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For r ∈ (0 , 1) , let E (r) be the complete elliptic integral of the second kind. In this paper, the authors obtain a accuracy approximation for E (r) , which improve the related inequalities satisfied by E (r) . [ABSTRACT FROM AUTHOR]

An extension of the bivariate homogeneous orthogonal polynomials can be introduced by using a linear functional with complex moments obtained from the series expansions of a bivariate function at the origin and infinity. They are used to construct the bivariate homogeneous two-point Padé approximant and to solve related problems. In this paper we study the connection between extended homogeneous bivariate orthogonal polynomials and symbolic Gaussian cubature formula for the approximation of bivariate integrals over domains with non-negative weight functions. By extension to the two-point case, a new symbolic Gaussian cubature is presented. A new numerical cubature is also developed. Finally, some numerical examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

We describe a theoretical framework to calculate depletion potential between colloid particles induced by non-adsorbing ideal polymer chains (s -species) and correlation functions in a coarse-grained one-component system of colloids (c -species). A Padé approximant is used to express the depletion potential as a pair potential with many-body contributions subsumed in it. The depletion potential and correlation functions of c -species are calculated using a self-consistent procedure. Results for several values of size ratio q = σ s σ c (σ s and σ c are, respectively diameters of the polymer chain and a colloid particle) and packing fractions of s - and c -species are reported. The spinodal curve and critical point of demixing transition are determined for several values of q. Calculated values are compared with values found from other theories and simulations. • A self consistent theory is developed to calculate the ideal polymer induced depletion potential and correlation functions. • Results are reported for several particle size ratio, colloidal and polymer packing fractions. • The fluid-fluid phase separation curves and critical points are determined and compared with simulation results. • Three-body potential and third virial coefficient are calculated and got an excellent agreement with the simulation results. [ABSTRACT FROM AUTHOR]

The nonlinear differential equations play a prominent role in the mathematical description of many phenomena that occur in our world. A similar set of equations appear in this paper that govern the homogeneous and heterogeneous chemical reactions in the nanofluid flowing between two non-parallel walls. Since the concentration of the homogeneous species is substantially high, quartic autocatalysis is considered for the analysis. It is found to be more effective than the cubic autocatalysis. Further, to avoid the deposition of nanoparticles on the surface, self-propelled microorganisms called gyrotactic microorganisms are allowed to swim in the nanofluid. This movement of microorganisms constitutes a major phenomenon called bioconvection. The set of governing equations thus formed are made dimensionless and the resulting system of equations are solved by Differential Transformation Method (DTM) with the help of Padé approximant that reduces the power series into rational function. This transformation helps in achieving a better convergence rate. The fluid flow analysis is interpreted through graphs and tables where it is observed that the heat source enhances the temperature of the nanofluid. Further, the homogeneous and heterogeneous chemical reaction parameters have significant impacts on the concentration of the reactants. Also, the outcomes indicated that the reaction profiles and motile density profiles increase with the increase in Schmidt number. [ABSTRACT FROM AUTHOR]

A petrophysical model that accurately relates bulk electrical conductivity (σ) to pore fluid conductivity (σw) is critical to the interpretation of geophysical measurements. Classical models are either only applicable over a limited salinity regime or incorrectly explain the nonlinear‐to‐linear behavior of the σ(σw) relationship. In this study, asymptotic limits at zero and infinite salinity are first established in which, σ is expressed as a linear function of σw with four parameters: cementation exponent (m), the equivalent value of volumetric surface electrical conductivity (σs), the volume fraction of overlapped diffuse layer (ϕod) and parameter χ representing the ratio of the volume fraction of the water phase to that of the solid phases in the surface conduction pathway. Subsequently, we bridge the gap between the two extremes by employing the Padé approximant (PA). Given that parameter χ exhibits a marginal influence on the σ(σw) curve, based on measurements for 15 samples, we identify its optimal value to be 0.4. After setting the optimal value of χ, we proceed to evaluate the performance of the PA model by comparing its estimates and estimates made by two existing models to measured values from 27 rock samples and eight sediment samples. The comparison confirms that the PA model estimates are more accurate than estimates made by existing models, particularly at low salinity and for samples with higher cation exchange capacity. The PA model is advantageous in scenarios involving the interpretation of electrical data in freshwater environments. Key Points: Asymptotic limits defining the linearity of bulk electrical conductivity with pore fluid salinity are established at zero and infinite salinityThe Padé approximant (PA) is used to interpolate and describe the non‐linear to linear behaviour between the two extreme electrical conductivity limitsThe PA model outperforms earlier models, particularly in the low salinity range and for high clay content samples [ABSTRACT FROM AUTHOR]

This study investigates the thermal properties of a longitudinally inclined moving porous fin with varying internal heat generation. The approach takes into account the combined influence of natural convection, radiation, and the wet condition while modelling the fin's energy equation. Using dimensionless terms, the governing energy balance equation is converted into an ordinary differential equation (ODE), which is then solved using the Differential transform method (DTM) and Pade approximant. The machine learning (ML) algorithms is also implemented for detecting temperature fluctuations in wetted fins. The ability of stacking ensemble ML model is employed to strengthen the reliability and accuracy of forecasts, which demonstratrates the improved regression predictions with absolute error rates ranging at 10−6. The coefficient of regression of 1 indicates the best fit for the data signifying efficient ML prediction. The graphical representations demonstrate how thermal factors influence temperature dispersion. The analysis reveals that the fin's temperature rises with increasing ambient temperature, nondimensional internal heat generation, generation number, power exponent, and Peclet number. However, under these conditions the temperature gradient reduces. Furthermore, greater values of the convective, radiative, wet porous, and inclination angle parameters result in lower fin temperatures, which aids in cooling while increasing the temperature gradient. [ABSTRACT FROM AUTHOR]

Steady solutions to the Navier–Stokes equations with internal temperature forcing are considered. The equations are solved in two dimensions using the Boussinesq approximation to couple temperature and density fluctuations. A perturbative Stokes expansion is used to prove that that steady flow variables are parametrically analytic in the size of the forcing. The Stokes expansion is complemented with analytic continuation, via functional Padé approximation. The zeros of the denominator polynomials in the Padé approximants are observed to agree with a numerical prediction for the location of singularities of the steady flow solutions. The Padé representations not only prove to be good approximations to the true flow solutions for moderate intensity forcing, but are also used to initialize a Newton solver to compute large amplitude solutions. The composite procedure is used to compute steady flow solutions with forcing several orders of magnitude larger than the fixed‐point method developed in previous work. [ABSTRACT FROM AUTHOR]

In this study, we introduce a novel advancement in the field of theoretical exploration. Specifically, we investigate the transfer and trapping of electronic excitations within a two-component disordered system confined to a finite volume. The implications of our research extend to energy transfer phenomena on spherical nanoparticles, characterized by randomly distributed donors and acceptors on their surface. Utilizing the three-body Padé approximant technique, previously employed in single-component systems, we apply it to address the challenge of trapping within our system. To validate the robustness of our model, we conduct Monte Carlo simulations on a donor–acceptor system positioned on a spherical nanoparticle. In particular, very good agreement between the model and Monte Carlo simulations has been found for donor fluorescence intensity decay. [ABSTRACT FROM AUTHOR]

In this study, we develop a modified version of the two-dimensional differential transform (TDDT) method for solving proportional delay partial differential equations (PDPDEs) that frequently arise in engineering and scientific models. This modification is achieved by integrating the TDDT method with the Laplace transform and the Padé approximant, thereby leveraging the strengths of each technique to improve overall performance. Theorems are provided in a general manner to cover various types of PDEs, with constant or variable coefficients. To validate the approach, we apply it to three test problems, demonstrating its effectiveness in extending the convergence domain of the traditional TDDT approach, reducing computational complexity, and yielding analytic solutions with fewer computational steps. Results indicate that the method is a viable alternative for addressing PDPDEs, especially in scenarios where traditional analytic solutions are challenging to obtain. This combination opens new avenues for efficiently solving complex delayed systems in engineering and science, potentially outperforming existing numerical and analytical techniques in both speed and reliability.

The modified or hybrid method is created by combining the differential transform method (DTM), Padé approximant, and Laplace transform. This method has been used frequently in many fields, especially in solving problems created by nonlinear equations. In this study, we examined the -analog of this method. We define the method as the -modified differential transformation method ( -MDTM). By creating -analogs of some problems solved by Laplace transform and DTM in the literature, we are solving them with -MDTM. In this sense, the study is an example of creating nonlinear -differential equations and solving them by applying the -modified method. [ABSTRACT FROM AUTHOR]

While (2+1)$(2+1)$‐dimensional black holes in the Einstein theory allow for only the anti‐de Sitter (AdS) asymptotic, when the higher curvature correction is tuned on, the asymptotically flat, de Sitter, and AdS cases are included. Here, a detailed study of the stability and quasinormal spectra of the scalar field perturbations around such black holes with all three asymptotics is proposed. Calculations of the frequencies are fulfilled with the help of the 6th order Wentzel–Kramers–Brillouin (WKB) method with Pade approximants, Bernstein polynomial method, and time‐domain integration. Results obtained by all three methods are in a very good agreement in their common range of applicability. When the multipole moment k$k$ is equal to zero, the purely imaginary, i.e., non‐oscillatory, modes dominate in the spectrum for all types of the asymptotic behavior, while the spectrum at higher k$k$ resembles that in four‐dimensional spacetime with the corresponding asymptotic. While (2 + 1)‐dimensional black holes in the Einstein theory allow for only the anti‐de Sitter (AdS) asymptotic, when the higher curvature correction is tuned on, the asymptotically flat, de Sitter, and AdS cases are included. Here, a detailed study of the stability and quasinormal spectra of the scalar field perturbations around such black holes with all three asymptotics is proposed. Calculations of the frequencies are fulfilled with the help of the 6th order Wentzel–Kramers–Brillouin (WKB) method with Pade approximants, Bernstein polynomial method, and time‐domain integration. Results obtained by all three methods are in a very good agreement in their common range of applicability. When the multipole moment k is equal to zero, the purely imaginary, i.e., non‐oscillatory, modes dominate in the spectrum for all types of the asymptotic behavior, while the spectrum at higher k resembles that in four‐dimensional spacetime with the corresponding asymptotic. [ABSTRACT FROM AUTHOR]

This work aims to present a reliable algorithm that can effectively generate accurate piecewise approximate analytical solutions for third- and fourth-order reaction-diffusion singular perturbation problems. These problems involve a discontinuous source term and exhibit both interior and boundary layers. The original problem was transformed into a system of coupled differential equations that are weakly interconnected. A zero-order asymptotic approximate solution was then provided, with known asymptotic analytical solutions for the boundary and interior layers, while the outer region solution was obtained analytically using an enhanced residual power series approach. This approach combined the standard residual power series method with the Padé approximation to yield a piecewise approximate analytical solution. It satisfies the continuity and smoothness conditions and offers higher accuracy than the standard residual power series method and other numerical methods like finite difference, finite element, hybrid difference scheme, and Schwarz method. The algorithm also provides error estimates, and numerical examples are included to demonstrate the high accuracy, low computational cost, and effectiveness of the method within a new asymptotic semi-analytical numerical framewor. [ABSTRACT FROM AUTHOR]

This work aims to present a reliable algorithm that can effectively generate accurate piecewise approximate analytical solutions for third- and fourth-order reaction-diffusion singular perturbation problems. These problems involve a discontinuous source term and exhibit both interior and boundary layers. The original problem was transformed into a system of coupled differential equations that are weakly interconnected. A zero-order asymptotic approximate solution was then provided, with known asymptotic analytical solutions for the boundary and interior layers, while the outer region solution was obtained analytically using an enhanced residual power series approach. This approach combined the standard residual power series method with the Padé approximation to yield a piecewise approximate analytical solution. It satisfies the continuity and smoothness conditions and offers higher accuracy than the standard residual power series method and other numerical methods like finite difference, finite element, hybrid difference scheme, and Schwarz method. The algorithm also provides error estimates, and numerical examples are included to demonstrate the high accuracy, low computational cost, and effectiveness of the method within a new asymptotic semi-analytical numerical framewor.

A new algorithm is proposed in this paper to explain how the modified homotopy perturbation approach can be successfully implemented based on the Pade approximants and the Laplace transform in order to acquire the accurate solutions of a nonlinear system of multi-photograph delay differential equations. The method that has been suggested has the benefit of giving exact solutions, and it is simple to implement analytically on the issues that have been presented. Examples have been provided to demonstrate that this strategy may be utilized and is successful in its application. The results show that the method that was described could be used to solve many different types of differential equations. [ABSTRACT FROM AUTHOR]

Theory of multistep excitation energy migration within the set of chemically identical chromophores distributed on the surface of a spherical nanoparticle is presented. The Green function solution to the master equation is expanded as a diagrammatic series. Topological reduction of the series leads to the expression for emission anisotropy decay. The solution obtained behaves very well over the whole time range and it remains accurate even for a high number of the attached chromophores. Emission anisotropy decay depends strongly not only on the number of fluorophores linked to the spherical nanoparticle but also on the ratio of critical radius to spherical nanoparticle radius, which may be crucial for optimal design of antenna-like fluorescent nanostructures. The results for mean squared excitation displacement are provided as well. Excellent quantitative agreement between the theoretical model and Monte–Carlo simulation results was found. The current model shows clear advantage over previously elaborated approach based on the Padé approximant. [ABSTRACT FROM AUTHOR]

The exponential function that appears in the formula of the stability radius of linear time-delay differential systems is approximated by its Padé approximant. This reduces the computation of the level sets of singular values in the stability radius formula to the computation of imaginary eigenvalues of special matrix polynomials. Then a bisection method is used for computing lower and upper bounds on the stability radius. A rounding error analysis is presented. Several numerical examples are given to demonstrate the feasibility and efficiency of the bisection method. [ABSTRACT FROM AUTHOR]

Previously, in Gatheral and Radoičić (Rational approximation of the rough Heston solution. Int. J. Theor. Appl. Finance, 2019, 22(3), 1950010), we derived a rational approximation of the solution of the rough Heston fractional ODE in the special case $ \lambda =0 $ λ = 0 , which corresponds to a pure power-law kernel. In this paper we extend this solution to the general case of the Mittag-Leffler kernel with $ \lambda \geq 0 $ λ ≥ 0. We provide numerical evidence of the convergence of the solution. [ABSTRACT FROM AUTHOR]

This work presents a reliable algorithm to obtain approximate analytical solutions for a strongly coupled system of singularly perturbed convection–diffusion problems, which exhibit a boundary layer at one end. The proposed method involves constructing a zero-order asymptotic approximate solution for the original system. This approximation results in the formation of two systems: a boundary layer system with a known analytical solution and a reduced terminal value system, which is solved analytically using an improved residual power series approach. This approach combines the residual power series method with Padé approximation and Laplace transformation, resulting in an approximate analytical solution with higher accuracy compared to the conventional residual power series method. In addition, error estimates are extracted, and illustrative examples are provided to demonstrate the accuracy and effectiveness of the method. [ABSTRACT FROM AUTHOR]

In this paper, using the latest Pantheon+ sample of Type Ia supernovae, baryon acoustic oscillation measurements, and observational Hubble data, we carry out a joint constraint on the Hubble constant H 0, the spatial curvature ΩK, and the sound horizon at the end of the drag epoch r d. To be model-independent, four cosmography models—i.e., the Taylor series in terms of redshift y 1 = z /(1 + z), y 2 = arctan (z) , y 3 = ln (1 + z) , and the Padé approximants—are used without the assumption of a flat Universe. The results show that H 0 is anticorrelated with ΩK and r d, indicating that smaller ΩK or r d would be helpful in alleviating the Hubble tension. The values of H 0 and r d are consistent with the estimate derived from the Planck cosmic microwave background data based on the flat ΛCDM model, but H 0 is in 2.3 ∼ 3.0 σ tension with that obtained by Riess et al. in all these cosmographic approaches. Meanwhile, a flat Universe is preferred by the present observations under all approximations except the third order of y 1 and y 2 of the Taylor series. Furthermore, according to the values of the Bayesian evidence, we found that the flat ΛCDM remains to be the most favored model by the joint data sets, and the Padé approximant of order (2,2), the third order of y 3 and y 1 are the top three cosmographic expansions that fit the data sets best, while the Taylor series in terms of y 2 are essentially ruled out. [ABSTRACT FROM AUTHOR]

Using Mathematica computer software, a numerical procedure called the modified Adomian decomposition method (MADM) is successfully implemented for obtaining exact solutions of some classes of Volterra integro-differential equations based on the ADM approximate series solutions, Laplace transform, and Pade approximants. The reliability and effectiveness of MADM are tested in some examples. The obtained results indicate that the implemented procedure is very effective and powerful for handling this kind of differential equation and is valid for a wide class of other types of differential equations. [ABSTRACT FROM AUTHOR]

This paper focuses on numerical approximation for fractional powers of elliptic operators on two-dimensional manifolds. Firstly, the parametric finite element method is employed to discretize the original problem. We then approximate fractional powers of the discrete elliptic operator by the product of rational functions, each of which is a diagonal Padé approximant for the corresponding power function. Rigorous error analysis is carried out and sharp error bounds are presented that show that the scheme is robust for |$\alpha \rightarrow 0^+$| and |$\alpha \rightarrow 1^-$|⁠. The cost of the proposed algorithm is solving some elliptic problems. Since the approach is exponentially convergent with respect to the number of solves, it is very efficient. Some numerical tests are given to confirm our theoretical analysis and the robustness of the algorithm. [ABSTRACT FROM AUTHOR]

A simple efficient method is presented for the determination of the intrinsic transmissivity tensor, as well as the intrinsic correction tensors at successive orders in the dimensionless slip parameter, that predicts the effective transmissivity tensorial coefficient for steady, one-phase, isothermal, creeping flow of a Newtonian fluid with slip boundary condition in a rough fracture. It is demonstrated that the solution of the first $N$ ancillary closure problems provides the slip correction tensors up to the $2N-1$ order, hence reducing the computational requirements by a factor of ${\sim }2$ compared with the conventional approach. In particular, the first-order correction tensor (i.e. a Klinkenberg-like tensor) can be obtained by solving the closure problem required for the computation of the intrinsic transmissivity tensor. In addition, symmetry and definiteness (positiveness or negativeness) properties of the individual tensors are analysed. It is shown that a Padé approximant, built on the correction tensors at the first three orders, outperforms the predictions for the effective transmissivity tensor. The new approach is illustrated and validated with numerical examples on model rough fractures. [ABSTRACT FROM AUTHOR]

We make use of Borel resummation to extract the exact time dependence from the divergent series found in the context of stochastic inflation. Correlation functions of self-interacting scalar fields in de Sitter spacetime are known to develop secular IR divergences via loops, and the first terms of the divergent series have been consistently computed both with standard techniques for curved spacetime quantum field theory and within the framework of stochastic inflation. We show that Borel resummation can be used to interpret the divergent series and to correctly infer the time evolution of the correlation functions. In practice, we adopt a method called Borel-Padé resummation where we approximate the Borel transformation by a Padé approximant. We also discuss the singularity structures of Borel transformations and mention possible applications to cosmology. [ABSTRACT FROM AUTHOR]

Temperature distribution, and efficiency of a rectangular profiled longitudinal fin are examined in this investigation with the impact of the magnetic field. By exploiting appropriate non‐dimensional terms, the heat transfer equation incorporating temperature‐dependent thermal conductivity, heat transfer coefficient, and Maxwell expression for the effect of the magnetic field yield a dimensionless nonlinear ordinary differential equation (ODE) with corresponding boundary conditions (BCs). Sumudu transform method with Pade approximant (STM‐PA) has been employed to obtain an analytical solution for the temperature profile of a longitudinal rectangular fin subjected to a uniform magnetic field under multi‐boiling heat transfer. The STM‐PA results are compared to the Runge‐Kutta Fehlberg's fourth‐fifth (RKF‐45) order technique for computational verification and are observed to be in good accordance. The behavior of dimensionless temperature profile has been explicated graphically for diverse values of non‐dimensional parameters such as thermal conductivity parameter, Hartmann number, and thermogeometric parameter. The results of this study show that as the thermal conductivity parameter enriches, the temperature profile of the longitudinal fin improves, whereas it declines for the Hartmann number and thermogeometric parameter. Under multi‐boiling heat transference, fin efficiency varies significantly depending on the impact of pertinent variables. [ABSTRACT FROM AUTHOR]

This study examines the stability regimes of three-dimensional interfacial gravity waves. The numerical results of the linear stability analysis extend the three-dimensional surface waves results of Ioualalen and Kharif (1994) to three-dimensional interfacial waves. An approach of the collocation type has been developed for this purpose. The equations of motion are reduced to an eigenvalue problem where the perturbations are spectrally decomposed into normal modes. The results obtained showed that the density ratio plays a stabilizing factor. In addition, the dominant instability is of three-dimensional structure, and it belongs to class I for all values of density ratio. [ABSTRACT FROM AUTHOR]

Abstract In this work, our main focus is to establish asymptotic expansions for the triple gamma function in terms of the triple Bernoulli polynomials. As application, an asymptotic expansion for hyperfactorial function is also obtained. Furthermore, using these asymptotic expansions, Padé approximants related to the triple gamma function are derived as a consequence. The results obtained are new, and their importance is demonstrated by deducing several interesting remarks and corollaries.

The present paper is strongly motivated by the brilliant work of Wazwaz [9, Sections 4 and 5] computing analytical approximation in the form of a series truncated at t8 and applying [4/4] Pade approximant to the series. In this paper, we make an attempt to workout analytical approximations in the form of the series truncated at t4 and apply suitable Pade approximations as well as asymptotic approximations with the following features: i) The solution contains exact non integral part. ii) The solution exhibits the population rapid rise along logistic curve followed by decay to zero in the long run. iii) The solution is reasonably comparable with that of Wazwaz [9] using the information from the series with terms only upto t4 . [ABSTRACT FROM AUTHOR]

This paper suggests a new technique for finding the analytical approximate solutions to two-dimensional kinetically reduced local Navier-Stokes equations. This new scheme depends combines the q-Homotopy analysis method (q- HAM), Laplace transform, and Padé approximant method. The power of the new methodology is confirmed by applying it to the flow problem of the lid-driven square cavity. The numerical results obtained by using the proposed method showed that the new technique has good convergence, high accuracy, and efficiency compared with the earlier studies. Moreover, the graphs and tables demonstrate the new approach's validity. [ABSTRACT FROM AUTHOR]

In this work, we have computed and implemented one‐body integrals concerning Gaussian confinement potentials over Gaussian basis functions. Then, we have set an equivalence between Gaussian and Hooke atoms and we have observed that, according to singlet and triplet state energies, both systems are equivalent for large confinement depth for a series of even number of electrons n = 2, 4, 6, 8 and 10. Unlike with harmonic potentials, Gaussian confinement potentials are dissociative for small enough depth parameter; this feature is crucial in order to model phenomena such as ionization. In this case, in addition to corresponding Taylor‐series expansions, the first diagonal and sub‐diagonal Padé approximant were also obtained, useful to compute the upper and lower limits for the dissociation depth. Hence, this method introduces new advantages compared to others. [ABSTRACT FROM AUTHOR]

The exponential analysis of 2n uniformly collected samples from an n-term exponential sum is equivalent to the reconstruction of a rational function of degree n −1 over n. The latter is by computing the Padé approximant of the z-transform of the sequence of samples. In practice, the samples are often noisy and 2n is replaced by N > 2ν with ν > n, leading to a least squares computation of the Padé approximant of degree ν −1 over ν. We show that the latter is a perturbed version of the one of degree n −1 over n and that the n exponential base terms can still be retrieved reliably. This has remained an open problem for many years, despite the fact that the least squares computation was used in most applications. [ABSTRACT FROM AUTHOR]

A new analytical approach for constructing approximate solutions to the elliptic Kepler equation is proposed. We first establish a high-accuracy initial approximation using the piecewise Padé approximation, subsequently we apply the Schröder method to further improve the accuracy of the initial approximation. In general, one Schröder iteration is sufficient to obtain a highly accurate approximate solution. This is a direct method that requires only solving a cubic equation and evaluating two trigonometric functions. The approximate, analytical solutions are compared with solutions by other numerical procedures to prove the accuracy and effectiveness of the proposed approach. • A new approach is proposed to construct accurate approximate solutions to the elliptic Kepler equation. • The Padé approximant is first used to construct an initial approximate solution. • Highly accurate approximate solutions can be established by applying just one Schröder iteration. • These approximations are valid for all eccentricity and eccentric anomaly. [ABSTRACT FROM AUTHOR]

We investigate approaches for the calculation of (resonance) Raman spectra in a real-time time-dependent density functional theory (RT-TDDFT) framework. Several short time approximations to the Kramers, Heisenberg, and Dirac polarizability tensor are examined with regard to the calculation of resonance Raman spectra: One relies on a Placzek type expansion of the electronic polarizability and the other one relies on the excited state gradient method. The first one is shown to be in agreement with an approach based on perturbation theory in the case of a weak δ-pulse perturbation. The latter is newly applied in a real time propagation framework, enabled by the use of Padé approximants to the Fourier transform which allow for a sufficient resolution in the frequency domain. An analysis of the performance of Padé approximants is given. All approaches were found to be in good agreement for uracil and R-methyloxirane. Moreover it is shown how RT-TDDFT can be used to calculate Raman excitation profiles efficiently. [ABSTRACT FROM AUTHOR]

We lay out the extension of range-separated density-functional theory to a four-component relativistic framework using a Dirac-Coulomb-Breit Hamiltonian in the no-pair approximation. This formalism combines a wave-function method for the long-range part of the electron-electron interaction with a density(-current) functional for the short-range part of the interaction. We construct for this formalism a short-range exchange local-density approximation based on calculations on a relativistic homogeneous electron gas with a modified Coulomb-Breit electron-electron interaction. More specifically, we provide the relativistic short-range Coulomb and Breit exchange energies per particle of the relativistic homogeneous electron gas in the form of Padé approximants which are systematically improvable to arbitrary accuracy. These quantities, as well as the associated effective Coulomb-Breit exchange hole, show the important impact of relativity on short-range exchange effects for high densities. [ABSTRACT FROM AUTHOR]

In this work, our main focus is to establish asymptotic expansions for the triple gamma function in terms of the triple Bernoulli polynomials. As application, an asymptotic expansion for hyperfactorial function is also obtained. Furthermore, using these asymptotic expansions, Padé approximants related to the triple gamma function are derived as a consequence. The results obtained are new, and their importance is demonstrated by deducing several interesting remarks and corollaries. [ABSTRACT FROM AUTHOR]

The temperature distribution in a longitudinal fin with magnetic field due to conductive‐convective‐radiative heat transfer is debriefed in this research article. Thermal properties of the fin material, such as thermal conductivity and heat transfer coefficient, have been considered to vary non‐linearly with local temperature whereas surface emissivity has been taken to be constant. The main governing equation of the current model is developed with the aid of Fourier's law of heat conduction, exponentially temperature‐dependent thermal conductivity, Maxwell expression for the effect of the magnetic field, and power‐law temperature‐dependent heat transfer coefficient. This equation is converted into a non‐dimensional form using dimensionless variables and then traced out numerically with the assist of Runge‐Kutta Fehlberg's fourth‐fifth method. Also, the transformed nonlinear energy equation is solved using a DTM‐Pade approximant, yielding an approximate closed‐form solution. The findings of the analytical and numerical investigation are depicted graphically. The outcomes have divulged that the convective and radiative parameters significantly decrease the temperature distribution and improve convective cooling from the fin surface. The rise in the Hartmann number is responsible for the decreasing of the temperature distribution and it aids in accelerating heat transfer. [ABSTRACT FROM AUTHOR]

We derive expressions for the gyro-averaging operator that is applicable to electrostatic fluctuations in a spatially inhomogeneous magnetic field. Both low and high wavenumber limits are considered. The gyro-averaging operator for the former case is represented by sums of Bessel functions with different orders. A simplified expression is provided as a Padé approximant in the low wavenumber limit. This form could be used in practical computations based on the gyrofluid formulation. In the high wavenumber limit, we find that the operator naturally involves fractional derivatives whose physical interpretations are yet to be explored. Discussions are made of a potential impact of this asymptotic expression in the high wavenumber limit. [ABSTRACT FROM AUTHOR]

This article focuses on analytic solutions for Newtonian fluid flow with slip and mass transpiration on a porous stretching sheet using the differential transform method and Pade approximants of an exceptionally nonlinear differential equation. The impacts of different parameters including mass transpiration (suction/injection), Navier's slip, and Darcy number parameters on the velocity of the liquid and tangential stress are discussed. A comprehensive comparison of our results with the previous one in the literature is made, and the results showed good agreement. An investigation is conducted of a combination of magnetic liquids that are conceivably pertinent for wound medicines, skin repair, and astute coatings for natural gadgets. It is found that there is a decrease in the velocity profiles and the boundary layer thickness for the case of suction. [ABSTRACT FROM AUTHOR]

This paper provides iterative solutions, via some variants of the extragradient approximants, associated with the pseudomonotone equilibrium problem (EP) and the fixed point problem (FPP) for a finite family of η-demimetric operators in Hilbert spaces. The classical extragradient algorithm is embedded with the inertial extrapolation technique, the parallel hybrid projection technique and the Halpern iterative methods for the variants. The analysis of the approximants is performed under suitable set of constraints and supported with an appropriate numerical experiment for the viability of the approximants. [ABSTRACT FROM AUTHOR]

The transparent boundary condition for the free Schrödinger equation on a rectangular computational domain requires implementation of an operator of the form ∂ t − i △ Γ where △ Γ is the Laplace-Beltrami operator. It is known that this operator is nonlocal in time as well as space which poses a significant challenge in developing an efficient numerical method of solution. The computational complexity of the existing methods scale with the number of time-steps which can be attributed to the nonlocal nature of the boundary operator. In this work, we report an effectively local approximation for the boundary operator such that the resulting complexity remains independent of number of time-steps. At the heart of this algorithm is a Padé approximant based rational approximation of certain fractional operators that handles corners of the domain adequately. For the spatial discretization, we use a Legendre-Galerkin spectral method with a new boundary adapted basis which ensures that the resulting linear system is banded. A compatible boundary-lifting procedure is also presented which accommodates the segments as well as the corners on the boundary. The proposed novel scheme can be implemented within the framework of any one-step time marching schemes. In particular, we demonstrate these ideas for two one-step methods, namely, the backward-differentiation formula of order 1 (BDF1) and the trapezoidal rule (TR). For the sake of comparison, we also present a convolution quadrature based scheme conforming to the one-step methods which is computationally expensive but serves as a golden standard. Finally, several numerical tests are presented to demonstrate the effectiveness of our novel method as well as to verify the order of convergence empirically. • We presented an efficient numerical realization of the transparent boundary operator ∂ t − i △ Γ on a rectangular domain. • We first present a convolution quadrature (CQ) based numerical recipe that turns out to be computationally expensive. • A novel Padé algorithm is then developed which removes all bottlenecks of the CQ approach and also of Menza's approach. • A new boundary adapted basis is presented for the spatial problem that ensures the bandedness of the linear systems. • Note that the computational complexity of our novel Padé algorithm remains independent of the number of time-steps. [ABSTRACT FROM AUTHOR]

The two-point Padé approximation problem is to find a ratio of two coprime polynomials with some constraints on their degrees to approximate a function whose power series expansions at the origin and at infinity are given. In this paper, we introduce the Hankel vector for the two-point Padé approximation problem and establish the intrinsic connections between the two-point Padé approximation problem and a certain Padé approximation problem at infinity determined by the Hankel vector of the former. These connections provide us with a new way to study the structural characteristics of the two-point Padé table and to deduce the three-term recursive relations for the numerators and denominators of three adjacent entries in the two-point Padé table. [ABSTRACT FROM AUTHOR]

The inverse scaling and squaring algorithm computes the logarithm of a square matrix |$A$| by evaluating a rational approximant to the logarithm at the matrix |$B:=A^{2^{-s}}$| for a suitable choice of |$s$|⁠. We introduce a dual approach and approximate the logarithm of |$B$| by solving the rational equation |$r(X)=B$|⁠ , where |$r$| is a diagonal Padé approximant to the matrix exponential at |$0$|⁠. This equation is solved by a substitution technique in the style of those developed by Fasi & Iannazzo (2020, Substitution algorithms for rational matrix equations. Elect. Trans. Num. Anal. , 53 , 500–521). The new method is tailored to the special structure of the diagonal Padé approximants to the exponential and in terms of computational cost is more efficient than the state-of-the-art inverse scaling and squaring algorithm. [ABSTRACT FROM AUTHOR]

We propose a novel grey-box model to capture the nonlinearity and the dynamics of cash flow model parameters. The grey-box model retains a simple white-box model structure, while their parameters are modelled as a black-box with a Padé approximant as a functional form. The growth rate of sales and firm age are used as exogenous variables because they are considered to have explanatory power for the parameter process. Panel data estimation methods are applied to investigate whether they outperform the pooled regression, which is widely used in the extant literature. We use the U.S. dataset to evaluate the performance of various models in predicting cash flow. Two performance measures are selected to compare the out-of-sample predictive power of the models. The results suggest that the proposed grey-box model can offer superior performance, especially in multi-period-ahead predictions. [ABSTRACT FROM AUTHOR]

The luminosity distance d L is possibly the most important distance scale in cosmology and therefore accurate and efficient methods for its computation is paramount in modern precision cosmology. Yet in most cosmological models the luminosity distance cannot be expressed by a simple analytic function in terms of the redshift z and the cosmological parameters, and is instead represented in terms of an integral. Although one can revert to numerical integration techniques utilizing quadrature algorithms to evaluate such an integral, the high accuracy required in modern cosmology makes this a computationally demanding process. In this paper, we use the Parker–Sochacki method (PSM) to generate a series approximate solution for the luminosity distance in spatially flat Λ CDM cosmology by solving a polynomial system of nonlinear differential equations. When compared with other techniques proposed recently, which are mainly based on the Padé approximant, the expression for the luminosity distance obtained via the PSM leads to a significant improvement in the accuracy in the redshift range 0 ≤ z ≤ 2.5 . Moreover, we show that this technique can be easily applied to other more complicated cosmological models, and its multistage approach can be used to generate analytic approximations that are valid on a wider redshift range. [ABSTRACT FROM AUTHOR]