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This paper presents a numerical technique to solve the generalized fractional Lane–Emden–Fowler equation. We use the Haar wavelet technique to approximate the solution of the generalized fractional Lane–Emden–Fowler equation. In this method, a differential equation transforms into a system of nonlinear equations, and this system is further solved to obtain Haar coefficients. We also showed the convergence of this method by establishing an upper bound. The rate of convergence is also discussed in our algorithm. Many numerical examples have been taken, and the results of those numerical experiments have been written in tabular form. We also documented the experiments graphically to show the efficiency and accuracy of this method.

The study of molybdenum disulfide, along with iron oxide, has various promising applications due to its thermal, electrical, and magnetic properties. In particular, for the efficient cooling in power devices, enhanced fluid transport in medical devices, chemical processing, and cancer therapy. The utility of both nanoparticles is vital. Hence, with this motivation, this study aims to investigate the characteristics of dissipative heat influenced by the inclusion of viscous and Joule dissipation in the flow of a hybrid nanofluid comprised of molybdenum disulfide $({\mathrm{\text{MoS}}}_2)$ and iron oxide $({\mathrm{\text{Fe}}}_3{\mathrm{\text{O}}}_4)$ in the base liquid, ethylene glycol $({\mathrm{\text{C}}}_2{\mathrm{\text{H}}}_6{\mathrm{\text{O}}}_2)$, within a coaxial permeable disk. The proposed model, encompassing various thermal properties, is transformed into a non-dimensional form by applying appropriate similarity rules. Traditional numerical techniques, such as the shooting-based fourth-order Runge–Kutta method, is employed to obtain solutions. Several profiles are computed for various values of contributing factors within their appropriate ranges employing the numerical scheme implemented in MATLAB. A comparative analysis with previous studies is presented, demonstrating a strong correlation in a specific case and thereby validating the current results.

In various industries, for advanced cooling, electronic device thermal management, and solar thermal systems ethylene glycol (EG)-based nanofluids are presented as efficient heat transfer agents. The proposed oxide tri-hybrid nanofluids comprising multiple oxide nanoparticles, i.e., Al₂O₃, TiO₂ and SiO₂ have significantly enhanced thermal properties and stability compared to mono and binary nanofluids. The current investigation aims at the transportation of heat characteristics of an EG-based tri-hybrid nanofluid over a curved Riga surface filled with a porous substance. The study focuses on thermal radiation effects. Moreover, the Riga plate is a magnetic device with alternating electrodes and magnets that provide a significant electromagnetic forcing mechanism for flow behavior. The designed mathematical model with their dimensional form needs transformation to their corresponding dimensionless form with the utility of similarity rules. Further, a numerical technique based on shooting is employed for the solution of the model for the attainment of physical factors. The physical properties of these factors are presented briefly through graphs followed by a comparative analysis.

In this research, we conducted an analysis using a stochastic neural network (SNN) approach to explore the dynamics of Ebola virus disease. Ebola virus, commonly referred to as Ebola hemorrhagic fever, is initially transmitted to humans through contact with wild and domesticated animals. Subsequently, it spreads rapidly among individuals through person-to-person transmission. To control the spread of the Ebola virus, we employed a five-compartmental mathematical model comprising the following classes: susceptible ($S$), exposed ($E$), infected ($I$), quarantined ($Q$) and recovered ($R$). We utilized the aforementioned SNN method for this purpose. The primary aim of this study is two-fold: first, to assess the convergence and accuracy of our proposed method, and second, to elucidate the impact of Ebola disease control measures. Lastly, we conducted a comparative analysis of our findings with those obtained from the numerical solvers ode15s and ode23e, demonstrating the precision and efficacy of our technique in addressing the current disease challenge.

Numerical schemes that are implemented by interpolation of exact solutions to a differential equation naturally preserve geometric properties of the differential equation. The solution interpolation method can be used for development of a new class of geometric integrators, which generally show better performances than standard method both quantitatively and qualitatively. Several examples including a linear convection equation and a nonlinear heat equation are included.

A lattice Boltzmann model for axisymmetric turbulent flows is developed. It is a further development of the enhanced axisymmetric lattice Boltzmann method (AxLAB®). The turbulent flow is efficiently and naturally simulated through incorporation of the standard subgrid-scale (SGS) stress model into the axisymmetric lattice Boltzmann equation in a consistent manner with the lattice gas dynamics. The model is verified by applying it to three typical cases in engineering: (i) pipe flow through an abrupt axisymmetric constriction, (ii) axisymmetric separated and reattached flow and (iii) pulsatile flows in a stenotic vessel. The numerical results obtained using the present method are compared with experimental data and other available numerical solutions, indicating good agreements. The model is simple and is able to predict axisymmetric turbulent flows at good accuracy.

To illustrate the role of Lorentz force on migration of nanopowders, CVFEM simulation has been reported in current research. The chamber contains hybrid nanomaterial and made up form porous media. Momentum equations have been modified for present paper with adding new source terms. The mentioned method works based on FEM in generation of mesh and calculation of gradient of scalars while it uses FVM approach for employing source terms. Testing with benchmark article shows the nice accuracy. Increase of permeability can enhance the speed of nanopowders and iso-temperature lines shapes become complicated. Impose of MHD creates new force against buoyancy and declines the velocity of the nanomaterial. Also, complication of isotherms declines with rise of Ha. With growth of Da, value of $\mathrm{\Psi }$ increases about 111% and 64.2% when $\mathrm{\text{Ha}}=0$ and 20, respectively. Also, augment of Ha results in reduction of velocity about 30% and 47.6% when $\mathrm{\text{Da}}=0.01$ and 100. Given $\mathrm{\text{Da}}=5\mathrm{\text{Ha}}=100$, Nu for $\mathrm{\text{Ra}}=\mathrm{\text{1e}}5$ is 6.83 times bigger than case with $log(\mathrm{\text{Ra}})=3$. Nu decreases to about 67.28% with increase of Ha when $\mathrm{\text{Da}}=100$, $\mathrm{\text{Ra}}=\mathrm{\text{1e}}5$. As Da increases, Nu rises about 62% when $\mathrm{\text{Ha}}=0$, $\mathrm{\text{Ra}}=\mathrm{\text{1e}}3$.

Numerical relativity became a powerful tool to investigate the dynamics of binary problems with black holes or neutron stars as well as the very structure of General Relativity. Although public numerical relativity codes are available to evolve such systems, a proper understanding of the methods involved is quite important. Here, we focus on the numerical solution of elliptic partial differential equations. Such equations arise when preparing initial data for numerical relativity, but also for monitoring the evolution of black holes. Because such elliptic equations play an important role in many branches of physics, we give an overview of the topic, and show how to numerically solve them with simple examples and sample codes written in C++ and Fortran90 for beginners in numerical relativity or other fields requiring numerical expertise.

The system of three gravitating particles on a line is considered. The forward and backward expansions of the solution in the Taylor series are used for numerical study of the problem. On the basis of the description of the solutions in the vicinity of pair collision points the pair collision transition operators are introduced. The dynamical problem is reduced to the study of the 2D-maps in the space of free parameters of the system in the vicinity of pair collision points. The first results of the computer simulations are presented.

A method is proposed for the quantum-mechanical determination of the eigenstates and eigenvalues of a particle (e.g. an electron) in a potential of general shape. For such systems the method allows one to calculate the exact solutions of the stationary Schrödinger equation. For the application of this method any well-known reference system is needed. A transition from the stationary reference system to the stationary system under investigation is performed by means of the time-dependent Schrödinger equation. The results are compared with various standard methods; alternative numerical methods will be discussed critically.

In this paper, we propose an efficient and fast numerical method of finding a stationary solution of large systems of aggregation–fragmentation equations of Smoluchowski type for concentrations of reacting particles. This method is applicable when the stationary concentrations steeply decrease with increasing aggregate size, which is fulfilled for the most important cases. We show that under rather mild restrictions, imposed on the kernel of the Smoluchowski equation, the following numerical procedure may be used: First, a complete solution for a relatively small number of equations (a “seed system”) is generated and then the result is exploited in a fast iterative scheme. In this way the new approach allows to obtain a steady-state solution for rather large systems of equations, by orders of magnitude faster than the standard schemes.

A novel numerical method of calculating the current transfer length (CTL) from several matrices to superconducting region has been proposed in this paper, there are two ways to calculate the CTL corresponding to two criteria of CTL: E-based criterion and I-based criterion, and the relationship between these criteria have been given again. The method of calculating the CTL from several matrices to superconducting region has high accuracy compared with the method introduced by Stenvall, and simulation and experiments of measuring the CTL have been carried out to compare with the ones calculated by the numerical method. The results show that the novel numerical method has high accuracy in calculating the CTL of superconducting tape and is useful for generalizing to calculate the CTL of superconducting cable.

The onset of thermal instability with hybrid nanoparticles Al₂O₃–Cu in nanofluid is investigated with the combined effects of variable gravity, variable heat source and Lorentz force in a porous medium. Its applications are in many areas like chemical engineering, geophysics, astrophysics, etc. Based on literature, many gravity variations are assumed in the present analysis, along with a space-dependent heat source/sink parameter that varies along the width of the channel. The finite difference-based Lobatto IIIa method has been applied to solve the system under no nanoparticle flux and rough boundary conditions, and approximate analytical results are generated for some special cases. The roughness parameters (${\lambda }_1,{\lambda }_2$), Darcy number (Da), gravity variation function ($G(z)$), and Chandrasekhar number (Q) delay the onset of convection and thus stabilize the system. It is also observed that an increase in the Lewis number Le, power index in variable heat source, and the nanoparticle Rayleigh concentration number Rn decreases the critical Rayleigh number $R_{\theta ,c}$ which destabilizes the system, and increases the critical wave number, which enlarges the convection cell size. In the case of exponential variation ($1-e^z$) in the gravity variation parameter, the system becomes stabilized due to a delay in the onset of convection. In addition, we have considered multilayer perceptron-artificial neural network (MLP-ANN) computation to predict the critical Rayleigh number as per function of important controlling parameters. The data set of 625 observations is chosen keeping 70% for testing, 15% for training and 15% for validation using efficient Levenberg−Marquardt back propagation algorithm with optimal accuracy measures i.e., root mean square deviation (RMSE), root mean relative error (RMRE) and $R^2$ (coefficient of determination). Finally, the regression plots are drawn that correlate, target and output data.

Numerical technique for examining the transport of nanofluid within the porous container has been applied. The tank has one curved hot wall which is located in the center of the outer cylinder. The single-phase approach to deriving properties of nanofluid was applied and Darcy law has been implemented to involve the porous term in the equation. The format of the equation has been converted to stream function format to remove the pressure terms and final equations were solved via CVFEM. The written code was verified according to the previous data of the published work. Outputs showed that loading alumina causes Nu to augment by 26.71% when Ha is zero and it can be increased about 41.22% when Ha = 15. Applying a magnetic field can reduce the Nu around 37.93% when $\mathrm{\text{Ra}}=700$. With increase of strength of rotating cell with the rise of Ra, Nu increases by about 38.22%. Changing the shape of alumina can increase the Nu by about 11.73% when $\mathrm{\text{Ha}}=15$ and $\mathrm{\text{Ra}}=700$.

This paper presents a theoretical analysis to the fracture behavior of the large single-domain YBCO superconductor with a center slant crack under electromagnetic force. The stress intensity factors are obtained by using the coupled finite element and infinite element numerical method. Numerical results are computed for two activation processes. For the zero-field cooling (ZFC) magnetization processes, the stress intensity factors increase as the applied field becomes large during field descent. For the field cooling (FC) magnetization processes, the stress intensity factors are obvious differences between the two cases when b_fc > 1 and b_fc ≤ 1. In addition, the predicted crack growth is the Mode-I fracture mainly. In this case, the numerical results developed in the present work are considered to be available for providing quantitative predictions of the fracture mechanism of superconductor both in theory and applications.

Thermal effects dramatically impact on the cavitation dynamics of cryogenic fluids. Thus, to study the thermal effect factors influencing cryogenic cavitation, numerical simulations were conducted considering an axisymmetric ogive and a 2D quarter caliber hydrofoil in liquid nitrogen and hydrogen, respectively. The modified Merkle cavitation model and filter-based turbulence model were applied to account for the thermodynamic properties of the fluid. The energy equation was modified considering the cavitation phase change effects. Compared to the experimental data, the numerical method satisfactorily predicts the cryogenic cavitation flows. Based on the numerical results, the thermal effect characteristics in the cavitation flow of cryogenic fluids were investigated. The thermal effects in cryogenic cavitation is obvious when vapor content in constant location is considerably low, where the cavity becomes more porous and the interface becomes less distinct. The factors influencing the thermal effects in cavitation such as the temperature, fluid type and velocity were analyzed. Findings showed that thermal effects of cavitation were prominent around the critical temperature of cryogenic liquids. Compared to the thermal effects in liquid nitrogen, those in liquid hydrogen were more distinct because of the changes in the density ratio, vapor pressure and other fluid properties. When the flow velocity is higher, the thermal effects of cavitation are suppressed as the pressure depression caused by evaporation is much smaller than the dynamic pressure.

Hybrid nanomaterial exergy drop and thermal manners were examined in this work with involving the porous media in middle part of the pipe. Moreover, to augment the rotational velocity, disturber device was applied in test section. The two sections near the entrance and outlet were in absence of disturber and porous zone because of importance of shun of backflow and reach of fully developed condition. Laminar flow with assume of homogeneous mixture for hybrid nanomaterial (Al₂O${}_3+\mathrm{\text{MWCNT}}$ and water) results in final model with involving the non-Darcy approach for permeable media. Finite volume approach was applied for achieving the outputs which are exergy loss (Xd) and hydrothermal behavior. Use of additives results in lower Xd more than 7.7% when $\mathrm{\text{Re}}=10\mathrm{\text{Da}}=\mathrm{\text{1e2}}$. With growth of Re, the Xd declines around a bit greater than 19%. Utilizing higher permeability allows the hybrid nanofluid moves quicker and exergy loss augments less than 10.5%.

The analysis brings out the investigation of the impact of thermal buoyancy on conducting the flow of an unsteady nanofluid within parallel moving walls embedded with a porous matrix. However, the medium is also embedded with permeable materials. Additionally, the impact of a uniform heat source is assumed to affect the designed model. The special attraction of the model is the variation of differently shaped nanoparticles using Hamilton–Crosser conductivity in which the base fluid is concatenated with the gold nanoparticles. The simulation is carried out for the governing equations numerically followed by requisite similarity rules used for the conversion of nonlinear problems of PDEs to ODEs. Further, shooting-based Runge–Kutta fourth-order scheme is imposed for the set of first-order ODEs. The behavior of several characterizing components within their range is presented for both the flow profiles via graphs and numerical results of the rate constants are deployed through the tabular form. Finally, the important outcomes are the particle concentration shows its greater impact in enhancing the fluid velocity neat the plate region and smooth retardation occurs at the central region further, the heat transfer rate retards significantly.

In developing current flow phenomena, the proposed study is carried out by the consideration of ternary hybrid nanofluid flow past a stretching surface. The flow became electrically conducting due to the inclusion of transverse magnetic field along the normal direction of the flow. Further, the energy profile is enhanced for the interaction of the radiative heat using Rosseland approximation. The innovation behind this approach is due to the assumption of the various shapes of the particles, i.e. spherical, cylindrical along with platelet-shaped nanoparticles and the corresponding physical properties are demonstrated. In view of physical phenomena, carbon nanotube, graphene, and aluminum oxide nanoparticles with water as a base liquid are used to prepare the hybrid nanofluid. Because of the complexity of the formulated model, the governing non-dimensional set of equations is handled by using traditional numerical technique following shooting-based “Runge–Kutta fourth-order” technique. Further, the significant role of the pertinent factors involved in the flow phenomena is the case of ternary nanofluid overshoots the fluid velocity in comparison to the case of nanofluid and hybrid nanofluid and the shear rate enhances for the increasing magnetization whereas the heat transfer rate attenuates significantly.

In the present scenario, the production process of various electronic gadgets including industrial products needs a higher cooling system to get its better features. Therefore, this study reveals the radiating ternary nanofluid flow through a wedged stretching surface. The forced convection of an electrically conducting fluid in association with the particle concentration enriches the study. The novelty of the proposed phenomena is due to the consideration of various shapes such as spherical, cylindrical, and platelet shaped solid particles combined with various physical models of electrical and thermal conductivity, viscosity, specific heat, and heat capacitance. The performance of the ternary nanofluid is presented using a carbon nanotube, graphene, and aluminum oxide in the base liquid water. To achieve a system of non-dimensional form of the governing equations, suitable transformations are adopted and further, shooting-based Runge-Kutta fourth-order technique is employed for the solution of this set of equations. Further, the important outcomes are deployed as follows: the combined effect of the nanoparticles and the base liquid augments the fluid velocity as well as the temperature distributions and thermal radiation also overshoots the temperature profile.