Frontiers in Industrial and Applied Mathematics

Flow and heat transfers along rough surfaces are investigated. A test facility is established, where rough surfaces generated by additive manufacturing can be tested. The computational work follows two goals. On the one hand, a computational tool is developed that can analyze the characteristics of a rough surface and generate rough surfaces with prescribed characteristics. On the other hand, Computational Fluid Dynamics (CFD) is applied for the analysis of flow and heat transfer along rough surfaces. The present focus is on the validation of turbulence models. Within this context, two alternative treatments, namely the wall functions (WF)-based approach and roughness resolving (RR) approach are assessed. Turbulence is modeled within a RANS (Reynolds Averaged Numerical Simulation) framework. All of the considered four turbulent viscosity models, using WF, showed a similar agreement with the measurements. Quantitatively, the realizable k-ε model is observed to deliver a better accuracy, in general, which is, then, also applied in RR calculations. The RR approach showed a fair qualitative performance, which was, however, quantitatively not as good as the WF approach. This is attributed to the idealized geometry on the one hand and possible limitations on the RANS turbulence modeling approach on the other hand. The analysis will be deepened in the future work.

In this study, we construct a linear recurrent fractal interpolation function (RFIF) with variable scaling parameters for data set with $$\alpha $$ α -stable noise (a generalization of Gaussian noise) on its ordinate, which captures the uncertainty at any missing or unknown intermediate point. The propagation of uncertainty in this linear RFIF is investigated, and a method for estimating parameters of the uncertainty at any interpolated value is provided. Moreover, a simulation study to visualize uncertainty for interpolated values is presented.

The field of non-commutative group-based cryptography has flourished in recent years, resulting in numerous non-commutative key exchange protocols, digital signature schemes, and secret sharing schemes. In this paper, we propose two 1-out-of-n oblivious transfer protocols using conjugation of group elements. The protocols are obtained by modifying the Ko-Lee key exchange protocol and the Anshel-Anshel-Goldfeld key exchange protocol.

In this paper, homotopy analysis method (HAM) is used to obtain the analytic solution for fragmentation population balance equation. Different sample problems are solved using HAM and their series solution is obtained. A detailed analysis of the series solution and the region of convergence of the solution is also studied. It is observed that the convergence region of the series solution can be adjusted with the help of certain parameters involved in HAM.

This paper presents several properties associated with the two-variable extension of the Chebyshev matrix polynomials of the second kind. In particular, we establish a three-term recurrence relation for these two-variable matrix polynomials and show that these two-variable matrix polynomials satisfy some second-order matrix differential equations. We derive their hypergeometric matrix representation and an expansion formula which links these generalized Chebyshev matrix polynomials with the Hermite matrix polynomials and the Laguerre matrix polynomials. We also drive their Volterra integral equation.

This paper is concerned with the blow-up phenomena and global existence of a fractional nonlinear reaction-diffusion equation with a non-local source term. Under sufficient conditions on the weight function a(x) and when the initial data is small enough, the global existence of solutions is proved using the comparison principle. We establish a finite time blow-up of the solution with large initial data by converting the fractional PDE into a simple ordinary differential inequality using the differential inequality technique. Moreover, by solving the obtained ordinary differential inequality, an upper bound of the blow-up time is also deduced.

In this paper, the methods for construction of multiplicative magic cubes of order n from the existing additive magic cube of order n has been introduced. Also, the modified Trenkler’s formula for the multiplicative magic cubes of odd and doubly even order has been instigated. Moreover, the newly defined power method has been proposed for the construction of multiplicative magic cubes. In all the methods, the conditions for obtaining multiplicative magic cubes consisting of either odd or even or composite numbers as their elements have been elaborated.

Real-life multi-attribute decision-making (MADM) has some major issues related to the space of the problem, inter-dependency among attributes, flexibility in the aggregation process, etc. So, our objective is to deal with these issues by adopting suitable tools and techniques like the q-rung orthopair fuzzy set (q-ROFS) for handling space-related difficulty. Dual Muirhead mean (DMM) is applied to address the inter-dependency among attributes, and for a flexible aggregation process, the Hamacher t-norm (TN) and t-conorm (TCN) are utilised. By fusing these approaches, this paper proposes two novel aggregation operators (AOs) named q-rung orthopair fuzzy Hamacher dual Muirhead mean (q-ROFHDMM) and q-rung orthopair fuzzy Hamacher weighted dual Muirhead mean (q-ROFHWDMM) operators. The essential properties of these AOs and special cases are explored as well. Finally, the q-ROFHWDMM operator has been used to construct a MADM method. The study also examines a practical example of selecting an enterprise resource planning (ERP) system, as well as sensitive and comparative analysis.

Linear, as well as weakly non-linear, analyses have been done to understand the onset of convection and heat and mass transport in a composite nanofluid horizontal layer heated from below under LTNE (local thermal non-equilibrium) effect. Two different types of nanoparticles are assumed to be suspended in the base fluid. Both the nanoparticles and the base fluid are taken to be at different temperature, and therefore, three temperature model is used for LTNE. Thermal Rayleigh number is evaluated analytically using Galerkin’s approach while non-linear analysis is done numerically. The effect of both top-heavy and bottom-heavy configurations of nanoparticles over convective instability is examined. It is found that the system is more stable in case of bottom-heavy configuration when compared to that of top-heavy case. Moreover, the effect of LTNE depends upon the concentration of nanoparticles significantly. A comparison between streamlines, isotherms and isohalines for both LTE (local thermal equilibrium) and LTNE cases is also presented.

This paper aims to examine the impact of the driver’s behavior with the downstream average flow on current traffic dynamics in the lattice hydrodynamic model. The influence of driver’s behavior and downstream traffic conditions with different sites are examined theoretically with the help of linear stability. It is observed that traffic flow stability can be improved by incorporating both driver’s behavior and the average flow of traffic downstream. Finally, numerical simulations show that present traffic dynamics may be improved by integrating the impacts of driver behavior and average downstream traffic conditions in order to alleviate traffic congestion. Also, it validates the theoretical findings.

Fractal functions provide a natural deterministic approximation of complex phenomena and also it has self-similarity. Recently, it has been recognized as an internal binary operation, called fractal convolution. In the present article, we obtain Bessel sequences of $$L^2(\mathcal {I} \times \mathcal {J})$$ L 2 ( I × J ) composed of product of fractal convolutions, using the identification of $$L^2(\mathcal {I} \times \mathcal {J})$$ L 2 ( I × J ) with the tensor product space $$L^2(\mathcal {I})\otimes L^2(\mathcal {J})$$ L 2 ( I ) ⊗ L 2 ( J ) , where $$\mathcal {I}$$ I and $$\mathcal {J}$$ J are real compact intervals.

Recently, Singh and Saini [Uniform approximation in $$L [0,\infty )$$ L [ 0 , ∞ ) -space by Ces $$\grave{a}$$ a ` ro means of Fourier–Laguerre series. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. (2021)] determined the degree of approximation of functions f belonging to $$L[0,\infty )$$ L [ 0 , ∞ ) by Ces $$\grave{a}$$ a ` ro means of its Fourier–Laguerre series for any $$x>0$$ x > 0 . In this paper, we obtain the error of approximation of functions $$f\in L[0,\infty )$$ f ∈ L [ 0 , ∞ ) using product mean $$C^{\gamma }.T (\gamma \ge 1)$$ C γ . T ( γ ≥ 1 ) of its Fourier–Laguerre series for any $$x>0$$ x > 0 . Further, we also discuss some particular cases of $$C^{\gamma }.T$$ C γ . T -means.

Hydrodynamics of wastewater, which is contaminated with helminth eggs is computationally and experimentally investigated, for laboratory conditions and for a small sewage treatment plant. In the computational analysis, the flow is mathematically modelled within the framework of a Eulerian–Lagrangian framework, where the continuous water phase is treated by an Eulerian, and the discrete particle phase (helminth eggs) is treated by a Lagrangian formulation. For turbulent flows, the Shear Stress Transport model is used to model the turbulence of the continuous phase. The effect of the latter on the discrete phase is modelled by a discrete random walk model. In modelling the momentum exchange between the phases, a special emphasis is placed upon the accurate determination of the drag coefficient for the helminth eggs. For this purpose, flow around individual eggs is analysed and laboratory measurements of other authors are inspected. Before applying these results, measurements are performed on a small sewage treatment plant using surrogate spheres, for validating the remaining aspects of the Eulerian–Lagrangian hydrodynamics modelling. Subsequently, the operation of the small sewage plant is analysed for wastewater containing helminth eggs for its optimization.

In majority of industrial and engineering applications, enhanced heat transfer with minimum entropy production is the major concern. With several theoretical and experimental works, it has been found that replacing the traditional heat transfer liquids with nanoliquid is one of the reliable ways to enhance the thermal transport with minimum loss of system energy. In this regard, the current article deals with the convective nanoliquid flow and the associated thermal dissipation as well as entropy generation rates in a porous annular enclosure saturated nanoliquid. The vertical surface of interior and exterior cylinders is maintained with sinusoidal thermal conditions with different phase deviations, while the horizontal boundaries are thermally insulated. The governing physical equations are solved by implementing finite difference method (FDM). The variation in buoyant nanoliquid flow and the corresponding heat transport rates along with local and global entropy production rates are systematically examined. For the numerical simulations, a vast range of parameters such as the Rayleigh (103 ≤ Ra ≤ 105) and Darcy (10–6 ≤ Da ≤ 10–2) numbers, phase deviation (0 ≤ γ ≤ π), and nanoparticle volume fraction (0 ≤ ϕ ≤ 0.05) are considered in this analysis. The contributions of heat transfer entropy and fluid friction entropy to global entropy production in the geometry are determined through the Bejan number. The numerical results reveal the impact of various parameters on control of convective flow, heat transfer, and entropy generation rates. Further, the results are in excellent agreement with standard benchmark simulations. The predicted results could provide some vital information in choosing the proper choice of parameters to enhance the system efficiency.

Outbreaks minimization has become the need of the hour. If we start clouding the list of infectious diseases, the list will point out that there is a rise in infectious diseases in the current era, and after the first case of any illness for knowing about its spread, testing methods are introduced. So testing plays a key role, and it is necessary to detect any infectious disease effects on humans. However, the peer influence effect of persons who have recovered from disease without visiting any doctor encourages other people to not get tested and take self-medication. They do not understand the need for tests and spread fake scenarios convincing others to follow them. The paper studies the impact of these individuals on the emergence of the disease by analyzing the mathematical model proposed in the situation, which is further analyzed and studied through simulation. The analysis section comprises local and global stability of the equilibrium points, primary reproduction number, and threshold analysis of the proposed model. Numerical simulation has provided a clear view of the qualitative analysis through the graphs and the plots.

Dengue fever is a mosquito-borne viral infection caused by the dengue virus (DENV) found worldwide in tropical and sub-tropical urban and non-urban areas. Dengue viruses (DENV) spread through the bite of an infected Ades species mosquito. There is not available any specific treatment or cure for this DENV infection. The dynamics of the secondary dengue virus infection considering the spatial mobility of dengue virus particles and cells can better be studied and analyzed with reaction–diffusion mathematical models. A reaction–diffusion mathematical model consisting of five simultaneous nonlinear partial differential equations to characterize the dynamics of secondary Dengue infection is studied in this paper. The spatial mobility of the dengue particles and cells is considered in the model. A numerical simulation technique based on the cubic B-splines collocation is proposed to approximate the solution of the considered model.

A non-linear analysis is done to analyze the heat and mass transport in a composite nanofluid layer confined between two parallel horizontal plates, heated from below. The Nusselt number for temperature and nanoparticle concentrations is obtained as a function of time. It is observed that the suspension of two different nanoparticles in a base fluid significantly affects the heat and mass transport. We observe that the modified diffusivity ratios and the Lewis numbers for the first and second types of nanofluids only affect the mass transportation of the first and second types of nanofluids, respectively.

In this paper, we study the existence and uniqueness results of the solutions for non-linear boundary value problems involving $$\varPsi $$ Ψ -Caputo fractional derivative. Furthermore, we prove some stability results of the given problem. The tools used in the analysis are relies on Banach fixed point theorem and $$\varPsi $$ Ψ -fractional Gronwall inequality.

In this paper, a new approach is proposed to study the fracture mechanics problems in 2-D arbitrary polarized piezoelectric media using X-FEM. The existing six-fold crack-tip enrichment functions defined for the generalized case of poling and alignment of crack in piezoelectric media are re-defined here by considering the localized solution of crack-tip field based on Lekhnitskii’s formalism in the transformed coordinate system obtained from material axes to crack-axes, whereas the existing crack-tip enrichment functions were developed other way round. Using the proposed enrichment functions, some benchmark problems such as center cracks, edge cracks, double-edge cracks, and macro–micro-collinear cracks have been studied under arbitrary poling direction, plain strain, and impermeable crack-face conditions. An excellent agreement of normalized intensity factors (IFs) has been obtained for all the cases with the results of existing six-fold enrichment functions.

In this article, we consider a time-dependent weakly coupled system of $$ m(\ge 2) $$ m ( ≥ 2 ) singularly perturbed convection-diffusion equations in the domain $$G:= \varOmega \times S$$ G : = Ω × S that has an interface $$ \varGamma _d:=\left\{ (d,t): t\in S\right\} , $$ Γ d : = ( d , t ) : t ∈ S , $$ d\in \varOmega :=(0,~1) $$ d ∈ Ω : = ( 0 , 1 ) and $$S:=(0,~ T].$$ S : = ( 0 , T ] . The source terms in the system of equations have discontinuities along $$\varGamma _d$$ Γ d . Also, the second-order term of each equation is multiplied by a small positive parameter. These parameters can be arbitrarily small and different in magnitude due to which overlapping boundary and interior layers appear in the solution. An appropriate Shishkin mesh is used to discretize the domain. At the mesh points that are not on the interface line, the problem is discretized using an upwind central difference scheme. For the mesh points on the interface line, a particular upwind central difference scheme is used. An appropriate decomposition of exact and numerical solutions is made to analyze the parameters-uniform convergence of the considered numerical scheme. The numerical approximations yielded by this scheme are parameters-uniformly convergent of first-order in time and almost first-order in space concerning the perturbation parameters. Numerical results are presented to validate the theoretical results.

Spectral methods are efficient, robust and highly accurate methods in numerical analysis. When it comes to approximating a discontinuous function with spectral methods, it produces spurious oscillations at the point of discontinuity, which is called Gibbs’ phenomenon. Gibbs’ phenomenon reduces the spectral accuracy of the method globally. Filtering is a widely used method to prevent the oscillations due to Gibbs’ phenomenon by which the accuracy of the spectral methods is regained up to an extent. In this work, we study the effects of various filters in time-dependent problems and do a comparison of numerical results.

Calendering is a finishing process used in many process industries like paper, textile and leather where the web passes through two or more rotating cylindrical bowls in touch with an aim to get special effects like smoothness, gloss and uniform flattening of the thin sheet. The key factor in the calendering process for getting desired results is pressure and temperature. Several unappealing elements such as damage to fabric and strength reduction of the fabric arise if pressure and temperature increase in excess. Temperature gradient calendering is used to overcome these undesirable factors. In this paper, the influence of parameters like cylindrical bowl temperature, dwell time and thermal diffusivity on the temperature of the fabric in the stiffness direction of the web inside the calender nip has been discussed for temperature gradient calenders of the textile industry using the heat balance integral method.

COVID-19 is a worldwide health threat that has resulted in a significant number of deaths and complicated healthcare management issues. To prevent the COVID-19 pandemic, there is a need to choose a safe and most effective vaccine. Several Multi-criteria Decision-Making (MADM) techniques and approaches have been selected to choose the optimal probable options. The purpose of this article is to deliver divergence measures for fuzzy sets. To validate these measures, some of the properties were also proved. The Multi-criteria Decision-Making method is employed to rank and hence select the best vaccine out of available alternatives. The proposed research allows the ranking of different vaccines based on specified criteria in a fuzzy environment to aid in the selection process. The results suggest that the proposed model provides a realistic way to select the best vaccine from the vaccines available. A case study on the selection of the best COVID-19 vaccine and its experimental results using fuzzy sets are discussed.

We report a linear instability mechanism of MHD mixed convection flow in a porous medium channel under a transverse magnetic field. The stability results are reported for an electrically conducting water-based electrolytes fluid. The governing equations are solved by a Chebyshev spectral collocation method. The linear disturbance equations formed a generalized eigenvalue problem. The results show that the basic flow contains the inflection point. The linear stability analysis shows that the growth of the disturbance reduces by increasing the strength of the magnetic field and decreasing the media permeability of the porous medium flow. The linear stability boundaries show that the relatively higher strength of the applied magnetic field stabilizes the flow, whereas an increase in the media permeability destabilizes the basic flow.

In this paper, we introduce the group action on a near ring $$\mathcal {N}$$ N and with it we study group action on fuzzy ideals of $$\mathcal {N},$$ N , $$\mathcal {G}$$ G -invariant fuzzy ideals, finite products of fuzzy ideals, and $$\mathcal {G}$$ G -primeness of fuzzy ideals of $$\mathcal {N}$$ N .

This paper investigates the effect of viscosity on the propagation of spherical shock waves in a dusty gas with a radiation heat flux and a density that grows exponentially. It is assumed that the dusty gas is a blend of fine solid particles and ideal gas. In a perfect gas, solid particles are uniformly distributed. To obtain several significant shock propagation properties, the solid particles are treated as a pseudo-fluid, and the mixture’s heat conduction is neglected. The flow’s equilibrium conditions are expected to be maintained in an optically thick gray gas model, and radiation is assumed to be of the diffusion type. The effects of modifying the viscosity parameter and time are explored, and non-similar solutions are found. The formal solution is determined by assuming that the shock wave’s velocity is variable and its total energy is not constant.

The effect of sinusoidal gravity modulation on the stability of natural convection in an inclined viscous fluid layer is studied using the energy stability theory. The variation of the critical value of the control parameter, the Rayleigh number, below which the basic flow is stable is discussed with the modulation parameters and the inclination of the fluid layer. An uncertain stability region is observed between the linear and the nonlinear marginal curves.

The analysis of prey–predator in an eco-epidemiological system has become the major concern of scientific research in the field of mathematics and disease dynamical studies. Our studies concern with three-tier species model system when infection is spreading among the prey populations. The impact of infection affecting population dynamics is more complicated studies in natural dynamics. Therefore, we investigate an eco-epidemiological model system’s local and global stability around the biologically feasible equilibrium point. In order to analyze the local and global stability of the model system, we perform a detailed numerical experiments. We analyze the resulting model through various mathematical characteristics like boundedness, global stability, local stability, and bifurcation. We further investigate time evaluation, phase portraits, and bifurcation diagrams and results show the complexity of the eco-epidemiological system. The analytical results are verified through simulations.

In firms, maintaining the quality of the product with carbon emission reduction is a big concern. To ensure the good quality of the product, so many retailers segregate perfect items from imperfect ones and made an attempt to reduce carbon emissions through green technologies. In the proposed model, the discount price of imperfect items is examined and the retailer’s joint decisions have been analyzed on reclamation of inventory and investment in reducing carbon emission under three environmental regulations such as carbon cap, carbon tax, and carbon cap-and-trade. These regulations and understanding of the customer for greener products invigorate retailers to invest in green technology. The total cost is minimized with respect to the optimal order quantity and annual investment on carbon emission reduction. Numerical examples and sensitive analysis are represented to understand the sturdiness of the model.

The interaction of surface and interface waves with a thin horizontal plate submerged in the lower layer of a two-layer fluid is studied under linearised theory of water waves. The associated boundary value problem is solved here by Fourier integral transform by reducing it to an integral equation involving the potential difference function across the plate. Application of multi-term Galerkin method to the solution of the integral equation leads to a simple, rapidly convergent numerical scheme and suitable expressions for different hydrodynamic quantities of interest. Numerical results for the reflection coefficients and the hydrodynamic force on the plate are presented to study the effect of different physical parameters. The present method is verified by recovering the published numerical results for a limiting case and through an energy balance relation.

Particle displacements and stresses are calculated for studying elastic wave propagation in a transversely isotropic homogeneous medium. A mesh consisting of rectangular elements is considered for discretization of two-dimensional domain. The spectral element method is applied through the non-uniformly distributed Gauss-Lobatto-Legendre nodes. The tensor product of high order Lagrangian interpolation polynomials is used as shape functions. Lagrangian interpolation polynomials along with Gauss-Lobatto-Legendre quadrature rule for numerical integration results in diagonal mass matrix which leads to an efficient fully explicit solver for time integration. Second order accurate, central difference method is applied for time discretization. The displacements and stress components are exhibited through time series at a point and snapshots in the domain. The influence of absorbing boundary conditions is demonstrated on the displacement components at different times. The validation of numerical solution is ensured through its comparison with known analytical solution for the two dimensional homogeneous transversely isotropic model.

In this manuscript, an upper bound estimate for the maximum modulus of a general class of polynomials with restricted zeros on a circle $$|z|=L$$ | z | = L , $$L\ge 1$$ L ≥ 1 , is obtained in terms of the maximum modulus of the same polynomials on $$|z|=1$$ | z | = 1 . It is observed that a result of Hussain [J. Pure Appl. Math., (2021) ( https://doi.org/10.1007/s13226-021-00169-7 )] is sharpened by our result. Also, this result generalizes and sharpens some other previously proved result.

The arrival and service data for a season was gathered from Sugar Mill in Meham, Haryana, to improve the service facilities for farmers and reduce queue waiting time through simulation. A suitable simulation model was developed utilizing the Monte Carlo technique to analyze the queue characteristics. Simulation revealed a significant reduction of 60% in waiting time with a marginal rise in the mill's sugarcane crushing limit.

This article proposes a novel adaptive step-size numerical method for solving initial value ordinary differential systems. The development of the proposed method is based on the theory of interpolation and collocation in which representation of the theoretical solution of the problem is assumed in the form of an appropriate interpolating polynomial. In order to bypass the first Dahlquist’s barrier on linear multistep methods, the proposed method considers five intra-step points in one-step block $$\left[ x_n,x_{n+1}\right] $$ x n , x n + 1 resulting in a hybrid method. Among these considered five intra-step points, the values of two intra-step points were fixed named as supporting off-step points and the optimized values of the other three intra-step points were obtained by minimizing the local truncation errors of the main formula at the point $$x_{n+1}$$ x n + 1 and other two additional formulas at supporting off-step points. The proposed method exhibits the property of self-starting as the formulation is immersed into a block structure which enhances the efficiency of the method. The resulting method is of order seven retaining the characteristic of $$\mathcal {A}$$ A -stability. The precision of numerical solution is intensified by drafting the proposed algorithm into an adaptive step-size formulation using an embedded-type procedure. The adaptive step-size method has been tested on some well-known stiff differential systems, viz., Robertson’s chemistry problem, Gear’s problem, the Brusselator system, Jacobi elliptic functions system, etc. The proposed method performs well in comparison to other iconic codes available in the literature.

The great opportunities of the new technology of artificial intelligence and the growing computational capacities together with interacting sensor technology leads to the next industrial revolution called Industry 4.0. In this field the combination of artificial intelligence with numerical simulation to develop a simplified model of a given system can be used for establishing a digital twin of the system for better control and more efficient performance. In this paper, the Artificial Neuronal Network (ANN) methodology is applied as well as a standard interpolation to develop two different simplified models of a 3D cavity flow. The problem is analyzed by Computational Fluid Dynamics (CFD). The CFD simulations are carried out using a commercial software for a case, for which experimental data from the literature exists. In general, the combination of CFD and ANN has been performed in different researches on different applications. Thus, the present paper focuses rather on the comparison of a standard interpolation procedure to ANN, utilizing two different error calculations.

A conforming Virtual Element Method along with a variational discretization concept for solving the optimization problem governed by an elliptic interface problem is presented. Elements with small edges and hanging nodes occur naturally while numerically solving interface problems. Conforming Finite Element Methods cannot handle these difficulties naturally. VEM has the attractive feature that it can tackle hanging nodes and is even robust with respect to small edges. We use these features of VEM to design a method that can tackle these difficulties naturally. The state, adjoint and control estimates have been derived in suitable norms. Numerical results verify our theoretical findings and show the robustness and flexibility of the proposed method.

In this paper, we introduce a class of novel $$C^1$$ C 1 -rational quartic spline zipper fractal interpolation functions (RQS ZFIFs) with variable scalings, where rational spline has a quartic polynomial in the numerator and a cubic polynomial in the denominator with two shape control parameters. We derive an upper bound for the uniform error of the proposed interpolant with a $$C^3$$ C 3 data generating function, and it is shown that our fractal interpolant has $$O(h^2)$$ O ( h 2 ) convergence and can be increased to $$O(h^3)$$ O ( h 3 ) under certain conditions. We restrict the scaling functions and shape control parameters so that the proposed RQS ZFIF is positive, when the given data set is positive. Using this sufficient condition, some numerical examples of positive RQS ZFIFs are presented to support our theory.

We present the solution of linear and nonlinear ordinary differential equations using collocation on finite elements. A heptic (septic) basis is derived and its properties are discussed. The phenomenon of superconvergence at the nodes is illustrated. An investigation of the global and nodal rates of convergence reveals remarkable agreement with a theorem proved by Carl R. de Boor in 1973.

The aim of the present study is to develop an iterative scheme of high convergence order with minimal computational cost. With this objective, a three-step method has been designed by utilizing only two Jacobian matrices, single matrix inversion, and three function evaluations. Under some standard assumptions, the proposed method is found to possess the sixth order of convergence. The iterative schemes with these characteristics are hardly found in the literature. The analysis is carried out to assess the computational efficiency of the proposed method, and further, outcomes are compared with the efficiencies of existing ones. In addition, numerical experiments are performed by applying the method to some practical nonlinear problems. The entire analysis remarkably favors the new technique compared with existing counterparts in terms of computational efficiency, stability, and CPU time elapsed during execution.

In this paper, we propose a new higher order iterative method to find multiple roots of nonlinear equations. The combination of Taylor’s series, Newton’s method and the composition approach are used to derive the new method. It requires three evaluations of the function and two evaluations of the derivative of the function per iteration. The theoretical convergence of the proposed method is proved in the main theorem which establishes sixth order of convergence. We compare the developed method with well-known equivalent existing methods by taking various numerical examples. The numerical results demonstrate the better efficiency of the developed method as compared to some standard iterative methods.

In this paper, the definition of the bipolar fuzzy (bf) point has been generalized, and using this, the concept of separation axioms has been introduced in bipolar fuzzy settings. Moreover, the relation between these separation axioms has been established.

In this present paper, we introduced the notion of $$\acute{C}iri\acute{c}$$ C ´ i r i c ´ type generalized $$\mathcal {B}$$ B -contraction for single mapping and for a pair of mappings which generalized Banach contraction principle in a way different from recent literature. Further, we proved some fixed point theorems using these notions. The newly established results are supported by illustrative examples. Finally, the results are applied to solve the Volterra type integral equations.

In this article, a study concerning the evolution of weak shock past plane and axis-symmetric bodies, in a two-dimensional steady supersonic flow field has been performed. The flow medium is considered to be a mixture of small solid particles and conducting fluid permeated with the transverse magnetic field. The wavefront analysis method is employed to derive transport equations describing the evolutionary behaviour of discontinuities in the plane and axis-symmetric cases. These equations are used to find a closed form expression for the shock formation distance and to determine the conditions ensuring that no shock will evolve at the wave head. In addition to this, the effect of mass concentration of solid particles, magnetic field strength, specific heat ratio and Mach number on the distance of shock formation is analysed and illustrated through figures.

A system of $$k(\ge 2)$$ k ( ≥ 2 ) linear singularly perturbed differential equations of reaction–diffusion type coupled through their reactive terms is considered with Robin type boundary conditions, and the system has discontinuous source terms. The highest order derivative term of each equation is multiplied by a small positive parameter and these parameters are assumed to be different in magnitude, due to which the overlapping and interacting interior and boundary layers may appear in the solution of the considered problem. A numerical scheme involving a central difference scheme for the differential equations and a cubic spline technique for the Robin boundary conditions is developed on an appropriate piecewise-uniform Shishkin mesh. Error analysis is done and the constructed scheme is proved to be almost second-order uniformly convergent with respect to each perturbation parameter. Numerical experiments are conducted to verify the theoretical findings.

A linear stability analysis is performed to investigate the effect of rotation and solute for a thin horizontal layer of water-based magnetic nanofluid ( $$W_{MNF}$$ W MNF ) and ester-based magnetic nanofluids ( $$E_{MNF}$$ E MNF ). The fluid is heated and salted from below, subject to rotation around the vertical axis. As stated in Buongiorno (J Heat Transf 128, 240–250, 2006, [1]), Brownian diffusion and thermophoresis are the significant slip mechanisms in nanofluids. In this work, we consider these two along with magnetophoresis since we are dealing with magnetic nanofluids. A numerical method is employed using MATLAB’s EIG function to solve the resulting eigenvalue problem. The effect of various parameters of the problem which govern the flow has been observed at the onset of convection in the gravity environment in a rigid-rigid boundary condition through neutral stability curves (NSCs). The effect of rotation is investigated using the Taylor number ( $$T_A$$ T A ). We analyse this significant parameter in rigid-free and free-free boundary conditions also with respect to both the environments (gravity and microgravity) and find that the increment in the value of $$T_A$$ T A contributes to system stability under both the environments in all the boundary conditions.

The COVID-19 pandemic has affected the global healthcare system in many countries. India has faced complex multidimensional problems concerning the healthcare system during the COVID-19 outbreak. This article explores some of the implications of COVID-19 on the health system. Also, we attempt to study health economics and other related issues. We have developed the susceptible-exposed-infection-recovered model, logistic growth model, time interrupted regression model, and a stochastic approach for these problems. These models focus on the effect of prevention measures and other interventions for a pandemic on the healthcare system. Our study suggests that the above models are appropriate for COVID-19 at break and effective models for the implications of the pandemic on the healthcare system.