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This paper aims to present dual solutions for the two-dimension copper oxide with silver (CuO–Ag) and zinc oxide with silver (ZnO–Ag) hybrid nanofluid flow past a permeable shrinking sheet in a dusty fluid with velocity slip.

The governing partial differential equations for the two dust particle phases are reduced to the pertinent ordinary differential equations using a similarity transformation. Closed-form analytical solutions for the reduced skin friction and reduced Nusselt number, as well as for the velocity and temperature profiles, were presented, both graphically and in tables, under specific non-dimensional physical parameters such as the suction parameter, Prandtl number, slip parameter and shrinking parameter, which are also presented in both figures and tables.

The results indicate that for the shrinking flow, the wall skin friction is higher in the dusty fluid when compared with the clear (viscous) fluid. In addition, the effect of the fluid–particle interaction parameter to the fluid phase can be seen more clearly in the shrinking flow. Furthermore, multiple (dual, upper and lower branch solutions) are found for the governing similarity equations and the upper branch solution expanded with higher values of the suction parameter. It can be confirmed that the lower branch solution is unstable.

In practice, the study of the stretching/shrinking flow is crucially important and useful. Both the problems of steady and unsteady flow of a dusty fluid have a wide range of possible applications in practice, such as in the centrifugal separation of particles, sedimentation and underground disposal of radioactive waste materials.

Even though the problem of dusty fluid has been broadly investigated, very limited results can be found for a shrinking sheet. Indeed, this paper has succeeded to obtain analytically dual solutions. The stability analysis can be performed by following many published papers on stretching/shrinking sheets. Finally, the critical values and plotting curves for obtaining single or dual solution are successfully presented.

This study aims to investigate the dual solutions for axisymmetric flow and heat transfer due to a permeable radially shrinking disk in copper oxide (CuO) and silver (Ag) hybrid nanofluids with radiation effect.

The partial differential equations that governed the problem will undergo a transformation into a set of similarity equations. Following this transformation, a numerical solution will be obtained using the boundary value problem solver, bvp4c, built in the MATLAB software. Later, analysis and discussion are conducted to specifically examine how various physical parameters affect both the flow characteristics and the thermal properties of the hybrid nanofluid.

Dual solutions are discovered to occur for the case of shrinking disk (λ < 0). Stronger suction triggers the critical values’ expansion and delays the boundary layer separation. Through stability analysis, it is determined that one of the solutions is stable, whereas the other solution exhibits instability, over time. Moreover, volume fraction upsurge enhances skin friction and heat transfer in hybrid nanofluid. The hybrid nanofluid’s heat transfer also heightened with the influence of radiation.

Flow over a shrinking disk has received limited research focus, in contrast to the extensively studied axisymmetric flow problem over a diverse set of geometries such as flat surfaces, curved surfaces and cylinder. Hence, this study highlights the axisymmetric flow due to a shrinking disk under radiation influence, using hybrid nanofluids containing CuO and Ag. Upon additional analysis, it is evidently shows that only one of the solutions exhibits stability, making it a physically dependable choice in practical applications. The authors are very confident that the findings of this study are novel, with several practical uses of hybrid nanofluids in modern industry.

The steady laminar wall jet flow over a stretching/shrinking surface in the presence of lateral suction or injection with a convective boundary condition is considered.

The partial differential equations for mass, momentum and energy conservation are changed to the system of ordinary differential equations through similarity solution transformations. Solutions, both numerical and asymptotic, to these similarity equations are found in some new ranges of parameters in the governing equations.

The equations are solved both asymptotically and numerically for a range of the transpiration parameter S and the flow parameter λ given in Mahros et al. (2023), thus greatly extending the range of these previous solutions. Asymptotic solutions for both large and small values of the Prandtl number σ are derived, showing good agreement with additional numerical integrations. It should be noted that in Mahros et al. (2023), only the case when σ=1 was treated. A solution for large λ when S=1 is obtained, showing a different asymptotic form to the case when S>0 in Mahros et al. (2023). Multiple solutions were seen by them for S<0 and the nature of the lower solution branch as S→0 from below is discussed. The question as to whether the lower branch solutions join as λ>0 when S<0 is resolved through obtaining an asymptotic solution λ small.

The accuracy of the solutions has been checked through a detailed comparison between the solutions obtained numerically and analytically, where excellent agreement has been found. This study is important for scientists working in the area of jet flows to become familiar with the flow properties and behaviour of jets.

This study aims to explore a novel model that integrates the Kairat-II equation and Kairat-X equation (K-XE), denoted as the Kairat-II-X (K-II-X) equation. This model demonstrates the connections between the differential geometry of curves and the concept of equivalence.

The Painlevé analysis shows that the combined K-II-X equation retains the complete Painlevé integrability.

This study explores multiple soliton (solutions in the form of kink solutions with entirely new dispersion relations and phase shifts.

Hirota’s bilinear technique is used to provide these novel solutions.

This study also provides a diverse range of solutions for the K-II-X equation, including kink, periodic and singular solutions.

This study provides formal procedures for analyzing recently developed systems that investigate optical communications, plasma physics, oceans and seas, fluid mechanics and the differential geometry of curves, among other topics.

The study introduces a novel Painlevé integrable model that has been constructed and delivers valuable discoveries.

The purpose of this paper is to study the new (3 + 1)-dimensional integrable fourth-order nonlinear equation which is used to model the shallow water waves.

By means of the Cole–Hopf transform, the bilinear form of the studied equation is extracted. Then the ansatz function method combined with the symbolic computation is implemented to construct the breather, multiwave and the interaction wave solutions. In addition, the subequation method tis also used to search for the diverse travelling wave solutions.

The breather, multiwave and the interaction wave solutions and other wave solutions like the singular periodic wave structure and dark wave structure are obtained. To the author’s knowledge, the solutions obtained are all new and have never been reported before.

The solutions obtained in this work have never appeared in other literature and can be regarded as an extension of the solutions for the new (3 + 1)-dimensional integrable fourth-order nonlinear equation.

The objective of this paper is to analyze the relationship between innovation and the development of artificial intelligence (AI) and digital technologies in Europe. The use of digital technologies among European companies is studied through a composite index, while the relationship between innovation and AI is studied through a log-linear regression model. The results of the model have made possible to develop interesting indications for economic and industrial policy.

The use of digital technologies among European companies is studied through a composite index of AI and information technology (ICT) (using the Fair and Sustainable Welfare methodology) with the aim of measuring territorial gaps and to know which European countries are more or less inclined to its use, while the relationship between innovation and AI is studied through a log-linear regression model.

In the paper, two different methodologies were used to analyze the relationship between innovation and the development of digital technologies in Europe. The synthetic indicator made possible to develop a taxonomy between the different countries, the log-linear model made possible to identify and explain the determinants of innovation.

The description of the biunivocal relationship between innovation and AI is a topical and relevant issue that is treated in the paper in an original way using a synthetic indicator and a log-linear model.

本文旨在探討在歐洲、創新與人工智能和數字技術的發展之間的關係。研究人員透過一個綜合指數、去探討歐洲公司之間數字技術的使用狀況。至於創新與人工智能之間的關係, 則以對數線性回歸模型來進行研究。從模型所得的結果, 為我們提供了建議、去訂定適切的經濟和產業政策。

研究人員透過一個人工智能和資訊科技的綜合指數, 去探討歐洲企業之間數字技術的使用狀況 (研究人員使用了公平和可持續福利方法論), 其目標為測量領土差距, 以及確定哪些歐洲國家、大體上傾向於使用數字技術;至於創新與人工智能之間的關係, 則以對數性回歸模型來進行研究。

本文使用了兩個不同的方法、去探討在歐洲、創新與數字技術發展之間的關係。有關的合成指標, 使研究人員可製定一個不同國家間的分類法;而有關的對數線性模型, 則讓研究人員可確立並說明創新的決定因素。

本文使用了合成指標和對數線性模型、去探討創新與人工智能之間的一對一的關係, 這是時下受到關注和適宜的課題;就研究法而言, 本研究確是新穎獨創的。

The purpose of this study is to numerically model the rigid pavement resting on two-parameter soil and to examine its modal parameters.

This study is carried out using a one-dimensional beam element with three rotational and three translational degrees of freedom based on the finite element method. MATLAB programming is used to perform the free vibration analysis of the rigid pavement.

Cyclic frequency and their corresponding mode shapes were determined. It has been investigated how cyclic frequency changes as a result of variations in the thickness, span length of pavement, shear modulus, modulus of subgrade, different boundary conditions and element discretization. Thickness of the pavement and span length has greater effect on the cyclic frequency. Maximum increase of 29.7% is found on increasing the thickness, whereas the cyclic frequency decreases by 63.49% on increasing span length of pavement.

The pavement's free vibration is the sole subject of the current investigation. This study limits for the preliminary design phase of rigid pavements, where a complete three-dimensional finite element analysis is unnecessary. The current approach can be extended to future research using a different method, such as finite element grilling technique, mesh-free technique on reinforced concrete pavements or jointed concrete pavements.

The finite element approach adopted in this paper involves six degrees of freedom for each node. Furthermore, to the best of the authors’ knowledge, no prior study has done seven separate parametric investigations on the modal analysis of rigid pavement resting on two-parameter soil.

In this study, a different approach is introduced to generate the kinetic sub-model for the modeling of solid-state pyrolysis reactions based on the thermogravimetric (TG) experimental data over a specified range of heating rates. Gene Expression Programming (GEP) is used to produce a correlation for the single-step global reaction rate as a function of determining kinetic variables, namely conversion, temperature, and heating rate.

For a case study on the coal pyrolysis, a coefficient of determination (R²) of 0.99 was obtained using the generated model according to the experimental benchmark data. Comparison of the model results with the experimental data proves the applicability, reliability, and convenience of GEP as a powerful tool for modeling purposes in the solid-state pyrolysis reactions.

The resulting kinetic sub-model takes advantage of particular characteristics, to be highly efficient, simple, accurate, and computationally attractive, which facilitates the CFD simulation of real pyrolizers under isothermal and non-isothermal conditions.

It should be emphasized that the above-mentioned manuscript is not under evaluation in any journals and submitted exclusively for consideration for possible publication in this journal. The generated kinetic model is in the final form of an algebraic correlation which, in comparison to the conventional kinetic models, suggests several advantages: to be relatively simpler, more accurate, and numerically efficient. These characteristics make the proposed model computationally attractive when used as a sub-model in CFD applications to simulate real pyrolizers under complex heating conditions.

This paper aims to study a non-Newtonian micropolar fluid flow over a bidirectional flexible surface for multiple exact solutions of momentum boundary layer and thermal transport phenomenon subject to wall mass flux, second-order slip and thermal jump conditions.

The coupled equations are transformed into ordinary differential equations using similarity variables. Analytical and numerical techniques are used to solve the coupled equations for single, dual or multiple solutions.

The results show that the stretching flow, shrinking flow, the wall drag, thermal profile and temperature gradient manifest large changes when treated for special effects of the standard parameters. The role of critical numbers is definitive in locating the domains for the existence of exact solutions. The nondimensional parameters, such as mass transfer parameter, bidirectional moving parameter, plate deformation strength parameter, velocity slips, material parameter, thermal jump and Prandtl number, are considered, and their physical effects are presented graphically. The presence of governing parameters exhibits special effects on the flow, microrotation and temperature distributions, and various exact solutions are obtained for the special parametric cases.

The originality and value of this work lie in its exploration of non-Newtonian micropolar fluid flow over a bidirectional flexible surface, highlighting the multiple exact solutions for momentum boundary layers and thermal transport under various physical conditions. The study provides insights into the effects of key parameters on flow and thermal behavior, contributing to the understanding of complex fluid dynamics.

To provide a new semi-analytical solution for the nonlinear Benjamin–Bona–Mahony (BBM) equation in the form of a convergent series. The results obtained through HPTM for BBM are compared with those obtained using the Sine-Gordon Expansion Method (SGEM) and the exact solution. We consider the initial condition as uncertain, represented in terms of an interval then investigate the solution of the interval Benjamin–Bona–Mahony (iBBM).

We employ the Homotopy Perturbation Transform Method (HPTM) to derive the series solution for the BBM equation. Furthermore, the iBBM equation is solved using HPTM to the initial condition has been considered as an interval number as the coefficient of it depends on several parameters and provides lower and upper interval solutions for iBBM.

The obtained numerical results provide accurate solutions, as demonstrated in the figures. The numerical results are evaluated to the precise solutions and found to be in good agreement. Further, the initial condition has been considered as an interval number as the coefficient of it depends on several parameters. To enhance the clarity, we depict our solutions using 3D graphics and interval solution plots generated using MATLAB.

A new semi-analytical convergent series-type solution has been found for nonlinear BBM and interval BBM equations with the help of the semi-analytical technique HPTM.