly streamflow forecast analysis that has been reported using the group method of data handling with wavelet decomposition (WGMDH) as a new forecasting attribute. Discrete wavelets decompose the perceived data into sub-series and forecast hydrological variables; these fittingly have been endorsed as inputs in the hybrid model. The forecast efficiency and estimations of the hybrid model are measured by the appropriate statistical techniques such as mean absolute error (RME), root mean square error (RMSE), and correlation coefficients (R) and compared to the group method of data handling (GMDH), least-square support vector machine and artificial neural network conventional models. The comparative analysis shows that the hybrid WGMDH model is more stable and more potent for forecasting river flow than other predictive models and significantly proved that the hybrid model is a robust alternate forecasting tool for TS data sets.

The current work is devoted to introduce a novel thermoelastic heat conduction model where the Moore-Gibson-Thompson (MGT) equation describes the heat equation. The constructed model is characterized by allowing limited velocities of heat wave propagation within the material, consistent with physical phenomena. The Green–Naghdi Type III model is improved by introducing the delay factor into the modified Fourier law. Also, from the presented model, some other models of thermoelasticity can be derived at specific states. Based on the suggested model, an infinite orthotropic material with a cylindrical hole exposed to time-dependent temperature variation was studied. It has also been considered that the coefficient of thermal conductivity varies with temperature, unlike in many other cases where this value is considered constant. The viscoelastic material of the investigated medium was assumed to be of the Kelvin–Voigt type. The Laplace transform method provides general solutions to the studied field variables equations. The effects of viscosity and thermal variability parameters on these fields are discussed and graphically presented. In addition, the numerical results were presented in tables, and a comparison with previous models was made to ensure the accuracy of the results of the proposed model.

This article aims to conduct a numerical investigation of phenomena induced by gas expansion in chemical propulsion nozzles. A numerical simulation of full-scale flat convergent-divergent nozzle geometry using the finite volume method on structured meshes is performed to predict the change in the convergent geometry on the boundary layer separation resulting from a shock/shock and shock/boundary layer. Two turbulence models are tested, namely, the k−ε and k−ω shear-stress transport (SST) models. Three steps are considered to achieve this work. First, 10 numerical schemes are tested to select the accurate one. The findings of the first step are used to predict the boundary layer separation in a supersonic overexpanded nozzle. The available experimental data from the NASA Langley Research Center are used to validate the results. The third step concerns investigating the impact of the convergent geometric profile on the downstream flow of the nozzle. The obtained results are analyzed and compared with the experimental data. These results show that convergent geometry may cause the formation of different shock structures and different points of flow separation and modifies several parameters of the flow and nozzle performance downstream the throat. The findings indicated that the convergent profile must be considered during the design phase when focusing on the problem of boundary layer separation in the supersonic overexpanded regime nozzles.

This work solves the problem of thin-film withdrawal and drainage of a steady incompressible couple stress fluid on the outer surface of a vertical cylinder. The governing equations for velocity and temperature distributions are subjected to the boundary conditions and solved with the help of homotopy analysis method. The obtained expressions for flow profile, temperature profile, average velocity, volume flow rate, and shear stress confirmed that the thin-film flow of couple stress fluid highly depends on involved parameters say Stokes number St , vorticity parameter λ, couple stress parameter η, and Brinkman number Br presented in the graphical description as well.

This article deals with a new modified heat conduction model with fractional order that includes the Caputo–Fabrizio differential operator (CF) and the thermal relaxation time. This new approach to the CF fractional derivative has attracted many researchers because it includes a nonsingular kernel. The nonlocal theory proposed by Eringen has also been applied to demonstrate the effect of scale-dependent thermoelastic materials. The problem of thermal isotropic semi-infinite space is addressed as an application of the presented model. The medium is exposed to regularly changing heat sources and is initially placed in a continuous external magnetic field. The system of governing equations was expressed in the field of the Laplace transform, and the problem in this field was solved by the state-space operation. The inverse of the transformed expressions of physical quantities is found numerically using Zakian’s algorithm. The effects of the nonlocal parameter, the fractal order parameter, and the magnetic field were graphically presented and analyzed in detail. Some of the previous investigations were extracted in some special cases.

Our key objective in the present work is to elaborate the concept of activation energy in chemically reactive flow with the help of modeling and computation. The model investigated is fluid flow over a vertical cylinder in the porous medium with chemical reaction and radiation effect. The similarity transform converted the resulting constitutive equations and partial differential equations (PDEs) into ordinary differential equations (ODEs). The resulting non-linear momentum, heat transfer, and mass transfer coupled equations are computed with the Range–Kutta–Fehlberg method. Both assisting and non-assisting buoyant flow conditions are considered, and observed numeric solutions vary with the transport properties. Characteristics of momentum, heat, and concentration under the applied boundary conditions are analyzed. In addition, the increment in activation energy parameters boosts the Lorentz force and mass transfer rate.

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

The fundamental objective of this article is to find exact solutions to the stochastic fractional-space Allen–Cahn equation, which is derived in the Itô sense by multiplicative noise. The exact solutions to this equation are required since it appears in many discipline areas including plasma physics, quantum mechanics and mathematical biology. The tanh–coth method is used to generate new hyperbolic and trigonometric stochastic and fractional solutions. The originality of this study is that the results produced here expand and improve on previously obtained results. Furthermore, we use Matlab package to display 3D surfaces of analytical solutions derived in this study to demonstrate the effect of stochastic term on the solutions of the stochastic-fractional-space Allen–Cahn equation.

For the stochastic heat equation (SHE), a very accurate spectral method is considered. To solve the SHE, we suggest using a shifted Legendre Gauss–Lobatto collocation approach in combination with a shifted Legendre Gauss–Radau collocation technique. A comprehensive theoretical formulation is offered, together with numerical examples, to demonstrate the technique’s performance and competency. The scheme’s superiority in tackling the SHE is demonstrated.

In the past few decades, many models have been proposed to address the shortcomings found in the classical theories of thermoelasticity and to allow limited speeds of heat waves. In this context, in the current paper a new generalized model of thermoelasticity based on the Moore–Gibson–Thompson (MGT) equation has been introduced. This new model can be derived by introducing the relaxation time factor into the third type of Green–Naghdi model (GN-III). In contrast to the previous works, it was taken into account that the physical properties of the material are dependent on temperature and on the viscous type. The viscoelastic medium has been assumed to obey the Kelvin–Voigt model. On the basis of the present model, thermo-viscoelastic interactions have been investigated in an unbounded orthotropic body with a cylindrical cavity. The surface of the cavity is restricted and exposed to a pulse-formed heat flow that dissolves exponentially. The characteristic thermal modulus of the material is assumed to be a linear function of temperature. The Laplace transform can be used to eliminate time dependency from control equations. Using a suitable approximate method, the transformed equations have been finally inverted by numerical inversion of the Laplace transform. Certain comparisons have been introduced to estimate the effects of the viscosity, pulsed heat, and thermal temperature-independent properties on all studied fields. A comparison with previous models of thermoelasticity is also performed in tables to verify the accuracy of the proposed model. We found from the results that the physical fields strongly depend on the viscoelastic parameter, the change of the thermal conductivity, and pulsed heat, so it is not possible to neglect their effect on the manufacturing process of machines and devices.

In this article, contact problem with fractional derivatives is studied. We use fractional derivative in the sense of Caputo. We deploy penalty function method to degenerate the obstacle problem into a system of fractional boundary value problems (FBVPs). The series solution of this system of FBVPs is acquired by using the variational iteration method (VIM). The performance as well as precision of the applied method is gauged by means of significant numerical tests. We further study the convergence and residual errors of the solutions by giving variation to the fractional parameter, and graphically present the solutions and residual errors accordingly. The outcomes thus obtained witness the high effectiveness of VIM for solving FBVPs.

In this article, the generalized plane Couette flow of Vogel’s model of incompressible, non-isothermal, couple stress fluid flowing steadily between two parallel walls is investigated. The governing equations are reduced to ordinary differential equations. To investigate the non-linear coupled system of differential equations, the optimal homotopy asymptotic method with DJ polynomial and asymptotic homotopy perturbation method have been used. Important flow properties are presented and discussed. We have obtained expressions for velocity, average velocity, shear stress, volume flux and temperature. The results gained employing these techniques are in the form of infinite series; thus, the results can be easily calculated. Comparison of various results, obtained through the suggested approaches, is carried out and an excellent agreement is achieved.

In this study, we propose a simple direct meshless scheme based on the Gaussian radial basis function for the one-dimensional linear and nonlinear convection–diffusion problems, which frequently occur in physical phenomena. This is fulfilled by constructing a simple ‘anisotropic’ space–time Gaussian radial basis function. According to the proposed scheme, there is no need to remove time-dependent variables during the whole solution process, which leads it to a really meshless method. The suggested meshless method is implemented to the challenging convection–diffusion problems in a direct way with ease. Numerical results show that the proposed meshless method is simple, accurate, stable, easy-to-program and efficient for both linear and nonlinear convection–diffusion equation with different values of Péclet number. To assess the accuracy absolute error, average absolute error and root-mean-square error are used.

In the present study, two new classes of convex functions are established with the aid of Raina’s function, which is known as the ψ-s-convex and ψ-quasi-convex functions. As a result, some refinements of the Hermite–Hadamard (ℋℋ{\mathcal{ {\mathcal H} {\mathcal H} }})-type inequalities regarding our proposed technique are derived via generalized ψ-quasi-convex and generalized ψ-s-convex functions. Considering an identity, several new inequalities connected to the ℋℋ{\mathcal{ {\mathcal H} {\mathcal H} }} type for twice differentiable functions for the aforesaid classes are derived. The consequences elaborated here, being very broad, are figured out to be dedicated to recapturing some known results. Appropriate links of the numerous outcomes apprehended here with those connecting comparatively with classical quasi-convex functions are also specified. Finally, the proposed study also allows the description of a process analogous to the initial and final condition description used by quantum mechanics and special relativity theory.

In this analysis, the baffling method is used to increase the efficiency of channel heat exchangers (CHEs). The present CFD (computational fluid dynamics)-based work aims to analyze the constant property, steady, turbulent, Newtonian, and incompressible fluid flow (air), in the presence of transverse-section, arc-shaped vortex generators (VGs) with two various geometrical models, i.e., arc towards the inlet section (called arc-upstream) and arc towards the outlet section (called arc-downstream), attached to the hot lower wall, in an in-line situation, through a horizontal duct. For the investigated range of Reynolds number (from 12,000 to 32,000), the order of the thermal exchange and pressure loss went from 1.599–3.309 to 3.667–21.103 times, respectively, over the values obtained with the unbaffled exchanger. The arc-downstream configuration proved its superiority in terms of thermal exchange rate by about 14% than the other shape of baffle. Due to ability to produce strong flows, the arc-downstream baffle has given the highest outlet bulk temperature.

Multi-term time-fractional partial differential equations (PDEs) have become a hot topic in the field of mathematical physics and are used to improve the modeling accuracy in the description of anomalous diffusion processes compared to the single-term PDE results. This research includes the numerical solutions of two-term time-fractional PDE models using an efficient and accurate local meshless method. Due to the advantages of the meshless nature and ease of applicability in higher dimensions, the demand for meshless techniques is increasing. This approach approximates the solution on a uniform or scattered set of nodes, resulting in well-conditioned and sparse coefficient matrices. Numerical tests are performed to demonstrate the efficacy and accuracy of the proposed local meshless technique.

In this article, the Sinh–Gordon function method and sub-equation method are used to construct traveling wave solutions of modified equal width equation. Thanks to the proposed methods, trigonometric soliton, dark soliton, and complex hyperbolic solutions of the considered equation are obtained. Common aspects, differences, advantages, and disadvantages of both analytical methods are discussed. It has been shown that the traveling wave solutions produced by both analytical methods with different base equations have different properties. 2D, 3D, and contour graphics are offered for solutions obtained by choosing appropriate values of the parameters. To evaluate the feasibility and efficacy of these techniques, a nonlinear evolution equation was investigated, and with the help of symbolic calculation, these methods have been shown to be a powerful, reliable, and effective mathematical tool for the solution of nonlinear partial differential equations.

The purpose of this proposed investigation is to study unsteady magneto hydrodynamic (MHD) mixed initial-boundary value problem for incompressible fractional Maxwell fluid model via oscillatory porous rectangular duct. Considering the modified Darcy’s law, the problem is simplified by using the method of the double finite Fourier sine and Laplace transforms. As a limiting case of the general solutions, the same results can be obtained for the classical Maxwell fluid. Also, the impact of magnetic parameter, porosity of medium, and the impact of various material parameters on the velocity profile and the corresponding tangential tensions are illuminated graphically. At the end, we will give the conclusion of the whole paper.

In this article, analytical exact and approximate solutions for fractional physical equations are obtained successfully via efficient analytical method called fractional residual power series method (FRPSM). The fractional derivatives are described in the Caputo sense. Three applications are discussed, showing the validity, accuracy and efficiency of the present method. The solution via FRPSM shows excellent agreement in comparison with the solutions obtained from other established methods. Also, the FRPSM can be used to solve other nonlinear fractional partial differential equation problems. The final results are presented in graphs and tables, which show the effectiveness, quality and strength of the presented method.

The main aim of this article is to use a new and simple algorithm namely the modified variational iteration algorithm-II (MVIA-II) to obtain numerical solutions of different types of fifth-order Korteweg-de Vries (KdV) equations. In order to assess the precision, stability and accuracy of the solutions, five test problems are offered for different types of fifth-order KdV equations. Numerical results are compared with the Adomian decomposition method, Laplace decomposition method, modified Adomian decomposition method and the homotopy perturbation transform method, which reveals that the MVIA-II exceptionally productive, computationally attractive and has more accuracy than the others.