Approximation by Durrmeyer Type Exponential Sampling Operators

We present a weighted-graph-theoretic approach to adaptively compute contributions from many-body approximations for smooth and accurate post-Hartree-Fock (pHF) ab initio molecular dynamics (AIMD) of highly fluxional chemical systems. This approach is ONIOM-like, where the full system is treated at a computationally feasible quality of treatment (density functional theory (DFT) for the size of systems considered in this publication), which is then improved through a perturbative correction that captures local many-body interactions up to a certain order within a higher level of theory (post-Hartree-Fock in this publication) described through graph-theoretic techniques. Due to the fluxional and dynamical nature of the systems studied here, these graphical representations evolve during dynamics. As a result, energetic "hops" appear as the graphical representation deforms with the evolution of the chemical and physical properties of the system. In this paper, we introduce dynamically weighted, linear combinations of graphs, where the transition between graphical representations is smoothly achieved by considering a range of neighboring graphical representations at a given instant during dynamics. We compare these trajectories with those obtained from a set of trajectories where the range of local many-body interactions considered is increased, sometimes to the maximum available limit, which yields conservative trajectories as the order of interactions is increased. The weighted-graph approach presents improved dynamics trajectories while only using lower-order many-body interaction terms. The methods are compared by computing dynamical properties through time-correlation functions and structural distribution functions. In all cases, the weighted-graph approach provides accurate results at a lower cost.

A fractional differential equation accurately represents nature when considering the symmetry properties that represent physical models. In this paper, we examine an approximate solution to the fractional coupled Korteweg–de Vries (CKdV) equation using Crank–Nicolson and quintic B-spline techniques on uniform mesh points. Fractional CKdV equation represents a mathematical model for shallow water wave surfaces. Quintic B-spline techniques are applied to spatial derivatives on uniform mesh points, while the temporal derivative is calculated using Caputo’s formula with the forward finite difference approach. Moreover, the Crank–Nicolson scheme is employed, as it is an unconditionally stable method that ensures reasonable accuracy. The discretization strategy used in this study is unconditionally stable when applied using the von Neumann technique. Additionally, the convergence analysis is examined; it has an order of [Formula: see text]. New numerical test problems demonstrate the method’s feasibility and the approximated solutions are shown to agree well with the exact solutions. Convergence order can be examined through graphical representations, which facilitate numerical analysis. Furthermore, the proposed method requires less memory storage for computing numerical solutions, making it more efficient.

The present article deals with the effect of convexity in the study of the well-known Whittaker iterative method, because an iterative method converges to a unique solution \(t^*\) of the nonlinear equation \(\psi(t)=0\) faster when the function's convexity is smaller. Indeed, fractional iterative methods are a simple way to learn more about the dynamic properties of iterative methods, i.e., for an initial guess, the sequence generated by the iterative method converges to a fixed point or diverges. Often, for a complex root search of nonlinear equations, the selective real initial guess fails to converge, which can be overcome by the fractional iterative methods. So, we have studied a Caputo fractional double convex acceleration Whittaker's method (CFDCAWM) of order at least (\(1+2\zeta\)) and its global convergence in broad ways. Also, the faster convergent CFDCAWM method provides better results than the existing Caputo fractional Newton method (CFNM), which has (\(1+\zeta\)) order of convergence. Moreover, we have applied both fractional methods to solve the nonlinear equations that arise from different real-life problems.

Mathematical modelling of combined pressure driven and electrokinetic effect in a channel with induced magnetic field: An exact solution

The present article examines the double diffusion mixed convective nanofluid flow over a wedge and cone in which slip effects as Brownian motion and the thermophoretic parameter are considered. Mixed convective flow over wedges and cones is applied in different technical fields such as fiber technology, nuclear cooling systems, surface treatment, spray deposition techniques, and polymer production. Double diffusion is increasingly significant in scientific fields like astronomy, geosciences, oceanography, and chemical processes. Mixed convection occurs in several applications such as electric devices, solar panels, nuclear power plants, heat exchangers, radiators, and atmospheric boundary layer fluxes. The governing equations of the study are highly coupled nonlinear partial differential equations with surface constraints. The process of similarity transformation used to converts partial differential equations into nondimensional ordinary differential equations, which can be resolved using the MATLAB Bvp5c function. The graphical representation and detailed analysis of variations in velocity, temperature, and concentration distributions of nanofluids are presented. As the mixed convection parameter increases, the fluid’s velocity rises, with a more pronounced effect observed in the cone compared to the wedge. As the Brownian diffusion parameter grows, the fluid temperature increases, and velocity is more pronounced in the wedge scenario compared to the cone. Moreover, an increase in the thermophoresis parameter results in an elevation in the fluid’s temperature. Additionally, the wedge has a higher concentration of fluid and nanoparticles compared to the wedge. Fluid concentration falls with increasing Schmidt number, but fluid nanoparticle concentration decreases with increasing Lewis number.

In this study, we begin by deriving a two-term strong converse inequality that characterizes the rate of approximation for generalized exponential sampling operators, utilizing the logarithmic moduli of smoothness. Subsequently, we combine the direct and converse estimates to establish the saturation property and identify the class of these approximation operators. Finally, we provide the results for the generalized exponential sampling series using specific kernels that meet the necessary assumptions. It should be noted that in the present study we follow a direct approach within the framework of Mellin analysis in the processes obtained.

In the present article, an efficient operational matrix based on the famous Laguerre polynomials is applied for the numerical solution of two-dimensional non-linear time fractional order reaction–diffusion equation. An operational matrix is constructed for fractional order differentiation and this operational matrix converts our proposed model into a system of non-linear algebraic equations through collocation which can be solved by using the Newton Iteration method. Assuming the surface layers are thermodynamically variant under some specified conditions, many insights and properties are deduced e.g., nonlocal diffusion equations and mass conservation of the binary species which are relevant to many engineering and physical problems. The salient features of present manuscript are finding the convergence analysis of the proposed scheme and also the validation and the exhibitions of effectiveness of the method using the order of convergence through the error analysis between the numerical solutions applying the proposed method and the analytical results for two existing problems. The prominent feature of the present article is the graphical presentations of the effect of reaction term on the behavior of solute profile of the considered model for different particular cases.

The present article deals with the local approximation results by means of Lipschitz maximal function, Ditzian-Totik modulus of smoothness and Lipschitz type space having two parameters for the summation-integral type operators defined by Mishra and Yadav (Tbilisi Mathematical Journal. 11(3), (2018), 175-91). Further, we determine the rate of convergence in the term of the with derivative of bounded variation and for the quantitative means of the defined operators, we establish the quantitative Voronovskaya type and Gr$\ddot{\text{u}}$ss type theorems. Moreover, the examples are given with graphical representation to support the main results.

In the present article, we explore a new static, spherically symmetric charged anisotropic fluid model of compact star in curvature coordinates. We consider metric potential g44 of Durgapal’s fifth solution [1] with a specific choice of electric field intensity E and physically acceptable expression of anisotropy factor Δ , which involve parameters K (charge) and Δ (anisotropy) respectively. The solution so obtained is utilized to construct the model for superdense star like neutron star. We have analysed that corresponding to X=0.1, K=2.8, a=1.6 and by assuming surface density , the mass of the compact star comes out to be with radius 14.51 kms, which closely resembles to that of PSRJ0348 + 0432. The solution is well behaved for the values of K satisfying 1≤K5. Our model is described analytically as well as with the help of graphical representations. Our solution is well behaved and free from any central singularity. It also satisfies all the energy conditions as well as the causality condition thus reconfirming the stability of our model.

In the present article, we present a system for monitoring the quality of air by measuring the temperature, air humidity and atmosphere pressure. The system uses Arduino microcontroller and BME280 for measuring and the obtained data is stored in database. The web based application provides real-time graphical representation of the data or the data for fixed period of time.

Max-product neural network and quasi-interpolation operators activated by sigmoidal functions

The present article is a continuation of the work done by Aral and Erbay [ 1 ]. We discuss the rate of convergence of the generalized Baskakov operators considered in the above paper with the aid of the second order modulus of continuity and the unified Ditzian Totik modulus of smoothness. A bivariate case of these operators is also defined and the degree of approximation by means of the partial and total moduli of continuity and the Peetre's K-functional is studied. A Voronovskaya type asymptotic result is also established. Further, we construct the associated Generalized Boolean Sum (GBS) operators and investigate the order of convergence with the help of mixed modulus of smoothness for the Bogel continuous and Bogel differentiable functions. Some numerical results to illustrate the convergence of the above generalized Baskakov operators and its comparison with the GBS operators are also given using Matlab algorithm.

This article deals with the study of approximation behavior of the exponential sampling type Kantorovich max-product neural network operators based on sigmoidal activation function. We study point-wise and uniform convergence theorems in the space of continuous functions for these operators and determine the rate of convergence by means of logarithmic modulus of continuity. We also investigate the approximation of p -th ( 1 ≤ p < ∞ ) Lebesgue integrable functions by the aforementioned operators and study the rate of convergence exploiting Peetre’s K − functional.

Exponential sampling type neural network Kantorovich operators based on Hadamard fractional integral

In the present article, we investigate some new inequalities of Steffensen type on an arbitrary time scale using the diamond‐α dynamic integrals, which are defined as a linear combination of the delta and nabla integrals. The obtained inequalities extend some known dynamic inequalities on time scales and unify and extend some continuous inequalities and their discrete analogues.